Nonlocal Impulsive Fractional Integral Boundary Value Problem for (ρ_k,ϕ_k)-Hilfer Fractional Integro-Differential Equations

Abstract

In this paper, we establish the existence and stability results for the $({\rho }_k,{\varphi }_k)$-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the $({\rho }_k,{\varphi }_k)$-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results.

1. Introduction

Fractional calculus ($\mathbb{FC}$) is discussed as the fractional integral operator ($\mathbb{FIO}$) and fractional derivative operator ($\mathbb{FDO}$), which have a long and illustrious history. $\mathbb{FC}$ is popularly used to analyze phenomena in the branch of mathematical analysis, which is noticed to be of outstanding assistance in modifying complex real-world problems in many fields, such as physical sciences [1], financial economics [2], dynamics of particles, fields and media [3], bio-engineering [4], Zika [5], HIV [6], COVID-19 [7], ecology [8], continuum mechanics [9], Navier–Stokes problem [10], social media addiction [11], and references cited therein. For more theoretical details on this topic, see: [12,13,14,15,16]. A variety of types of $\mathbb{FDO}$s are regularly settled in the sense of $\mathbb{FIO}$s. Various types of $\mathbb{FDO}$s with different kernel terms, such as Riemann–Liouville ($\mathbb{RL}$), Caputo, Hadamard, Katugampola, Hilfer, and others, are shown in the literature survey on $\mathbb{FC}$.

Recently, in 2018, the concept of $\mathbb{FDO}$ with respect to another function was developed by Sousa and Oliveira [17], which is known as the $\varphi$-Hilfer $\mathbb{FDO}$. Some existence and stability results of the solutions for fractional differential equations ($\mathbb{FDE}$s) were created in the context of $\varphi$-Hilfer $\mathbb{FDO}$ [18,19,20,21,22] and the references therein. After that, in 2021, Kucche and Mali [23] introduced and demonstrated some properties of the ($\rho ,\varphi$)-Hilfer $\mathbb{FDO}$. They applied Banach’s type to analyzed the uniqueness result for the non-linear $\mathbb{FDE}$s under ($\rho ,\varphi$)-Hilfer $\mathbb{FDO}$:

$$\left\{\begin{array}{c}{\displaystyle {}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }u\left(t\right)=f(t,u\left(t\right)),t\in (a,b],\alpha \in (0,\rho ),\beta \in [0,1],}\hfill \\ {\displaystyle {}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\rho -{\gamma }_r;\varphi }u\left(a\right)=u_a\in \mathbb{R},{\gamma }_r=\alpha +\beta (\rho -\alpha ),\rho >0,}\hfill \end{array}\right.$$

(1)

where ${}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }$ is the ($\rho ,\varphi$)-Hilfer $\mathbb{FDO}$ of order $\alpha$ and type $\beta$ with $\alpha \in (0,1)$, and $f\in \mathcal{C}\left(\right[a,b]\times \mathbb{R},\mathbb{R})$, $0\le a<b<\infty$. It is worth noting the ($\rho ,\varphi$)-Hilfer $\mathbb{FDO}$, which can be generalized as various known $\mathbb{FDO}$s (see more details in Remark 2).

The physical and social sciences are explained by applying impulsive differential equations with integer order and fractional order. They are also applicable to dynamical systems, such as evolutionary processes, which show instantaneous state changes at some points. The qualitative theory of impulsive $\mathbb{FDE}$s, such as existence theory and stability results, has been widely employed in engineering and applied sciences throughout the last several decades (see [24,25,26]). Many researchers will attempt to operate in the area of impulsive $\mathbb{FDE}$s with impulses and have presented essential and interesting results through the years that have contributed greatly to the mathematical analysis of $\mathbb{FDE}$s with impulses effect. In 2009, Benchohra and Slimani [27] studied a variety of conditions for the existence of the solutions for the impulsive Caputo-type $\mathbb{FDE}$ with initial condition by using Banach’s, Leray–Schauder’s, and Schaefer’s fixed point theorems. Later, the impulsive $\mathbb{FDE}$s in [27] have been extended and studied for their existence results in Banach spaces by Benchohra and Seba [28]. In 2012, Wang et al. [29] investigated the piecewise continuous solutions to the problem in [27,28]. The existence, uniqueness, and Ulam’s stability results of solutions for the impulsive boundary value problems ($\mathbb{BVP}$s) are obtained by using a fixed point theorem via generalized Gronwall inequalities. In 2014, Wang and Lin [30] investigated the existence of solutions to impulsive Caputo $\mathbb{FDE}$s under anti-periodic boundary conditions via constant coefficients. The formula of solutions to the problem in [30] was constructed in the sense of Mittag–Leffler kernels. At the same time, the Lipschitz and non-linear growth conditions were used to establish the existence results of solutions to the problem in [30]. In 2017, Zuo et al. [31] established the existence and uniqueness results for impulsive anti-periodic $\mathbb{BVP}$s through fractional integro-differential equation ($\mathbb{FIDE}$) with constant coefficient based on Banach’s and Krasnoselskii’s types. In 2020, Kucche et al. [32] developed the existence results of solutions for the non-linear $\varphi$-Hilfer impulsive $\mathbb{FDE}$ with initial condition:

$$\left\{\begin{array}{c}{\displaystyle {}_{}^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }u\left(t\right)=f(t,u\left(t\right)),t\in {\mathcal{J}}_k\subseteq (a,T],t\ne t_k,T>a,}\hfill \\ {\displaystyle \Delta {}_{}^{}{\mathcal{I}}_{a^{+}}^{1-\gamma ;\varphi }u\left(t_k\right)={\xi }_k\in \mathbb{R},k=1,2,\dots ,m,}\hfill \\ {\displaystyle {}_{}{\mathcal{I}}_{a^{+}}^{1-\gamma ;\varphi }u\left(a\right)=\delta \in \mathbb{R},\gamma =\alpha +\beta -\alpha \beta ,}\hfill \end{array}\right.$$

(2)

where ${}_{}^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }$ is the $\varphi$-Hilfer $\mathbb{FDO}$ of order $\alpha \in (0,1)$ and type $\beta \in [0,1]$, ${}_{}^{}{\mathcal{I}}_{a^{+}}^{1-\gamma ;\varphi }$ is the $\varphi$-$\mathbb{RL}$-$\mathbb{FIO}$ of order $1-\gamma >0$. In addition, they extended the problem (2) to the non-local $\varphi$-Hilfer $\mathbb{FDE}$. In 2022, based on Banach’s and Schauder’s types, Salim et al. [33] proved the existence and uniqueness of solutions for the non-linear implicit $\rho$-generalized $\varphi$-Hilfer $\mathbb{FDE}$-$\mathbb{BVP}$s via retardation and anticipation:

$$\left\{\begin{array}{c}{\displaystyle {}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }u\left(t\right)=f(t,u_t\left(t\right),{}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }u\left(t\right)),t\in (a,b],}\hfill \\ {\displaystyle a_1{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\rho (1-\xi );\varphi }u\left(a^{+}\right)+a_2{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\rho (1-\gamma );\varphi }u\left(b\right)=a_3,\gamma =\frac{1}{\rho }(\beta (\rho -\alpha )+\alpha ),}\hfill \\ {\displaystyle u\left(t\right)=\omega \left(t\right),t\in [a-\lambda ,a],\lambda >0,a_1,a_2,a_3\in \mathbb{R},}\hfill \\ {\displaystyle u\left(t\right)=\overline{\omega }\left(t\right),t\in [b,b+\overline{\lambda }],\overline{\lambda }>0,a_1+a_2\ne 0,}\hfill \end{array}\right.$$

(3)

where ${}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }$ and ${}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\rho (1-\gamma );\varphi }$ are the $\rho$-generalized $\varphi$-Hilfer $\mathbb{FDO}$ of order $\alpha \in (0,\rho )$, $\rho >0$ and type $\beta \in [0,1]$, and the $\rho$-generalized $\varphi$-Hilfer $\mathbb{FIO}$ of order $\rho (1-\gamma )\in (0,\rho )$, respectively, and $u_t\left(\tau \right)=u(t+\tau )$ for $\tau \in [-\lambda ,\overline{\lambda }]$. Note that several works have been published using concentrated and important tools in mathematical analysis. We suggest modern works on impulsive $\mathbb{FDE}$s on existence, uniqueness, and Ulam’s stability and the reference given therein [34,35,36,37,38,39,40,41,42,43,44,45,46].

To motivate the enrichment of novel literature for interested researchers, in this paper, we establish qualitative results of the solutions for the following non-linear impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$s with non-local multi-point fractional integral boundary conditions ($\mathbb{NMP}$-$\mathbb{FIBC}$s) as:

$$\left\{\begin{array}{c}{\displaystyle {}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}u\left(t\right)={\lambda }_{k}u\left(t\right)+f(t,u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}u\left(t\right)),t\in {\mathcal{J}}_k,t\ne t_k,}\hfill \\ {\displaystyle \Delta {}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}u\left(t_k\right)={\phi }_k\left(u\left(t_k\right)\right),k=1,2,\dots ,m,}\hfill \\ {\displaystyle \sum _{i=0}^m{\kappa }_{i}u\left({\eta }_i\right)=\sum _{j=0}^n{\omega }_j{}_{{\rho }_j}{\mathcal{I}}_{t_j^{+}}^{{\mu }_j;{\varphi }_j}u\left({\xi }_j\right)+A,{\eta }_i\in (t_i,t_{i+1}],{\xi }_j\in (t_j,t_{j+1}].}\hfill \end{array}\right.$$

(4)

where ${}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}$ denotes the $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FDO}$ of order ${\alpha }_k$ and type ${\beta }_k$ on ${\mathcal{J}}_k$, $0<{\alpha }_k<1$, $0\le {\beta }_k\le 1$, ${\rho }_k>0$, ${\mathcal{J}}_k=(t_k,t_{k+1}]\subseteq (0,T]$, $k=0,1,\dots ,m$, ${\mathcal{J}}_0=[a,t_1]$, $\mathcal{J}=[a,b]$, $0\le a=t_0<t_1<\cdots <t_m<t_{m+1}=b$, ${}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{q;{\varphi }_k}$ is the $({\rho }_k,{\varphi }_k)$-$\mathbb{RL}$ $\mathbb{FIO}$ of order $q\in \{{\delta }_k,{\theta }_k,{\mu }_k\}>0$, $k=0,1,\dots ,m$, ${\phi }_k\in \mathcal{C}(\mathbb{R},\mathbb{R})$, $\Delta {}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}u\left(t_k\right)={}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}u\left(t_k^{+}\right)-{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{{\rho }_{k-1}(1-{\gamma }_{k-1});{\varphi }_{k-1}}u\left(t_k^{-}\right)$ where ${}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}u\left(t_k^{+}\right)={lim}_{h\to 0^{+}}{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}u(t_k+h)$, $k=1,2,\dots ,m$, $f\in \mathcal{C}(\mathcal{J},{\mathbb{R}}^3,\mathbb{R})$, ${\kappa }_i$, ${\omega }_j\in \mathbb{R}$, ${\eta }_i\in (t_i,t_{i+1}]$, ${\xi }_j\in (t_j,t_{j+1}]$, $i=0,1,\dots ,m$, $j=0,1,\dots ,n$, $\lambda <0$ and $A\in \mathbb{R}$. For the sake of use, the problem (4) can be called the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s.

The remaining sections of this work are structured as follows: in Section 2, some concepts of the ($\rho ,\varphi$)-Hilfer fractional operators related to our discussion are defined along with some essential lemmas are proved. Additionally, the solution of the linear variant of the ($\rho ,\varphi$)-Hilfer fractional Cauchy problem (11) is derived in the form of the generalized Mittag–Leffler kernel. After that, an equivalent integral equation to the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4). The essential lemma is very important to transform the proposed problem (4) into a fixed-point problem. In Section 3, presenting the first main results of the problem (4), the uniqueness result is proved by Banach’s type and the existence result is studied by a fixed point theorem due to O’Regan. In addition, a variety of Ulam’s stability results for problem (4) are investigated in Section 4. Finally, Section 5 shows illustrative examples to verify the main results.

2. Preliminaries

This section introduces fundamental concepts and constructs several properties of the $(\rho ,\varphi )$-Hilfer fractional calculus relevant to our results.

2.1. The $(\rho ,\varphi )$-Hilfer Fractional Calculus and Its Properties

Definition 1

([47]). Let $f\in L^1([a,b],b)$ and an increasing function $\varphi \left(t\right):[a,b]\to \mathbb{R}$ via ${\varphi }^{\prime }\left(t\right)\ne 0$ for $t\in [a,b]$. The $(\rho ,\varphi )$-$\mathbb{RL}$-$\mathbb{FIO}$ of a function f of order $\alpha >0$ is defined by

$${}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ;\varphi }f\left(t\right)=\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\left(\varphi \left(t\right)-\varphi \left(s\right)\right)}^{\frac{\alpha }{\rho }-1}{\varphi }^{\prime }\left(s\right)f\left(s\right)ds,\rho >0,$$

where ${\Gamma }_{\rho }(·)$ is the ρ-Gamma function which is introduced by Diaz and Pariguan [48],

$${\Gamma }_{\rho }\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-\frac{t^{\rho }}{\rho }}dt,z\in \mathbb{C},Re\left(z\right)>0,\rho >0.$$

(5)

Some other useful properties of (5) are well known as follows:

$${\Gamma }_{\rho }(z+\rho )=z{\Gamma }_{\rho }\left(z\right),{\Gamma }_{\rho }\left(\rho \right)=1,{\Gamma }_{\rho }\left(z\right)={\left(\rho \right)}^{\frac{z}{\rho }-1}\Gamma \left(\frac{z}{\rho }\right),\Gamma \left(z\right)=\underset{\rho \to 1}{lim}{\Gamma }_{\rho }\left(z\right).$$

(6)

Definition 2

([23]). Let $f\in {\mathcal{C}}^n([a,b],\mathbb{R})$, $\varphi \in {\mathcal{C}}^n([a,b],\mathbb{R})$, ${\varphi }^{\prime }\left(t\right)\ne 0$, for $t\in [a,b]$, α, $\rho \in {\mathbb{R}}^{+}$, and $\beta \in [0,1]$. The $(\rho ,\varphi )$-Hilfer $\mathbb{FDO}$ of a function f of order α and type β is given by

$${}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }f\left(t\right)={}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\beta (\rho n-\alpha );\varphi }{\delta }_{\varphi }^n{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{(1-\beta )(\rho n-\alpha );\varphi }f\left(t\right),$$

(7)

where ${\delta }_{\varphi }^n={\left(\frac{\rho }{{\varphi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n$ and $n=⌈\frac{\alpha }{\rho }⌉$.

Remark 1.

The ($\rho ,\varphi$)-Hilfer $\mathbb{FDO}$ can be rewritten in the sense of the ($\rho ,\varphi$)-$\mathbb{RL}$-$\mathbb{FDO}$ as follows:

$${}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }f\left(t\right)={}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\gamma -\alpha ;\varphi }{\left(\frac{\rho }{{\varphi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{n\rho -\gamma ;\varphi }f\left(t\right)={}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\gamma -\alpha ;\varphi }\left({}_{\rho }^{\mathbb{RL}}{\mathfrak{D}}_{a^{+}}^{\gamma ;\varphi }f\right)\left(t\right),$$

where

$${}_{\rho }^{\mathbb{RL}}{\mathfrak{D}}_{a^{+}}^{\gamma ;\varphi }f\left(t\right)={\left(\frac{\rho }{{\varphi }^{\prime }\left(t\right)}\frac{d}{dt}\right)}^n{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{n\rho -\gamma ;\varphi }f\left(t\right),$$

and $n\rho -\gamma =(1-\beta )(n\rho -\alpha )$ with $\frac{\gamma }{\rho }\in (n-1,n]$.

Remark 2.

It is noticed that:

(${\mathbb{A}}_1$)

If we take $\beta =0$ in (7), then we have the ($\rho ,\varphi$)-$\mathbb{RL}$-$\mathbb{FDO}$ defined in [23], while if $\varphi \left(t\right)=t$ with $\beta =0$, then we obtain the ρ-$\mathbb{RL}$-$\mathbb{FDO}$ defined in [49].

(${\mathbb{A}}_2$)

If we take $\beta =1$ in (7), then we have the ($\rho ,\varphi$)-Caputo $\mathbb{FDO}$ defined in [23], while if we take $\varphi \left(t\right)=t$ with $\beta =0$, then we obtain the ρ-Caputo $\mathbb{FDO}$ defined in [23].

(${\mathbb{A}}_3$)

If we take $\varphi \left(t\right)=t^{\sigma }$ in (7), then we have the ρ-Hilfer–Katugampola $\mathbb{FDO}$, that is:

(i) 

If we take $\varphi \left(t\right)=t^{\sigma }$ with $\beta =0$ in (7), then we have the ρ-Katugampola $\mathbb{FDO}$ defined in [50].

($ii$)

If we take $\varphi \left(t\right)=t^{\sigma }$ with $\beta =1$ in (7), then we have the ρ-Caputo–Katugampola $\mathbb{FDO}$ defined in [50].

(${\mathbb{A}}_4$)

If we take $\varphi \left(t\right)=logt$ in (7), then we have the ρ-Hilfer–Hadamard $\mathbb{FDO}$, that is:

(i) 

If we take $\varphi \left(t\right)=logt$ with $\beta =0$ in (7), then we have the ρ-Hadamard $\mathbb{FDO}$ defined in [23].

($ii$)

If we take $\varphi \left(t\right)=logt$ with $\beta =1$ in (7), then we have the ρ-Caputo–Hadamard $\mathbb{FDO}$ defined in [23].

Some important basic properties, which are used throughout this paper, are as follows:

Lemma 1

([23]). Let α, $\rho >0$ and $\beta \in \mathbb{R}$, such that $\frac{\beta }{\rho }>-1$. Then, we have

(i) 

${}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ;\varphi }\left[{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{\frac{\beta }{\rho }}\right]=\frac{{\Gamma }_{\rho }(\beta +\rho )}{{\Gamma }_{\rho }(\beta +\rho +\alpha )}{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{\frac{\beta +\alpha }{\rho }}$.

(ii) 

${}_{\rho }^{RL}{\mathfrak{D}}_{a^{+}}^{\alpha ;\varphi }\left[{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{\frac{\beta }{\rho }}\right]=\frac{{\Gamma }_{\rho }(\beta +\rho )}{{\Gamma }_{\rho }(\beta +\rho -\alpha )}{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{\frac{\beta -\alpha }{\rho }}$.

(iii) 

${}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ;\varphi }{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\beta ;\varphi }f\left(t\right)={}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha +\beta ;\varphi }f\left(t\right)={}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\beta ;\varphi }{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ;\varphi }f\left(t\right)$.

Lemma 2

([33]). If $f\in {\mathcal{C}}^n([a,b],\mathbb{R}$, $\frac{\alpha }{\rho }\in (n-1,n)$, $\beta \in [0,1]$, where $n\in \mathbb{N}$ and $\rho >0$, then

$$\left({}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ;\varphi }{}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }f\right)\left(t\right)=f\left(t\right)-\sum _{i=1}^n\frac{{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{\gamma -i}}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\left[{\delta }_{\varphi }^{n-i}\left({}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\rho (n-\gamma );\varphi }f\left(a\right)\right)\right],$$

where $\gamma =\frac{1}{\rho }\left(\beta \left(\rho n-\alpha \right)+\alpha \right)$ and $n=⌈\frac{\alpha }{\rho }⌉$.

Next, we provide the Mittag–Leffler functions ${\mathbb{E}}_{\alpha }(·)$ and ${\mathbb{E}}_{\alpha ,\beta }(·)$ that will be employed throughout in this paper.

Lemma 3

([51,52]). Take $\alpha \in (0,1)$, $c>0$. Hence, ${\mathbb{E}}_{\alpha }(·)$ and ${\mathbb{E}}_{\alpha ,c}(·)$ are non-negative functions, and for each $u<0$, ${\mathbb{E}}_{\alpha }\left(u\right)\le 1$, ${\mathbb{E}}_{\alpha ,c}\left(u\right)\le 1/\Gamma \left(c\right)$, with

$${\mathbb{E}}_{\alpha }\left(u\right)=\sum _{n=0}^{\infty }\frac{u^n}{\Gamma (\alpha n+1)}and{\mathbb{E}}_{\alpha ,c}\left(u\right)=\sum _{n=0}^{\infty }\frac{u^n}{\Gamma (\alpha n+c)},u\in \mathbb{R}.$$

For the sake of easy for calculation in this paper, we define the symbols:

$${\Phi }_{\varphi }^{c-1}(t,a)={\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{c-1}.$$

(8)

Lemma 4.

Assume that $\alpha >0$, $c>0$, $\rho >0$, $q>0$, and $\lambda \in \mathbb{R}$. Then, we obtain

$${}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ,\varphi }\left[{\Phi }_{\varphi }^{q-1}(t,a){\mathbb{E}}_{c,q}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(t,a)\right)}^c\right)\right]={\rho }^{-\frac{\alpha }{\rho }}{\Phi }_{\varphi }^{q+\frac{\alpha }{\rho }-1}(t,a){\mathbb{E}}_{c,q+\frac{\alpha }{\rho }}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(t,a)\right)}^c\right),$$

(9)

where ${\Phi }_{\varphi }^{(·)}(t,a)$ and ${\mathbb{E}}_{u,v}(·)$ are given as in (8) and Lemma 3, respectively.

Proof. 

By applying Definition 1 and Lemma 3 we have

$$\begin{array}{ccc}& & {}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ,\varphi }\left[{\Phi }_{\varphi }^{q-1}(t,a){\mathbb{E}}_{c,q}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(t,a)\right)}^c\right)\right]\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(s,t)\left[{\Phi }_{\varphi }^{q-1}(s,a){\mathbb{E}}_{c,q}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(s,a)\right)}^c\right)\right]{\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s){\Phi }_{\varphi }^{q-1}(s,a)\left[\sum _{n=0}^{\infty }\frac{{\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(s,a)\right)}^c\right)}^n}{\Gamma \left(cn+q\right)}\right]{\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}\sum _{n=0}^{\infty }\frac{{\left(\lambda {\rho }^{-c}\right)}^n}{\Gamma \left(cn+q\right)}{\int }_a^t{\left(\varphi \left(t\right)-\varphi \left(s\right)\right)}^{\frac{\alpha }{\rho }-1}{\left(\varphi \left(s\right)-\varphi \left(a\right)\right)}^{cn+q-1}{\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& \sum _{n=0}^{\infty }\frac{{\left(\lambda {\rho }^{-c}\right)}^n}{\Gamma \left(cn+q\right)}{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ,\varphi }\left[{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{\frac{\rho (cn+q)}{\rho }-1}\right].\hfill \end{array}$$

By using (ii) of Lemma 1, we have

$$\begin{array}{ccc}& & {}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ,\varphi }\left[{\Phi }_{\varphi }^{q-1}(t,a){\mathbb{E}}_{c,q}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(t,a)\right)}^c\right)\right]\hfill \\ & =& \sum _{n=0}^{\infty }\frac{{\left(\lambda {\rho }^{-c}\right)}^n}{\Gamma \left(cn+q\right)}\left[\frac{{\Gamma }_{\rho }\left(\rho (cn+q)\right)}{{\Gamma }_{\rho }(\rho (cn+q)+\alpha )}{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{\frac{\rho (cn+q)+\alpha }{\rho }-1}\right]\hfill \\ & =& \sum _{n=0}^{\infty }\frac{{\left(\lambda {\rho }^{-c}\right)}^n}{\Gamma \left(cn+q\right)}\left[\frac{{\rho }^{cn+q-1}\Gamma (cn+q)}{{\rho }^{cn+q+\frac{\alpha }{\rho }-1}\Gamma (cn+q+\frac{\alpha }{\rho })}{\left(\varphi \left(t\right)-\varphi \left(a\right)\right)}^{cn+q+\frac{\alpha }{\rho }-1}\right],\hfill \end{array}$$

which provides the desired (9). □

Lemma 5.

Suppose that $\alpha >0$, $c>0$, $\rho >0$, $\lambda \in \mathbb{R}$, and $h\in \mathcal{C}\left(\right[a,b\left]\right)$. Then, we have

$$\begin{array}{ccc}\hfill & & {}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ,\varphi }\left[{\int }_a^t{\Phi }_{\varphi }^{c-1}(t,s){\mathbb{E}}_{c,c}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(t,s)\right)}^c\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\right]\hfill \\ & & ={\rho }^{-\frac{\alpha }{\rho }}{\int }_a^t{\Phi }_{\varphi }^{c+\frac{\alpha }{\rho }-1}(s,r){\mathbb{E}}_{c,c+\frac{\alpha }{\rho }}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(s,r)\right)}^c\right)h\left(r\right){\varphi }^{\prime }\left(r\right)dr,\hfill \end{array}$$

(10)

where ${\Phi }_{\varphi }^{(·)}(t,a)$ and ${\mathbb{E}}_{u,v}(·)$ are given as in (8) and Lemma 3, respectively.

Proof. 

By applying Definition 1 and Lemma 3 we have

$$\begin{array}{ccc}& & {}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ,\varphi }\left[{\int }_a^t{\Phi }_{\varphi }^{c-1}(t,s){\mathbb{E}}_{c,c}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(t,s)\right)}^c\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\right]\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)\left[{\int }_a^s{\Phi }_{\varphi }^{c-1}(s,r){\mathbb{E}}_{c,c}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(s,r)\right)}^c\right)h\left(r\right){\varphi }^{\prime }\left(r\right)dr\right]{\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& {\int }_a^t\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_r^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)\left[{\Phi }_{\varphi }^{c-1}(s,r){\mathbb{E}}_{c,c}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(s,r)\right)}^c\right)\right]{\varphi }^{\prime }\left(s\right)dsh\left(r\right){\varphi }^{\prime }\left(r\right)dr\hfill \\ & =& {\int }_a^t{}_{\rho }^{}{\mathcal{I}}_{r^{+}}^{\alpha ,\varphi }\left[{\Phi }_{\varphi }^{c-1}(s,r){\mathbb{E}}_{c,c}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }^{}(s,r)\right)}^c\right)\right]h\left(r\right){\varphi }^{\prime }\left(r\right)dr.\hfill \end{array}$$

By using Lemma 4, the equality (10) is obtained. □

2.2. The Linear $(\rho ,\varphi )$-Hilfer Fractional Cauchy Problem

Consider the linear variant of the $(\rho ,\varphi )$-Hilfer fractional Cauchy problem with constant coefficient as follows:

$$\left\{\begin{array}{c}{\displaystyle {}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }u\left(t\right)=\lambda u\left(t\right)+h\left(t\right),\alpha \in (n-1,n),\beta \in [0,1],t\in (a,b],}\hfill \\ {\displaystyle {\delta }_{\varphi }^{n-i}{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\rho (n-\gamma );\varphi }u\left(a\right)=c_i,i=1,2,\dots ,n,\alpha \le \gamma =\frac{1}{\rho }\left(\beta \left(\rho n-\alpha \right)+\alpha \right),}\hfill \end{array}\right.$$

(11)

where ${}_{\rho }^H{\mathfrak{D}}_{a^{+}}^{\alpha ,\beta ;\varphi }$ denotes the $(\rho ,\varphi )$-Hilfer $\mathbb{FDO}$ of order $\alpha$ and type $\beta$, ${}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\rho (n-\gamma ),\varphi }$ denotes the $(\rho ,\varphi )$-$\mathbb{RL}$-$\mathbb{FIO}$ of order $\rho (n-\gamma )>0$, $c_j\in \mathbb{R}$, $j=1,2,\dots ,n$, and $\lambda <0$. By applying the Picard’s successive approximation technique, we derive to construct an explicit solution to the problem (11) in form of the Mittag–Leffler kernel.

Lemma 6.

Let $h\in \mathcal{C}\left(\right[a,b],\mathbb{R})$, $\lambda \in \mathbb{R}$, $\alpha \in (n-1,n)$, $\beta \in [0,1]$, and $\rho >0$. Then, the explicit solution of the problem (11) is provided by

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }^{\frac{\alpha }{\rho }}}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s){\mathbb{E}}_{\frac{\alpha }{\rho },\frac{\alpha }{\rho }}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }(t,s)\right)}^{\frac{\alpha }{\rho }}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i}{{\rho }^{\gamma -n}}{\Phi }_{\varphi }^{\gamma -i}(t,a){\mathbb{E}}_{\frac{\alpha }{\rho },\gamma -i+1}\left(\lambda {\left({\rho }^{-1}{\Phi }_{\varphi }(t,a)\right)}^{\frac{\alpha }{\rho }}\right).\hfill \end{array}$$

(12)

Proof. 

Assume u is a solution of the problem (11). By applying Lemma 2, The corresponding an integral equation of the problem (11) can be represented as

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \sum _{i=1}^n\frac{{\Phi }_{\varphi }^{\gamma -i}(t,a)c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}+\frac{\lambda }{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)u\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds.\hfill \end{array}$$

The method of successive approximation is applied to develop an explicit form for the solution in our results. Define

$$\begin{array}{ccc}\hfill u_0\left(t\right)& =& \sum _{i=1}^n\frac{{\Phi }_{\varphi }^{\gamma -i}(t,a)c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)},\hfill \\ \hfill u_k\left(t\right)& =& u_0\left(t\right)+\frac{\lambda }{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)u_{k-1}\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds,k=1,2,3,\dots .\hfill \end{array}$$

For $k=1$, by using Definition 1, we obtain

$$\begin{array}{ccc}\hfill u_1\left(t\right)& =& u_0\left(t\right)+\frac{\lambda }{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)u_0\left(s\right){\varphi }^{\prime }\left(s\right)ds+\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds+\sum _{i=1}^n\frac{{\Phi }_{\varphi }^{\gamma -i}(t,a)c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\hfill \\ & & +\frac{\lambda }{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)\left(\sum _{i=1}^n\frac{{\Phi }_{\varphi }^{\gamma -i}(s,a)c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds+\sum _{i=1}^n\frac{{\Phi }_{\varphi }^{\gamma -i}(t,a)c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\hfill \\ & & +\sum _{i=1}^n\frac{\lambda c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\left[\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)\left({\Phi }_{\varphi }^{\frac{\rho (\gamma -i+1)}{\rho }-1}(s,a)\right){\varphi }^{\prime }\left(s\right)ds\right]\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds+\sum _{i=1}^n\frac{{\Phi }_{\varphi }^{\gamma -i}(t,a)c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\hfill \\ & & +\sum _{i=1}^n\frac{\lambda c_i}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ;\varphi }\left[{\Phi }_{\varphi }^{\frac{\rho (\gamma -i+1)}{\rho }-1}(t,a)\right].\hfill \end{array}$$

From (i) of Lemma 1, we have

$$\begin{array}{ccc}\hfill u_1\left(t\right)& =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}+\sum _{i=1}^n\frac{\lambda c_i{\Phi }_{\varphi }^{\frac{\rho (\gamma -i+1)+\alpha }{\rho }-1}(t,a)}{{\rho }^{i-n}{\Gamma }_{\rho }(\rho (\gamma -i+1)+\alpha )}\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}}\left(\frac{1}{{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}+\frac{\lambda {\Phi }_{\varphi }^{\frac{\alpha }{\rho }}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+\alpha )}\right)\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}}\left(\sum _{j=1}^2\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{(j-1)\frac{\alpha }{\rho }}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+(j-1)\alpha )}\right).\hfill \end{array}$$

By the same process, for $k=2$, one has

$$\begin{array}{ccc}\hfill u_2\left(t\right)& =& u_0\left(t\right)+\frac{\lambda }{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)u_1\left(s\right){\varphi }^{\prime }\left(s\right)ds+\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds+\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\hfill \\ & & +\frac{\lambda }{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)\{\frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^s{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(s,r)h\left(r\right){\varphi }^{\prime }\left(r\right)dr\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(s,a)}{{\rho }^{i-n}}\left(\sum _{j=1}^2\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{(j-1)\frac{\alpha }{\rho }}(s,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+(j-1)\alpha )}\right)\}{\varphi }^{\prime }\left(s\right)ds\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds+\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}\hfill \\ & & +\frac{\lambda }{\rho {\Gamma }_{\rho }\left(2\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{2\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i}{{\rho }^{i-n}}\left(\sum _{j=1}^2\frac{{\lambda }^j}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+(j-1)\alpha )}\right)\hfill \\ & & \times \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s){\Phi }_{\varphi }^{\frac{\rho (\gamma -i+1)+(j-1)\alpha }{\rho }-1}(s,a){\varphi }^{\prime }\left(s\right)ds\hfill \\ & =& \frac{1}{\rho {\Gamma }_{\rho }\left(\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds+\frac{\lambda }{\rho {\Gamma }_{\rho }\left(2\alpha \right)}{\int }_a^t{\Phi }_{\varphi }^{\frac{2\alpha }{\rho }-1}(t,s)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}+\sum _{i=1}^n\frac{c_i}{{\rho }^{i-n}}\left\{\sum _{j=1}^2\frac{{\lambda }^j{}_{\rho }^{}{\mathcal{I}}_{a^{+}}^{\alpha ;\varphi }\left[{\Phi }_{\varphi }^{\frac{\rho (\gamma -i+1)+(j-1)\alpha }{\rho }-1}(t,a)\right]}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+(j-1)\alpha )}\right\}\hfill \\ & =& \frac{1}{\rho }{\int }_a^t\left(\frac{{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)}{{\Gamma }_{\rho }\left(\alpha \right)}+\frac{\lambda {\Phi }_{\varphi }^{\frac{2\alpha }{\rho }-1}(t,s)}{{\Gamma }_{\rho }\left(2\alpha \right)}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}+\sum _{i=1}^n\frac{c_i}{{\rho }^{i-n}}\left(\sum _{j=1}^2\frac{{\lambda }^j{\Phi }_{\varphi }^{\frac{\rho (\gamma -i+1)+j\alpha }{\rho }-1}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+j\alpha )}\right)\hfill \\ & =& \frac{1}{\rho }{\int }_a^t\left(\frac{{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)}{{\Gamma }_{\rho }\left(\alpha \right)}+\frac{\lambda {\Phi }_{\varphi }^{\frac{2\alpha }{\rho }-1}(t,s)}{{\Gamma }_{\rho }\left(2\alpha \right)}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}}\left(\frac{1}{{\Gamma }_{\rho }\left(\rho (\gamma -i+1)\right)}+\sum _{j=1}^2\frac{{\lambda }^j{\Phi }_{\varphi }^{\frac{j\alpha }{\rho }}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+j\alpha )}\right)\hfill \\ & =& \frac{1}{\rho }{\int }_a^t\left(\sum _{j=1}^2\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{\frac{j\alpha }{\rho }-1}(t,s)}{{\Gamma }_{\rho }\left(j\alpha \right)}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}}\left(\sum _{j=1}^3\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{(j-1)\frac{\alpha }{\rho }}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+(j-1)\alpha )}\right).\hfill \end{array}$$

For $k=1,2,\dots$, we obtain that

$$\begin{array}{ccc}\hfill u_k\left(t\right)& =& \frac{1}{\rho }{\int }_a^t\left(\sum _{j=1}^k\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{\frac{j\alpha }{\rho }-1}(t,s)}{{\Gamma }_{\rho }\left(j\alpha \right)}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}}\left(\sum _{j=1}^{k+1}\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{(j-1)\frac{\alpha }{\rho }}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+(j-1)\alpha )}\right).\hfill \end{array}$$

Taking $k\to \infty$ with changing the summation index in the last expression, $j\to j+1$, we obtain

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{\rho }{\int }_a^t\left(\sum _{j=1}^{\infty }\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{\frac{j\alpha }{\rho }-1}(t,s)}{{\Gamma }_{\rho }\left(j\alpha \right)}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}}\left(\sum _{j=1}^{\infty }\frac{{\lambda }^{j-1}{\Phi }_{\varphi }^{(j-1)\frac{\alpha }{\rho }}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+(j-1)\alpha )}\right)\hfill \\ & =& \frac{1}{\rho }{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)\left(\sum _{j=0}^{\infty }\frac{{\lambda }^j{\Phi }_{\varphi }^{\frac{j\alpha }{\rho }}(t,s)}{{\Gamma }_{\rho }\left((j+1)\alpha \right)}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{i-n}}\left(\sum _{j=0}^{\infty }\frac{{\lambda }^j{\Phi }_{\varphi }^{j\frac{\alpha }{\rho }}(t,a)}{{\Gamma }_{\rho }(\rho (\gamma -i+1)+j\alpha )}\right).\hfill \end{array}$$

By using the property (6), we obtain

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }^{\frac{\alpha }{\rho }}}{\int }_a^t{\Phi }_{\varphi }^{\frac{\alpha }{\rho }-1}(t,s)\left(\sum _{j=0}^{\infty }\frac{{\lambda }^j{\Phi }_{\varphi }^{\frac{j\alpha }{\rho }}(t,s)}{{\rho }^{\frac{j\alpha }{\rho }}\Gamma \left(\frac{\alpha }{\rho }j+\frac{\alpha }{\rho }\right)}\right)h\left(s\right){\varphi }^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=1}^n\frac{c_i{\Phi }_{\varphi }^{\gamma -i}(t,a)}{{\rho }^{\gamma -n}}\left(\sum _{j=0}^{\infty }\frac{{\lambda }^j{\Phi }_{\varphi }^{j\frac{\alpha }{\rho }}(t,a)}{{\rho }^{\frac{j\alpha }{\rho }}\Gamma \left(\frac{\alpha }{\rho }j+\gamma -i+1\right)}\right).\hfill \end{array}$$

Applying Lemma 3, we find that the explicit solution (12). □

2.3. An Auxiliary Lemma

Let us denote the weighted space

$${\mathcal{C}}_{{\varphi }_k}^{1-{\gamma }_k}(\mathcal{J},\mathbb{R})=\left\{u:(a,b]\to \mathbb{R}|u\left(a^{+}\right)\mathrm{exists}\mathrm{and}{\Phi }_{\varphi }^{1-\gamma }(t,a)u\left(t\right)\in \mathcal{C}(\mathcal{J},\mathbb{R})\right\},\gamma \in (0,1],$$

where ${\mathcal{C}}_{{\varphi }_k}^{1-{\gamma }_k}={\mathcal{C}}_{{\varphi }_k}^{1-{\gamma }_k}(\mathcal{J},\mathbb{R})$. Next, we provide the weighted space of piecewise continuous functions as follows:

$$\begin{array}{ccc}\hfill {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}(\mathcal{J},\mathbb{R})& =& \{u:(a,b]\to \mathbb{R}|u\in {\mathcal{C}}_{{\varphi }_k}^{1-{\gamma }_k},k=0,1,2,\dots ,m,\hfill \\ & & {}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}u\left(t_k^{+}\right),{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{{\rho }_{k-1}(1-{\gamma }_{k-1});{\varphi }_{k-1}}u\left(t_k^{-}\right)\mathrm{exist}\mathrm{and}\hfill \\ & & {}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{{\rho }_{k-1}(1-{\gamma }_{k-1});{\varphi }_{k-1}}u\left(t_k^{-}\right)={}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{{\rho }_{k-1}(1-{\gamma }_{k-1});{\varphi }_{k-1}}u\left(t_k\right),k=1,\dots ,m\}.\hfill \end{array}$$

Observe that ${\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}={\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}(\mathcal{J},\mathbb{R})$ is a Banach space equipped with

$${\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}=\underset{t\in \mathcal{J}}{sup}\left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)u\left(t\right)\right|.$$

For the easy to prove, we set the symbol that will be used throughout this paper.

$${\mathcal{P}}_k\left(j\right)=\prod _{l=j}^{k-1}{\mathbb{E}}_{\frac{{\alpha }_l}{{\rho }_l}}\left({\lambda }_l{\left({\rho }_l^{-1}{\Phi }_{{\varphi }_l}(t_{l+1},t_l)\right)}^{\frac{{\alpha }_l}{{\rho }_l}}\right),\left|{\mathcal{P}}_k\left(j\right)\right|\le 1,$$

(13)

$$\begin{array}{ccc}\hfill \Xi & =& \sum _{i=0}^m\frac{{\kappa }_i{\mathcal{P}}_i\left(0\right){\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\hfill \\ & & -\sum _{j=0}^n\frac{{\omega }_j{\mathcal{P}}_j\left(0\right){\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right).\hfill \end{array}$$

(14)

Lemma 7.

Assume that ${\alpha }_k\in (0,1)$, ${\beta }_k\in [0,1]$, ${\rho }_k>0$, ${\gamma }_k=({\beta }_k({\rho }_k-{\alpha }_k)+{\alpha }_k)/{\rho }_k$, ${\lambda }_k<0$, ${\varphi }_k\in \mathcal{C}(\mathcal{J},\mathbb{R})$ with ${\varphi }_k^{\prime }>0$, $k=0,1,2\dots ,m$, ${\kappa }_i$, ${\omega }_r\in \mathbb{R}$, $i=0,1,2\dots ,m$, $j=0,1,2\dots ,n$, $h\in {\mathcal{C}}_{{\varphi }_k}^{1-{\gamma }_k}$, and $\Xi \ne 0$. Then, the following impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s:

$$\left\{\begin{array}{c}{\displaystyle {}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}u\left(t\right)={\lambda }_{k}u\left(t\right)+h\left(t\right),t\in {\mathcal{J}}_k\subseteq \mathcal{J},t\ne t_k,}\hfill \\ {\displaystyle \Delta {}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}u\left(t_k\right)={\phi }_k\left(u\left(t_k\right)\right),k=1,2,\dots ,m,}\hfill \\ {\displaystyle \sum _{i=0}^m{\kappa }_{i}u\left({\eta }_i\right)=\sum _{j=0}^n{\omega }_j{}_{{\rho }_j}{\mathcal{I}}_{t_j^{+}}^{{\mu }_j;{\varphi }_j}u\left({\xi }_j\right)+A,{\eta }_i\in (t_i,t_{i+1}],{\xi }_j\in (t_j,t_{j+1}],}\hfill \end{array}\right.$$

(15)

is corresponding to the following integral equation, $u\in {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$,

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)h\left(s\right){\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +[\sum _{r=0}^{k-1}\frac{{\mathcal{P}}_k(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^k{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_k\left(r\right)]\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\hfill \\ & & +\frac{1}{\Xi }(\sum _{j=0}^n\frac{{\omega }_j}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)h\left(s\right){\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\left\{\right[\sum _{r=0}^{j-1}\frac{{\mathcal{P}}_j(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^j{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_j\left(r\right)\left]\frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right\}\hfill \\ & & +A-\sum _{i=0}^m\frac{{\kappa }_i}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & -\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{{\mathcal{P}}_i(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^i{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_i\left(r\right)\left]\frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right\})\hfill \\ & & \times \frac{{\mathcal{P}}_k\left(0\right){\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right).\hfill \end{array}$$

(16)

Proof. 

Let $u\in {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$ be a solution of the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4). We consider the following several cases.

For $t_1\in [t_0,t_1]$, we obtain

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }_0^{\frac{{\alpha }_0}{{\rho }_0}}}{\int }_{t_0}^t{\Phi }_{{\varphi }_0}^{\frac{{\alpha }_0}{{\rho }_0}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},\frac{{\alpha }_0}{{\rho }_0}}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t,s)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)h\left(s\right){\varphi }_0^{\prime }\left(s\right)ds\hfill \\ & & +u_0\frac{{\Phi }_{{\varphi }_0}^{{\gamma }_0-1}(t,t_0)}{{\rho }_0^{{\gamma }_0-1}}{\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},{\gamma }_0}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t,t_0)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right).\hfill \end{array}$$

Taking the operator ${}_{{\rho }_0}{\mathcal{I}}_{t_0^{+}}^{{\rho }_0\left(1-{\gamma }_0\right);{\varphi }_0}$ into the above equation with Lemmas 4 and 5, which implies that

$$\begin{array}{ccc}\hfill {}_{{\rho }_0}{\mathcal{I}}_{t_0^{+}}^{{\rho }_0\left(1-{\gamma }_0\right);{\varphi }_0}u\left(t\right)& =& \frac{1}{{\rho }_0^{\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}}{\int }_{t_0}^t{\Phi }_{{\varphi }_0}^{\frac{{\alpha }_0}{{\rho }_0}-{\gamma }_0}(t,s){\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t,s)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)h\left(s\right){\varphi }_0^{\prime }\left(s\right)ds\hfill \\ & & +u_0{\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0}}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t,t_0)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)\hfill \end{array}$$

In particular, for $t=t_1$, it follows that

$$\begin{array}{ccc}& & {}_{{\rho }_0}{\mathcal{I}}_{t_0^{+}}^{{\rho }_0\left(1-{\gamma }_0\right);{\varphi }_0}u\left(t_1\right)\hfill \\ & =& \frac{1}{{\rho }_0^{\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}}{\int }_{t_0}^{t_1}{\Phi }_{{\varphi }_0}^{\frac{{\alpha }_0}{{\rho }_0}-{\gamma }_0}(t_1,s){\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,s)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)h\left(s\right){\varphi }_0^{\prime }\left(s\right)ds\hfill \\ & & +u_0{\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0}}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,t_0)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)\hfill \end{array}$$

For $t\in (t_1,t_2]$, we obtain

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }_1^{\frac{{\alpha }_1}{{\rho }_1}}}{\int }_{t_1}^t{\Phi }_{{\varphi }_1}^{\frac{{\alpha }_1}{{\rho }_1}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({\rho }_1^{-1}{\Phi }_{{\varphi }_1}(t,s)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)h\left(s\right){\varphi }_1^{\prime }\left(s\right)ds\hfill \\ & & +{}_{{\rho }_1}^{}{\mathcal{I}}_{t_1^{+}}^{\rho (1-{\gamma }_1);{\varphi }_1}u\left(t_1^{+}\right)\frac{{\Phi }_{{\varphi }_1}^{{\gamma }_1-1}(t,t_1)}{{\rho }_1^{{\gamma }_1-1}}{\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},{\gamma }_1}\left({\lambda }_1{\left({\rho }_1^{-1}{\Phi }_{{\varphi }_1}(t,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right).\hfill \end{array}$$

By using the impulsive condition, that is ${}_{{\rho }_1}^{}{\mathcal{I}}_{t_1^{+}}^{{\rho }_1(1-{\gamma }_1);{\varphi }_1}u\left(t_1^{+}\right)={}_{{\rho }_0}^{}{\mathcal{I}}_{t_0^{+}}^{{\rho }_0(1-{\gamma }_0);{\varphi }_0}u\left(t_1^{-}\right)+{\phi }_1\left(u\left(t_1\right)\right)$, we obtain

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }_1^{\frac{{\alpha }_1}{{\rho }_1}}}{\int }_{t_1}^t{\Phi }_{{\varphi }_1}^{\frac{{\alpha }_1}{{\rho }_1}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({\rho }_1^{-1}{\Phi }_{{\varphi }_1}(t,s)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)h\left(s\right){\varphi }_1^{\prime }\left(s\right)ds\hfill \\ & & +\left[{}_{{\rho }_0}^{}{\mathcal{I}}_{t_0^{+}}^{{\rho }_0(1-{\gamma }_0);{\varphi }_0}u\left(t_1^{-}\right)+{\phi }_1\left(u\left(t_1\right)\right)\right]\frac{{\Phi }_{{\varphi }_1}^{{\gamma }_1-1}(t,t_1)}{{\rho }_1^{{\gamma }_1-1}}{\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},{\gamma }_1}\left({\lambda }_1{\left({\rho }_1^{-1}{\Phi }_{{\varphi }_1}(t,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)\hfill \\ & =& \frac{1}{{\rho }_1^{\frac{{\alpha }_1}{{\rho }_1}}}{\int }_{t_1}^t{\Phi }_{{\varphi }_1}^{\frac{{\alpha }_1}{{\rho }_1}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({\rho }_1^{-1}{\Phi }_{{\varphi }_1}(t,s)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)h\left(s\right){\varphi }_1^{\prime }\left(s\right)ds\hfill \\ & & +[\frac{1}{{\rho }_0^{\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}}{\int }_{t_0}^{t_1}{\Phi }_{{\varphi }_0}^{\frac{{\alpha }_0}{{\rho }_0}-{\gamma }_0}(t_1,s){\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,s)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)h\left(s\right){\varphi }_0^{\prime }\left(s\right)ds\hfill \\ & & +u_0{\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0}}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,t_0)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)+{\phi }_1\left(u\left(t_1\right)\right)]\frac{{\Phi }_{{\varphi }_1}^{{\gamma }_1-1}(t,t_1)}{{\rho }_1^{{\gamma }_1-1}}{\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},{\gamma }_1}\left({\lambda }_1{\left({\rho }_1^{-1}{\Phi }_{{\varphi }_1}(t,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right).\hfill \end{array}$$

Taking ${}_{{\rho }_1}{\mathcal{I}}_{t_1^{+}}^{{\rho }_1\left(1-{\gamma }_1\right);{\varphi }_1}$ into the above equation with Lemmas 4 and 5, one has

$$\begin{array}{ccc}& & {}_{{\rho }_1}{\mathcal{I}}_{t_1^{+}}^{{\rho }_1\left(1-{\gamma }_1\right);{\varphi }_1}u\left(t\right)\hfill \\ & =& \frac{1}{{{\rho }_1}^{\frac{{\alpha }_1}{{\rho }_1}+1-{\gamma }_1}}{\int }_{t_1}^t{\Phi }_{{\varphi }_1}^{\frac{{\alpha }_1}{{\rho }_1}-{\gamma }_1}(t,s){\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},\frac{{\alpha }_1}{{\rho }_1}+1-{\gamma }_1}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t,s)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)h\left(s\right){\varphi }_1^{\prime }\left(s\right)ds\hfill \\ & & +[\frac{1}{{\rho }_0^{\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}}{\int }_{t_0}^{t_1}{\Phi }_{{\varphi }_0}^{\frac{{\alpha }_0}{{\rho }_0}-{\gamma }_0}(t_1,s){\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,s)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)h\left(s\right){\varphi }_0^{\prime }\left(s\right)ds\hfill \\ & & +u_0{\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0}}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,t_0)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)+{\phi }_1\left(u\left(t_1\right)\right)]{\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right).\hfill \end{array}$$

In particular, for $t=t_2$, we have

$$\begin{array}{ccc}& & {}_{{\rho }_1}{I}_{t_1^{+}}^{{\rho }_1\left(1-{\gamma }_1\right);{\varphi }_1}u\left(t_2\right)\hfill \\ & =& \frac{1}{{{\rho }_1}^{\frac{{\alpha }_1}{{\rho }_1}+1-{\gamma }_1}}{\int }_{t_1}^{t_2}{\Phi }_{{\varphi }_1}^{\frac{{\alpha }_1}{{\rho }_1}-{\gamma }_1}(t_2,s){\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},\frac{{\alpha }_1}{{\rho }_1}+1-{\gamma }_1}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t_2,s)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)h\left(s\right){\varphi }_1^{\prime }\left(s\right)ds\hfill \\ & & +[\frac{1}{{\rho }_0^{\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}}{\int }_{t_0}^{t_1}{\Phi }_{{\varphi }_0}^{\frac{{\alpha }_0}{{\rho }_0}-{\gamma }_0}(t_1,s){\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,s)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)h\left(s\right){\varphi }_0^{\prime }\left(s\right)ds\hfill \\ & & +u_0{\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0}}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,t_0)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)+{\phi }_1\left(u\left(t_1\right)\right)]{\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t_2,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right).\hfill \end{array}$$

For $t\in (t_2,t_3]$ with ${}_{{\rho }_2}^{}{\mathcal{I}}_{t_2^{+}}^{{\rho }_2(1-{\gamma }_2);{\varphi }_2}u\left(t_2^{+}\right)={}_{{\rho }_1}^{}{\mathcal{I}}_{t_1^{+}}^{{\rho }_1(1-{\gamma }_1);{\varphi }_1}u\left(t_2^{-}\right)+{\phi }_2\left(u\left(t_2\right)\right)$, we obtain

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }_2^{\frac{{\alpha }_2}{{\rho }_2}}}{\int }_{t_2}^t{\Phi }_{{\varphi }_2}^{\frac{{\alpha }_2}{{\rho }_2}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_2}{{\rho }_2},\frac{{\alpha }_2}{{\rho }_2}}\left({\lambda }_2{\left({\rho }_2^{-1}{\Phi }_{{\varphi }_2}(t,s)\right)}^{\frac{{\alpha }_2}{{\rho }_2}}\right)h\left(s\right){\varphi }_2^{\prime }\left(s\right)ds\hfill \\ & & +\{\frac{1}{{{\rho }_1}^{\frac{{\alpha }_1}{{\rho }_1}+1-{\gamma }_1}}{\int }_{t_1}^{t_2}{\Phi }_{{\varphi }_1}^{\frac{{\alpha }_1}{{\rho }_1}-{\gamma }_1}(t_2,s){\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1},\frac{{\alpha }_1}{{\rho }_1}+1-{\gamma }_1}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t_2,s)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)h\left(s\right){\varphi }_1^{\prime }\left(s\right)ds\hfill \\ & & +{\int }_{t_0}^{t_1}{\Phi }_{{\varphi }_0}^{\frac{{\alpha }_0}{{\rho }_0}-{\gamma }_0}(t_1,s){\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0},\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,s)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)h\left(s\right){\varphi }_0^{\prime }\left(s\right)ds\hfill \\ & & \times \frac{{\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t_2,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)}{{\rho }_0^{\frac{{\alpha }_0}{{\rho }_0}+1-{\gamma }_0}}+u_0{\mathbb{E}}_{\frac{{\alpha }_0}{{\rho }_0}}\left({\lambda }_0{\left({\rho }_0^{-1}{\Phi }_{{\varphi }_0}(t_1,t_0)\right)}^{\frac{{\alpha }_0}{{\rho }_0}}\right)\hfill \\ & & \times {\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t_2,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)+{\phi }_1\left(u\left(t_1\right)\right){\mathbb{E}}_{\frac{{\alpha }_1}{{\rho }_1}}\left({\lambda }_1{\left({{\rho }_1}^{-1}{\Phi }_{{\varphi }_1}(t_2,t_1)\right)}^{\frac{{\alpha }_1}{{\rho }_1}}\right)\hfill \\ & & +{\phi }_2\left(u\left(t_2\right)\right)\}\frac{{\Phi }_{{\varphi }_2}^{{\gamma }_2-1}(t,t_2)}{{\rho }_2^{{\gamma }_2-1}}{\mathbb{E}}_{\frac{{\alpha }_2}{{\rho }_2},{\gamma }_2}\left({\lambda }_2{\left({\rho }_2^{-1}{\Phi }_{{\varphi }_2}(t,t_2)\right)}^{\frac{{\alpha }_2}{{\rho }_2}}\right).\hfill \end{array}$$

Repeating the previous procedure, for $t\in {\mathcal{J}}_k$, $k=0,1,\dots ,m$, it follows form

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)h\left(s\right){\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +[\sum _{r=0}^{k-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}\prod _{i=r+1}^{k-1}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},1}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}(t_{i+1},t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_j}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^k{\phi }_r\left(u\left(t_r\right)\right)\prod _{i=r}^{k-1}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},1}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}(t_{i+1},t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)]\hfill \\ & & \times \frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)+u_0\prod _{i=0}^{k-1}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},1}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}(t_{i+1},t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\hfill \\ & & \times \frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right).\hfill \end{array}$$

(17)

By using the symbol (13), the form (17) can be rewritten

$$\begin{array}{ccc}\hfill u\left(t\right)& =& \frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)h\left(s\right){\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +[\sum _{r=0}^{k-1}\frac{{\mathcal{P}}_k(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\hfill \\ & & \times h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^k{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_k\left(r\right)]\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\hfill \\ & & +u_0\frac{{\mathcal{P}}_k\left(0\right){\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right).\hfill \end{array}$$

(18)

From the non-local condition, ${\sum }_{i=0}^m{\kappa }_{i}u\left({\eta }_i\right)={\sum }_{j=0}^n{\omega }_j{}_{{\rho }_j}{\mathcal{I}}_{t_j^{+}}^{{\mu }_j;{\varphi }_j}u\left({\xi }_j\right)+A$, we obtain

$$\begin{array}{ccc}& & \sum _{i=0}^m{\kappa }_{i}u\left({\eta }_i\right)\hfill \\ & =& \sum _{i=0}^m\frac{{\kappa }_i}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{{\mathcal{P}}_i(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\hfill \\ & & \times h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^i{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_i\left(r\right)\left]\frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right\}\hfill \\ & & +u_0\sum _{i=0}^m\frac{{\kappa }_i{\mathcal{P}}_i\left(0\right){\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right).\hfill \end{array}$$

(19)

and

$$\begin{array}{ccc}& & \sum _{j=0}^n{\omega }_j{}_{{\rho }_j}{\mathcal{I}}_{t_j^{+}}^{{\mu }_j;{\varphi }_j}u\left({\xi }_j\right)\hfill \\ & =& \sum _{j=0}^n\frac{{\omega }_j}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)h\left(s\right){\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\left\{\right[\sum _{r=0}^{j-1}\frac{{\mathcal{P}}_j(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\hfill \\ & & \times h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_j\left(r\right)\left]\frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right\}\hfill \\ & & +u_0\sum _{j=0}^n\frac{{\omega }_j{\mathcal{P}}_j\left(0\right){\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right).\hfill \end{array}$$

(20)

Solving the above system (19) and (20), we obtain

$$\begin{array}{ccc}& & u_0\hfill \\ & =& \frac{1}{\Xi }(\sum _{j=0}^n\frac{{\omega }_j}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)h\left(s\right){\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\left\{\right[\sum _{r=0}^{j-1}\frac{{\mathcal{P}}_j(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\hfill \\ & & \times h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_j\left(r\right)\left]\frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right\}\hfill \\ & & +A-\sum _{i=0}^m\frac{{\kappa }_i}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & -\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{{\mathcal{P}}_i(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\hfill \\ & & \times h\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^i{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_i\left(r\right)\left]\frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right\}),\hfill \end{array}$$

where $\Xi$ is given by (14). Inserting $u_0$ into (18), we obtain the solution (16).

Conversely, it is easy to show by direct calculation that the solution $u\left(t\right)$ is defined by (16) satisfies the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (15). □

3. Existence Results

By Lemma 7, we define an operator $\mathcal{Q}:{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}\to {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$ as

$$\begin{array}{ccc}\hfill \left(\mathcal{Q}u\right)\left(t\right)& =& \frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right){\mathcal{F}}_u\left(s\right){\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +[\sum _{r=0}^{k-1}\frac{{\mathcal{P}}_k(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^k{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_k\left(r\right)]\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\hfill \\ & & +\frac{1}{\Xi }(\sum _{j=0}^n\frac{{\omega }_j}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right){\mathcal{F}}_u\left(s\right){\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\left\{\right[\sum _{r=0}^{j-1}\frac{{\mathcal{P}}_j(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^j{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_j\left(r\right)\left]\frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right\}\hfill \\ & & +A-\sum _{i=0}^m\frac{{\kappa }_i}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right){\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & -\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{{\mathcal{P}}_i(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^i{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_i\left(r\right)\left]\frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right\})\hfill \\ & & \times \frac{{\mathcal{P}}_k\left(0\right){\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right).\hfill \end{array}$$

(21)

where ${\mathcal{F}}_u\left(t\right)=f(t,u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}u\left(t\right))$. It should be noted that $\mathcal{Q}$ has fixed points if and only if the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) has solutions.

We define the notations of constants that will be used throughout this paper.

$$\begin{array}{ccc}\hfill {\Delta }_1& =& \frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right],\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\Delta }_2& =& \frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left[m+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)\right],\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\Delta }_3& =& 1+\frac{1}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left(\sum _{j=0}^n\left|{\omega }_j\right|+\sum _{i=0}^m\left|{\kappa }_i\right|\right).\hfill \end{array}$$

(24)

3.1. Uniqueness Result via Banach’s Fixed Point Theorem

Lemma 8

(Banach’s fixed point theorem [53]). Assume that $\mathcal{B}$ is a non-empty closed subset of $\mathcal{X}$ where $\mathcal{X}$ is a Banach. Then, any contraction mapping $\mathcal{Q}$ from $\mathcal{B}$ into itself has a unique fixed point.

Theorem 1.

Let $f\in \mathcal{C}(\mathcal{J}\times {\mathbb{R}}^3,\mathbb{R})$ and ${\phi }_k\in \mathcal{C}(\mathbb{R},\mathbb{R})$ for $k=1,2\dots ,m$. Suppose that

(${\mathbb{H}}_1$)

There exist constants ${\mathbb{L}}_1$, ${\mathbb{L}}_2$, ${\mathbb{L}}_3>0$ so that

$$\begin{array}{ccc}& & |f(t,u_1,v_1,w_1)-f(t,u_2,v_2,w_2)|\hfill \\ & & \le {\Phi }_{{\rho }_k}^{{\gamma }_k-1}(t,t_k)\left[{\mathbb{L}}_1\parallel u_1-u_2{\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{L}}_2\parallel v_1-v_2{\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{L}}_3{\parallel w_1-w_2\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\right],\hfill \end{array}$$

for all $t\in \mathcal{J}$ and $u_i$, $v_i$, $w_i\in \mathbb{R}$, $i=1,2$.

(${\mathbb{H}}_2$)

There exists a constant ${\mathbb{N}}_1>0$ so that

$$|{\phi }_k\left(u\right)-{\phi }_k\left(v\right)|\le {\mathbb{N}}_1{\Phi }_{{\rho }_k}^{{\gamma }_k-1}(t,t_k){\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}},u,v\in \mathbb{R},k=1,2,\dots ,m.$$

Then, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) has a unique solution if

$${\Delta }_1\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right)+{\Delta }_2{\mathbb{N}}_1<1,$$

(25)

where ${\Phi }_1^{\ast }$ and ${\Phi }_2^{\ast }$ are given by

$${\Phi }_1^{\ast }:=\frac{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right){\Phi }_{{\rho }_m}^{\frac{{\delta }_m}{{\rho }_m}+{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}({\rho }_m{\gamma }_m+{\delta }_m)},{\Phi }_2^{\ast }:=\frac{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right){\Phi }_{{\rho }_m}^{\frac{{\theta }_m}{{\rho }_m}+{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}({\rho }_m{\gamma }_m+{\theta }_m)}.$$

(26)

Proof. 

Transformation the problem (4) into a fixed point problem, $u=\mathcal{Q}u$, where $\mathcal{Q}$ is define by (21). We know that the fixed points of $\mathcal{Q}$ are solutions to the problem (4). We separate the procedure into two steps.

Step 1: We show that $\mathcal{Q}{\mathcal{B}}_{{{\rm Y}}_1}\subset {\mathcal{B}}_{{{\rm Y}}_1}$.

Let ${sup}_{t\in \mathcal{J}}|f(t,0,0,0)|:={\mathbb{F}}_1<\infty$ and ${\mathbb{I}}_1:=max\{|{\phi }_k\left(0\right)|:k=1,2,\dots ,m\}$. Define ${\mathcal{B}}_{{{\rm Y}}_1}:=\{u\in {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}{:\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le {{\rm Y}}_1\}$ with the radius

$${{\rm Y}}_1\ge \frac{{\Delta }_1{\mathbb{F}}_1+{\Delta }_2{\mathbb{I}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}}{1-\left[{\Delta }_1\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right)+{\Delta }_2{\mathbb{N}}_1\right]}.$$

Obviously, the set ${\mathcal{B}}_{{{\rm Y}}_1}$ is a bounded, closed, and convex subset of ${\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$.

For every $u\in {\mathcal{B}}_{{{\rm Y}}_1}$, we obtain

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\mathcal{Q}u\right)\left(t\right)\right|\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s)\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)\right|{\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +[\sum _{r=0}^{k-1}\frac{\left|{\mathcal{P}}_k(r+1)\right|}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\right|\hfill \\ & & \times \left|{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^k\left|{\phi }_r\left(u\left(t_r\right)\right)\right|\left|{\mathcal{P}}_k\left(r\right)\right|]\frac{1}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|\hfill \end{array}$$

$$\begin{array}{ccc}& & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s)\left|{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)\right|{\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\left\{\right[\sum _{r=0}^{j-1}\frac{\left|{\mathcal{P}}_j(r+1)\right|}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\right|\hfill \\ & & \times \left|{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j\left|{\phi }_r\left(u\left(t_r\right)\right)\right|\left|{\mathcal{P}}_j\left(r\right)\right|]\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right|\}+\left|A\right|+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s)\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{\left|{\mathcal{P}}_i(r+1)\right|}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^i\left|{\phi }_r\left(u\left(t_r\right)\right)\right|\left|{\mathcal{P}}_i\left(r\right)\right|\left]\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right|\right\})\hfill \\ & & \times \frac{\left|{\mathcal{P}}_k\left(0\right)\right|}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|.\hfill \end{array}$$

Since,

$$\begin{array}{ccc}\hfill \left|{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}u\left(t\right)\right|& =& \frac{1}{{\rho }_k{\Gamma }_{{\rho }_k}\left({\delta }_k\right)}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\delta }_k}{{\rho }_k}-1}(t,s){\varphi }_k^{\prime }\left(s\right)\left|u\left(s\right)\right|ds\hfill \\ & \le & \frac{{\Gamma }_{{\rho }_k}\left({\rho }_k{\gamma }_k\right)}{{\Gamma }_{{\rho }_k}({\rho }_k{\gamma }_k+{\delta }_k)}{\Phi }_{{\rho }_k}^{\frac{{\delta }_k}{{\rho }_k}+{\gamma }_k-1}(t,t_k){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}},\hfill \\ \hfill \left|{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}u\left(t\right)\right|& \le & \frac{{\Gamma }_{{\rho }_k}\left({\rho }_k{\gamma }_k\right)}{{\Gamma }_{{\rho }_k}({\rho }_k{\gamma }_k+{\theta }_k)}{\Phi }_{{\rho }_k}^{\frac{{\theta }_k}{{\rho }_k}+{\gamma }_k-1}(t,t_k){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}.\hfill \end{array}$$

By applying (${\mathbb{H}}_1$)–(${\mathbb{H}}_2$), we have the following inequalities

$$\begin{array}{ccc}\hfill \left|{\mathcal{F}}_u\left(t\right)\right|& \le & \left|f(t,u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}u\left(t\right))-f(t,0,0,0)\right|+\left|f(t,0,0,0)\right|\hfill \\ & \le & {\mathbb{L}}_1\left|u\left(t\right)\right|+{\mathbb{L}}_2\left|{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}u\left(t\right)\right|+{\mathbb{L}}_3\left|{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}u\left(t\right)\right|+\left|f(t,0,0,0)\right|\hfill \\ & \le & \left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \left|{\phi }_k\left(u\left(t_k\right)\right)\right|& \le & \left|{\phi }_k\left(u\left(t_k\right)\right)-{\phi }_k\left(0\right)\right|+\left|{\phi }_k\left(0\right)\right|\le {\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1.\hfill \end{array}$$

(28)

By using the properties ${\mathbb{E}}_{\alpha }\left(z\right)\le 1$ and ${\mathbb{E}}_{\alpha ,\beta }\left(z\right)\le 1/\Gamma \left(\beta \right)$ with (27) and (28), we obtain that

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\mathcal{Q}u\right)\left(t\right)\right|\hfill \\ & \le & \frac{\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}\Gamma \left(\frac{{\alpha }_k}{{\rho }_k}\right)}{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k){\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\varphi }_k^{\prime }\left(s\right)ds\hfill \end{array}$$

$$\begin{array}{ccc}& & +\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}[\sum _{r=0}^{k-1}\frac{\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +k\left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]]\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|\left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\Gamma \left(\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}\right)}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\left\{\right[\sum _{r=0}^{j-1}\frac{\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +j\left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]\left]\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}\Gamma \left({\gamma }_j+\frac{{\mu }_j}{{\rho }_j}\right)}\right\}\hfill \\ & & +\left|A\right|+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|\left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}\Gamma \left(\frac{{\alpha }_i}{{\rho }_i}\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +i\left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){∥u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]\left]\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}\right\})\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}.\hfill \end{array}$$

By using (6) and the property (i) in Lemma 1, we have

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\mathcal{Q}u\right)\left(t\right)\right|\hfill \\ & \le & \frac{1}{{\rho }_k{\Gamma }_{{\rho }_k}\left({\alpha }_k\right)}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\varphi }_k^{\prime }\left(s\right)ds{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]\hfill \\ & & +\frac{1}{{\Gamma }_{{\rho }_k}\left({\beta }_k({\rho }_k-{\alpha }_k)+{\alpha }_k\right)}[\sum _{r=0}^{k-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & \times \left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]+k\left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]]\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j\right)}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\varphi }_j^{\prime }\left(s\right)ds\right[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\hfill \\ & & +{\mathbb{F}}_1]+\sum _{j=0}^n\{\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}[\sum _{r=0}^{j-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]\hfill \\ & & +j\left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]\left]\right\}+\left|A\right|\hfill \\ & & +\frac{1}{\left|\Xi \right|}\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i{\Gamma }_{{\rho }_i}\left({\alpha }_i\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\varphi }_r^{\prime }\left(s\right)ds\left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]\hfill \end{array}$$

$$\begin{array}{ccc}& & +\sum _{i=0}^m\left\{\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\right[\sum _{r=0}^{i-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]\hfill \\ & & +i\left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]\left]\right\})\frac{1}{{\Gamma }_{{\rho }_k}\left({\beta }_k({\rho }_k-{\alpha }_k)+{\alpha }_k\right)}\hfill \\ & \le & \{\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right]\}\hfill \\ & & \times \left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]+\frac{1}{{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}\{m\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\right)\}\hfill \\ & & \times \left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}\hfill \\ & =& \{\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right]\}\hfill \\ & & \times \left[\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{F}}_1\right]+\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[m\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)]\hfill \\ & & \times \left[{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\mathbb{I}}_1\right]+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\hfill \\ & =& \left[{\Delta }_1\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right)+{\Delta }_2{\mathbb{N}}_1\right]{\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+{\Delta }_1{\mathbb{F}}_1+{\Delta }_2{\mathbb{I}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\hfill \\ & \le & {{\rm Y}}_1.\hfill \end{array}$$

Hence, ${\parallel \mathcal{Q}u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le {{\rm Y}}_1$ which implies that $\mathcal{Q}{\mathcal{B}}_{{{\rm Y}}_1}\subset {\mathcal{B}}_{{{\rm Y}}_1}$.

Step 2: We show that $\mathcal{Q}$ is a contraction.

For each u, $v\in {\mathcal{B}}_{{{\rm Y}}_1}$ and for any $t\in \mathcal{J}$, we obtain that

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\left(\mathcal{Q}u\right)\left(t\right)-\left(\mathcal{Q}v\right)\left(t\right)\right)\right|\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s)\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)-{\mathcal{F}}_v\left(s\right)\right|{\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +[\sum _{r=0}^{k-1}\frac{\left|{\mathcal{P}}_k(r+1)\right|}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\right|\hfill \\ & & \times \left|{\mathcal{F}}_u\left(s\right)-{\mathcal{F}}_v\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^k\left|{\phi }_r\left(u\left(t_r\right)\right)-{\phi }_r\left(v\left(t_r\right)\right)\right|\left|{\mathcal{P}}_k\left(r\right)\right|]\hfill \\ & & \times \frac{1}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|+\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s)\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)-{\mathcal{F}}_v\left(s\right)\right|{\varphi }_j^{\prime }\left(s\right)ds+\sum _{j=0}^n\left\{\right[\sum _{r=0}^{j-1}\frac{\left|{\mathcal{P}}_j(r+1)\right|}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)-{\mathcal{F}}_v\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^j\left|{\phi }_r\left(u\left(t_r\right)\right)-{\phi }_r\left(v\left(t_r\right)\right)\right|\left|{\mathcal{P}}_j\left(r\right)\right|]\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right|\}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s)\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right|\left|{\mathcal{F}}_u\left(s\right)-{\mathcal{F}}_v\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{\left|{\mathcal{P}}_i(r+1)\right|}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)\right|\hfill \\ & & \times \left|{\mathcal{F}}_u\left(s\right)-{\mathcal{F}}_v\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^i\left|{\phi }_r\left(u\left(t_r\right)\right)-{\phi }_r\left(v\left(t_r\right)\right)\right|\left|{\mathcal{P}}_i\left(r\right)\right|]\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right|\left\}\right)\frac{\left|{\mathcal{P}}_k\left(0\right)\right|}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|.\hfill \end{array}$$

By using (${\mathbb{H}}_1$)–(${\mathbb{H}}_2$), we have

$$\begin{array}{ccc}\hfill \left|{\mathcal{F}}_u\left(t\right)-{\mathcal{F}}_v\left(t\right)\right|& \le & |f(t,u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}u\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}u\left(t\right))\hfill \\ & & -f(t,v\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}v\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}v\left(t\right))|\hfill \\ & \le & \left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}},\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill \left|{\phi }_k\left(u\left(t_k\right)\right)-{\phi }_k\left(v\left(t_k\right)\right)\right|& \le & {\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}.\hfill \end{array}$$

(30)

By using Lemma 1, Lemma 3 and (29) and (30), implies that

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\left(\mathcal{Q}u\right)\left(t\right)-\left(\mathcal{Q}v\right)\left(t\right)\right)\right|\hfill \\ & \le & \{\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\right)\left]\right\}\hfill \\ & & \times \left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+\frac{{\mathbb{N}}_1{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m){\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}}{{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}\{m\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\right)\}\hfill \\ & =& \{\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right]\}\hfill \\ & & \times \left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right){\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+\frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[m\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)]{\mathbb{N}}_1{\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\hfill \\ & =& \left[{\Delta }_1\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right)+{\Delta }_2{\mathbb{N}}_1\right]{\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}},\hfill \end{array}$$

this yields that ${∥\mathcal{Q}u-\mathcal{Q}v∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le [{\Delta }_1\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right)+{\Delta }_2{\mathbb{N}}_1]{\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}$. Since condition (25) holds, then $\mathcal{Q}$ is a contraction map. Hence, by Lemma 8, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) has a unique solution. □

3.2. Existence Result via O’Regan’s Fixed Point Theorem

Lemma 9

(O’Regan’s fixed point theorem [54]). Let $\mathcal{K}$ be an open set in a closed, convex set $\mathcal{B}$ of a Banach space $\mathcal{X}$, with $\overline{\mathcal{K}}$ and $\partial \mathcal{K}$ representing the closure and boundary of $\mathcal{K}$, respectively. Moreover, it is assumed that $0\in \mathcal{K}$ and $\mathcal{Q}:\overline{\mathcal{K}}\to \mathcal{B}$ is such that $\mathcal{Q}\left(\overline{\mathcal{K}}\right)$ is bounded and that $\mathcal{Q}={\mathcal{Q}}_1+{\mathcal{Q}}_2$, where ${\mathcal{Q}}_1:\mathcal{K}\to \mathcal{B}$ is continuous and completely continuous and ${\mathcal{Q}}_2:\mathcal{K}\to \mathcal{B}$ is non-linear contraction, that is, there exists a non-negative non-decreasing function $\psi :[0,\infty )\to [0,\infty )$, such that $\psi \left(z\right)<z$ for $z>0$, and $\parallel {\mathcal{Q}}_{2}u-{\mathcal{Q}}_{2}v\parallel \le \psi (\parallel u-v\parallel )$ for all u, $v\in \mathcal{K}$. Then, either ($C_1$) $\mathcal{Q}$ has a fixed point $u\in \overline{\mathcal{K}}$; or ($C_2$) there exist a point $u\in \partial \mathcal{K}$ and $\sigma \in (0,1)$, such that $u=\sigma \mathcal{Q}\left(u\right)$.

Theorem 2.

Let $f\in \mathcal{C}(\mathcal{J}\times {\mathbb{R}}^3,\mathbb{R})$ and ${\phi }_k\in \mathcal{C}(\mathbb{R},\mathbb{R})$, $k=1,2\dots ,m$. Suppose that

(${\mathbb{H}}_3$)

There exists ${\mathbb{M}}_1>0$ such that $|{\phi }_k\left(u\right)|\le {\mathbb{M}}_1$ for all $u\in \mathbb{R}$, $k=1,2,\dots ,m$.

(${\mathbb{H}}_4$)

There exist a continuous non-decreasing function $\psi :[0,\infty )\to (0,\infty )$ and $p_1$, $p_2$, $p_3\in \mathcal{C}([0,T],{\mathbb{R}}^{+})$, such that

$$|f(t,u,v,w)|\le p_1\left(t\right)\psi \left({\Phi }_{{\rho }_k}^{{\gamma }_k-1}(t,t_k)|u|\right)+{\Phi }_{{\rho }_k}^{{\gamma }_k-1}(t,t_k)\left[p_2\left(t\right)|v|+p_3\left(t\right)|w|\right],$$

(31)

for any $(t,u,v,w)\in \mathcal{J}\times {\mathbb{R}}^3$.

(${\mathbb{H}}_5$)

There exist a continuous non-decreasing function $q_1:[0,\infty )\to [0,\infty )$ and $b_1>0$, such that

$$\begin{array}{ccc}\hfill |{\phi }_k\left(u\right)-{\phi }_k\left(v\right)|& \le & q_1\left({\Phi }_{{\rho }_k}^{{\gamma }_k-1}(t,t_k)|u-v|\right)\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill q_1\left({\Phi }_{{\rho }_k}^{{\gamma }_k-1}(t,t_k)|u|\right)& \le & b_1{\Phi }_{{\rho }_k}^{{\gamma }_k-1}(t,t_k)|u|,\hfill \end{array}$$

(33)

for any u, $v\in \mathbb{R}$, $k=1,2,\dots ,m$ satisfying ${\Delta }_2{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)b_1<1$ where ${\Delta }_2$ is defined by (23).

(${\mathbb{H}}_6$)

The following condition holds

$$\underset{{{\rm Y}}_2\in (0,\infty )}{sup}\frac{{{\rm Y}}_2}{{\Delta }_1{p}_1^{\ast }\psi \left({{\rm Y}}_2\right)+{\Delta }_2{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}}>\frac{1}{1-\left[{\Delta }_1(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast })\right]},$$

(34)

with $\left[{\Delta }_1(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast })\right]<1$ and ${\Delta }_i$, $(i=1,2)$, are given by (22) and (23), respectively, $p_1^{\ast }={sup}_{t\in \mathcal{J}}|p_1\left(t\right)|$, $p_2^{\ast }={sup}_{t\in \mathcal{J}}|p_2\left(t\right)|$, $p_3^{\ast }={sup}_{t\in \mathcal{J}}|p_3\left(t\right)|$.

Then, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) has at least one solution.

Proof. 

We separate $\mathcal{Q}:{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}\to {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$ by (21) into two operators ${\mathcal{Q}}_1$ and ${\mathcal{Q}}_2$ as $\left(\mathcal{Q}u\right)\left(t\right)=\left({\mathcal{Q}}_{1}u\right)\left(t\right)+\left({\mathcal{Q}}_{2}u\right)\left(t\right)$ for $t\in \mathcal{J}$, where

$$\begin{array}{ccc}\hfill \left({\mathcal{Q}}_{1}u\right)\left(t\right)& =& \frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right){\mathcal{F}}_u\left(s\right){\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=0}^{k-1}\frac{{\mathcal{P}}_k(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}\hfill \\ & & \times {\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)+(\sum _{j=0}^n\frac{{\omega }_j}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right){\mathcal{F}}_u\left(s\right){\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\{\sum _{r=0}^{j-1}\frac{{\mathcal{P}}_j(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & \times \frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\}-\sum _{i=0}^m\frac{{\kappa }_i}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\hfill \\ & & \times {\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds-\sum _{i=0}^m\frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\{\sum _{r=0}^{i-1}\frac{{\mathcal{P}}_i(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\hfill \\ & & \times {\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_u\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\left\}\right)\frac{{\mathcal{P}}_k\left(0\right){\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{\Xi {\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right),\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill \left({\mathcal{Q}}_{2}u\right)\left(t\right)& =& \frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\sum _{r=1}^k{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_k\left(r\right)\hfill \\ & & +\frac{1}{\Xi }(\sum _{j=0}^n\left[\frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\sum _{r=1}^j{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_j\left(r\right)\right]\hfill \\ & & +A-\sum _{i=0}^m\left[\frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\sum _{r=1}^i{\phi }_r\left(u\left(t_r\right)\right){\mathcal{P}}_i\left(r\right)\right])\hfill \\ & & \times \frac{{\mathcal{P}}_k\left(0\right){\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right).\hfill \end{array}$$

(36)

Let ${\mathcal{B}}_{{{\rm Y}}_2}=\{u\in {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}{:\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le {{\rm Y}}_2\}$ satisfying

$$\frac{{{\rm Y}}_2}{{\Delta }_1{p}_1^{\ast }\psi \left({{\rm Y}}_2\right)+{\Delta }_2{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}}>\frac{1}{1-\left[{\Delta }_1(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast })\right]}.$$

From Theorem 1, we can prove that ${\mathcal{Q}}_1$ is continuous. By using (${\mathbb{H}}_4$), we show that ${\mathcal{Q}}_1\left({\mathcal{B}}_{{{\rm Y}}_2}\right)$ is bounded. For any $u\in {\mathcal{B}}_{{{\rm Y}}_2}$, we obtain

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left({\mathcal{Q}}_{1}u\right)\left(t\right)\right|\hfill \\ & \le & \left[p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2\right]\{\frac{{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{{\rho }_k{\Gamma }_{{\rho }_k}\left({\alpha }_k\right)}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\varphi }_k^{\prime }\left(s\right)ds\hfill \\ & & +\frac{1}{{\Gamma }_{{\rho }_k}\left({\beta }_k({\rho }_k-{\alpha }_k)+{\alpha }_k\right)}[\sum _{r=0}^{k-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\Gamma \left(\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}\right)}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathcal{F}}_u\left(s\right){\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i{\Gamma }_{{\rho }_i}\left({\alpha }_i\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\varphi }_r^{\prime }\left(s\right)ds+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\hfill \\ & & \times \sum _{r=0}^{i-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds\left)\right]\}\hfill \\ & \le & {\Delta }_1\left[p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2\right],\hfill \end{array}$$

which implies that ${\mathcal{Q}}_1\left({\mathcal{B}}_{{{\rm Y}}_2}\right)\le {\Delta }_1[p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2]$.

Next, we will show that ${\mathcal{Q}}_1$ maps bounded set ${\mathcal{B}}_{{{\rm Y}}_2}$ into equicontinuous set of ${\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$. Let ${\tau }_1$, ${\tau }_2\in {\mathcal{J}}_k$, for $k=0,1,\dots ,m$ with ${\tau }_1<{\tau }_2$ and $u\in {\mathcal{B}}_{{{\rm Y}}_2}$. Then, we obtain

$$\begin{array}{ccc}& & \left|\left({\mathcal{Q}}_{1}u\right)\left({\tau }_2\right)-\left({\mathcal{Q}}_{1}u\right)\left({\tau }_1\right)\right|\hfill \\ & \le & \left[p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2\right]\{\frac{\left|{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}}({\tau }_2,t_m)-{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}}({\tau }_1,t_m)\right|}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}\hfill \\ & & +\frac{\left|{\Phi }_{{\varphi }_m}^{\frac{{\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m}{{\rho }_m}-1}({\tau }_2,t_m)-{\Phi }_{{\varphi }_m}^{\frac{{\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m}{{\rho }_m}-1}({\tau }_1,t_m)\right|}{{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}[\sum _{r=0}^{k-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}}(t_{r+1},t_r)}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}\left)\right]\}.\hfill \end{array}$$

It is easy to see that the above result is independent of variable $u\in {\mathcal{B}}_{{{\rm Y}}_2}$, which implies that $\left|\left({\mathcal{Q}}_{1}u\right)\left({\tau }_2\right)-\left({\mathcal{Q}}_{1}u\right)\left({\tau }_1\right)\right|\to 0$ as ${\tau }_2\to {\tau }_1$. Since, ${\mathcal{Q}}_1$ maps bounded set ${\mathcal{B}}_{{{\rm Y}}_2}$ into equicontinuous set of ${\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$. Thus, by the Arzelá-Ascoli theorem, we get that ${\mathcal{Q}}_1$ is completely continuous.

Next, we will show that ${\mathcal{Q}}_2$ is a non-linear contraction. Define a continuous non-decreasing function $\psi :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ by $\psi \left(ϵ\right)={\Delta }_2{b}_1ϵ$, for all $ϵ\ge 0$. Clearly, $\psi \left(t\right)$ satisfies $\psi \left(0\right)=0$ and by applying ${\Delta }_2{b}_1<1$, we obtain $\psi \left(ϵ\right)<ϵ$ for all $ϵ>0$. For every u, $v\in {\mathcal{B}}_{{{\rm Y}}_2}$, we have

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\left({\mathcal{Q}}_{2}u\right)\left(t\right)-\left({\mathcal{Q}}_{2}v\right)\left(t\right)\right)\right|\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|\sum _{r=1}^k\left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left({\phi }_r\left(u\left(t_r\right)\right)-{\phi }_r\left(v\left(t_r\right)\right)\right)\right|\hfill \\ & & \times \left|{\mathcal{P}}_k\left(r\right)\right|+\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{\left|\Xi \right|}\left(\sum _{j=0}^n\right[\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right|\hfill \\ & & \times \sum _{r=1}^j\left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left({\phi }_r\left(u\left(t_r\right)\right)-{\phi }_r\left(v\left(t_r\right)\right)\right)\right|\left|{\mathcal{P}}_j\left(r\right)\right|]+\sum _{i=0}^m[\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right|\sum _{r=1}^i\left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left({\phi }_r\left(u\left(t_r\right)\right)-{\phi }_r\left(v\left(t_r\right)\right)\right)\right|\left|{\mathcal{P}}_i\left(r\right)\right|\left]\right)\hfill \\ & & \times \frac{\left|{\mathcal{P}}_k\left(0\right)\right|}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\rho }_m^{{\gamma }_m-1}\Gamma \left({\gamma }_m\right)}\sum _{r=1}^m{q}_1{(\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}})+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\right[\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}\Gamma \left({\gamma }_j+\frac{{\mu }_j}{{\rho }_j}\right)}\hfill \\ & & \times \sum _{r=1}^j{q}_1{(\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\left)\right]+\sum _{i=0}^m\left[\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}\sum _{r=1}^i{q}_1{(\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}})\right])\frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\rho }_m^{{\gamma }_m-1}\Gamma \left({\gamma }_m\right)}\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\rho }_m^{{\gamma }_m-1}\Gamma \left({\gamma }_m\right)}\left[m+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}\Gamma \left({\gamma }_j+\frac{{\mu }_j}{{\rho }_j}\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}\right)\right]b_1{\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\hfill \\ & =& {\Delta }_2{b}_1{\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}.\hfill \end{array}$$

By setting $\psi \left(ϵ\right)={\Delta }_2{b}_1ϵ$, note that $\psi \left(0\right)=0$ and $\psi \left(ϵ\right)<ϵ$ for all $ϵ>0$. So,

$${∥{\mathcal{Q}}_{2}u-{\mathcal{Q}}_{2}v∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le {\psi (\parallel u-v\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}),$$

which yields that ${\mathcal{Q}}_2$ is non-linear contraction.

Next, we will show that $\mathcal{Q}\left({\mathcal{B}}_{{{\rm Y}}_2}\right)$ is bounded. By using (${\mathbb{H}}_3$), for any $u\in {\mathcal{B}}_{{{\rm Y}}_2}$, we obtain

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left({\mathcal{Q}}_{2}u\right)\left(t\right)\right|\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|\sum _{r=1}^k\left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k){\phi }_r\left(u\left(t_r\right)\right)\right|\left|{\mathcal{P}}_k\left(r\right)\right|\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\right[\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{\left|{\mu }_j\right|}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\right|{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)\hfill \\ & & \times \sum _{r=1}^j\left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k){\phi }_r\left(u\left(t_r\right)\right)\right|\left|{\mathcal{P}}_j\left(r\right)\right|]+\left|A\right|+\sum _{i=0}^m[\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}\hfill \\ & & \times \left|{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right|{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)\sum _{r=1}^i\left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k){\phi }_r\left(u\left(t_r\right)\right)\right|\left|{\mathcal{P}}_i\left(r\right)\right|\left]\right)\hfill \\ & & \times \frac{\left|{\mathcal{P}}_k\left(0\right)\right|}{{\rho }_k^{{\gamma }_k-1}}\left|{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\right|\hfill \\ & \le & \{\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left[m+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)\right]\hfill \\ & & +\left|A\right|\}{\mathbb{M}}_1\hfill \\ & =& \left\{{\Delta }_2+\frac{\left|A\right|}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\right\}{\mathbb{M}}_1.\hfill \end{array}$$

This yields that $\mathcal{Q}\left({\mathcal{B}}_{{{\rm Y}}_2}\right)$ is bounded with the boundedness of the set ${\mathcal{Q}}_1\left({\mathcal{B}}_{{{\rm Y}}_2}\right)$.

Finally, we will show that ($C_2$) of Theorem 2 does not true. On the contrary, assume that ($C_2$) holds. Then, there exists $\sigma \in (0,1)$ and for every $u\in {\mathcal{B}}_{{{\rm Y}}_2}$ so that $u=\sigma \mathcal{Q}u$. Hence, we have ${\parallel u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le {{\rm Y}}_2$ and

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)u\left(t\right)\right|\hfill \\ & =& \sigma \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\mathcal{Q}u\right)\left(t\right)\right|\hfill \\ & \le & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(\left({\mathcal{Q}}_{1}u\right)\left(t\right)+\left({\mathcal{Q}}_{2}u\right)\left(t\right)\right)\right|\hfill \\ & \le & \frac{p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2}{{\rho }_m{\Gamma }_{{\rho }_m}\left({\alpha }_m\right)}{\int }_{t_m}^T{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-1}(T,s){\varphi }_m^{\prime }\left(s\right)ds{\Phi }_{{\varphi }_m}^{1-{\gamma }_m}(T,t_m)\hfill \\ & & +\frac{p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2}{{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}\sum _{r=0}^{m-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds+(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j\right)}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\varphi }_j^{\prime }\left(s\right)ds\hfill \end{array}$$

$$\begin{array}{ccc}& & +\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i{\Gamma }_{{\rho }_i}\left({\alpha }_i\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\sum _{r=0}^{i-1}\frac{1}{{\rho }_r{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(1-{\gamma }_r)\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r+{\rho }_r(1-{\gamma }_r)}{{\rho }_r}-1}(t_{r+1},s){\varphi }_r^{\prime }\left(s\right)ds)\frac{p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}\hfill \\ & & +\frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}[m+\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\beta }_j({\rho }_j-{\alpha }_j)+{\alpha }_j+{\mu }_j\right)}\hfill \\ & & +\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i}{{\rho }_i}-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\beta }_i({\rho }_i-{\alpha }_i)+{\alpha }_i\right)}\left)\right]{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\beta }_m({\rho }_m-{\alpha }_m)+{\alpha }_m\right)}\hfill \\ & \le & \left[p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2\right]\{\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}\hfill \\ & & +\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}+\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}\hfill \\ & & +\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right]\}+\frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[m\hfill \\ & & +\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)]{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\hfill \\ & =& {\Delta }_1\left[p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2\right]+{\Delta }_2{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)},\hfill \end{array}$$

which yields that

$${{\rm Y}}_2\le {\Delta }_1\left[p_1^{\ast }\psi \left({{\rm Y}}_2\right)+(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast }){{\rm Y}}_2\right]+{\Delta }_2{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}.$$

Then, we have

$$\frac{{{\rm Y}}_2}{{\Delta }_1{p}_1^{\ast }\psi \left({{\rm Y}}_2\right)+{\Delta }_2{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}}\le \frac{1}{1-\left[{\Delta }_1(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast })\right]},$$

which contradicts the condition (${\mathbb{H}}_6$). Hence, ${\mathcal{Q}}_1$ and ${\mathcal{Q}}_2$ fulfill all the conditions of Lemma 9. Therefore, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) has at least one solution. □

4. Stability Results

First of all, we give the following inequalities for analyzing Ulam’s stability of the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4). Let $\mathcal{T}\in \mathcal{C}(\mathcal{J},{\mathbb{R}}^{+})$ be a non-decreasing function, $ϵ>0$, $\tau \ge 0$, $z\in \mathcal{B}$, such that for $t\in {\mathcal{J}}_k$, $k=1,2,\dots ,m$, the following inequalities are fulfilled:

$$\left\{\begin{array}{c}{\displaystyle \left|{}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}z\left(t\right)-{\lambda }_{k}z\left(t\right)-f(t,z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}z\left(t\right))\right|\le ϵ,}\hfill \\ {\displaystyle \left|{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{\rho (1-{\gamma }_k);{\varphi }_k}z\left(t_k^{+}\right)-{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{\rho (1-{\gamma }_{k-1});{\varphi }_{k-1}}z\left(t_k^{-}\right)-{\phi }_k\left(z\left(t_k\right)\right)\right|\le ϵ,}\hfill \end{array}\right.$$

(37)

$$\left\{\begin{array}{c}{\displaystyle \left|{}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}z\left(t\right)-{\lambda }_{k}z\left(t\right)-f(t,z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}z\left(t\right))\right|\le \mathcal{T}\left(t\right),}\hfill \\ {\displaystyle \left|{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{\rho (1-{\gamma }_k);{\varphi }_k}z\left(t_k^{+}\right)-{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{\rho (1-{\gamma }_{k-1});{\varphi }_{k-1}}z\left(t_k^{-}\right)-{\phi }_k\left(z\left(t_k\right)\right)\right|\le \tau ,}\hfill \end{array}\right.$$

(38)

$$\left\{\begin{array}{c}{\displaystyle \left|{}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}z\left(t\right)-{\lambda }_{k}z\left(t\right)-f(t,z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}z\left(t\right))\right|\le ϵ\mathcal{T}\left(t\right),}\hfill \\ {\displaystyle \left|{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{\rho (1-{\gamma }_k);{\varphi }_k}z\left(t_k^{+}\right)-{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{\rho (1-{\gamma }_{k-1});{\varphi }_{k-1}}z\left(t_k^{-}\right)-{\phi }_k\left(z\left(t_k\right)\right)\right|\le ϵ\tau .}\hfill \end{array}\right.$$

(39)

Definition 3.

The impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is called Ulam–Hyers ($\mathcal{UH}$) stable, if there exists ${\mathfrak{C}}_{\mathcal{F}}>0$, such that for any $ϵ>0$ and for each $z\in \mathcal{B}$ of (37) there is $u\in \mathcal{B}$ of (4) that satisfies

$$\left|z\left(t\right)-u\left(t\right)\right|\le {\mathfrak{C}}_{\mathcal{F}}ϵ,t\in \mathcal{J}.$$

(40)

Definition 4.

The impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is called generalized Ulam-Hyers ($\mathcal{GUH}$) stable, if there exists $\mathcal{T}\in \mathcal{C}({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ with $\mathcal{T}\left(0\right)=0$, such that for any $ϵ>0$ and for each $z\in \mathcal{B}$ of (38) there is $u\in \mathcal{B}$ of (4) that satisfies

$$\left|z\left(t\right)-u\left(t\right)\right|\le \mathcal{T}\left(ϵ\right),t\in \mathcal{J}.$$

(41)

Definition 5.

The impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is called Ulam–Hyers–Rassias $\mathcal{UHR}$ stable with respect to $(\tau ,\mathcal{T})$, if there exists ${\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}>0$ such that for any $ϵ>0$ and for each $z\in \mathcal{B}$ of (39) there is $u\in \mathcal{B}$ of (4) that satisfies

$$\left|z\left(t\right)-u\left(t\right)\right|\le {\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}ϵ(\tau +\mathcal{T}\left(t\right)),t\in \mathcal{J}.$$

(42)

Definition 6.

The impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is called generalized Ulam–Hyers–Rassias $\mathcal{GUHR}$ stable with respect to $(\tau ,\mathcal{T})$, if there exists ${\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}>0$ such that for any $z\in \mathcal{B}$ of (38) there is $u\in \mathcal{B}$ of (4) that satisfies

$$\left|z\left(t\right)-u\left(t\right)\right|\le {\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}(\tau +\mathcal{T}\left(t\right)),t\in \mathcal{J}.$$

(43)

Remark 3.

It is easy to see that: (i) Definition 3⇒ Definition 4, ($ii$) Definition 5⇒ Definition 6, and ($iii$) Definition 5 with $\tau +\mathcal{T}\left(t\right)=1$⇒ Definition 3.

Remark 4.

$z\in \mathcal{B}$ is a solution of (37) if there is $x\in \mathcal{B}$ and $x_k$, $k=1,2,\dots ,m$, (which depends on z), such that

(i) 

$|x(t)|\le ϵ$, $|x_k|\le ϵ$, $t\in \mathcal{J}$,

(ii) 

${}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}z\left(t\right)={\lambda }_{k}z\left(t\right)+f(t,z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}z\left(t\right))+x\left(t\right)$, $t\in \mathcal{J}$,

(iii) 

${}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{\rho (1-{\gamma }_k);{\varphi }_k}z\left(t_k^{+}\right)-{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{\rho (1-{\gamma }_{k-1});{\varphi }_{k-1}}z\left(t_k^{-}\right)={\phi }_k\left(z\left(t_k\right)\right)+x_k$.

Remark 5.

$z\in \mathcal{B}$ is a solution of (38) if there is $x\in \mathcal{B}$ and $x_k$, $k=1,2,\dots ,m$, (which depends on z), such that

(i) 

$|x(t)|\le \mathcal{T}\left(t\right)$, $|x_k|\le \tau$, $t\in \mathcal{J}$,

(ii) 

${}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}z\left(t\right)={\lambda }_{k}z\left(t\right)+f(t,z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}z\left(t\right))+x\left(t\right)$, $t\in \mathcal{J}$,

(iii) 

${}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{\rho (1-{\gamma }_k);{\varphi }_k}z\left(t_k^{+}\right)-{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{\rho (1-{\gamma }_{k-1});{\varphi }_{k-1}}z\left(t_k^{-}\right)={\phi }_k\left(z\left(t_k\right)\right)+x_k$.

Remark 6.

$z\in \mathcal{B}$ is a solution of (39) if there is $x\in \mathcal{B}$ and $x_k$, $k=1,2,\dots ,m$, (which depends on z), such that

(i) 

$|x(t)|\le ϵ\mathcal{T}\left(t\right)$, $|x_k|\le ϵ\tau$, $t\in \mathcal{J}$,

(ii) 

${}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}z\left(t\right)={\lambda }_{k}z\left(t\right)+f(t,z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\delta }_k;{\varphi }_k}z\left(t\right),{}_{{\rho }_k}^{}{\mathcal{I}}_{t_k}^{{\theta }_k;{\varphi }_k}z\left(t\right))+x\left(t\right)$, $t\in \mathcal{J}$,

(iii) 

${}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{\rho (1-{\gamma }_k);{\varphi }_k}z\left(t_k^{+}\right)-{}_{{\rho }_{k-1}}^{}{\mathcal{I}}_{t_{k-1}^{+}}^{\rho (1-{\gamma }_{k-1});{\varphi }_{k-1}}z\left(t_k^{-}\right)={\phi }_k\left(z\left(t_k\right)\right)+x_k$.

4.1. Ulam–Hyers Stability Results

We construct the proof of the following lemma, which gives a base for obtaining a solution to the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4).

Theorem 3.

Let ${\alpha }_k\in (0,1)$, ${\beta }_k\in [0,1]$, ${\rho }_k>0$, ${\gamma }_k=({\beta }_k({\rho }_k-{\alpha }_k)+{\alpha }_k)/{\rho }_k$, ${\lambda }_k\in \mathbb{R}$, ${\varphi }_k\in \mathcal{C}(\mathcal{J},\mathbb{R})$ with ${\varphi }_k^{\prime }>0$ for $k=1,2,\dots ,m$. Assume that $f\in \mathcal{C}(\mathcal{J}\times {\mathbb{R}}^3,\mathbb{R})$ and ${\phi }_k\in \mathcal{C}(\mathbb{R},\mathbb{R})$ ($k=1,2\dots ,m$). Suppose that (${\mathbb{H}}_1$)–(${\mathbb{H}}_2$) hold. Then, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is $\mathcal{UH}$ stable if

$$\left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right)<1.$$

Proof. 

Suppose that $x\in {\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}$ is a solution of the problem (37). From Lemma (15) with Remark 4 ($ii$)–($iii$), we obtain

$$\left\{\begin{array}{c}{\displaystyle {}_{{\rho }_k}^H{\mathfrak{D}}_{t_k^{+}}^{{\alpha }_k,{\beta }_k;{\varphi }_k}z\left(t\right)={\lambda }_{k}z\left(t\right)+{\mathcal{F}}_z\left(t\right)+x\left(t\right),t\in {\mathcal{J}}_k\subseteq \mathcal{J},t\ne t_k,}\hfill \\ {\displaystyle \Delta {}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\rho }_k(1-{\gamma }_k);{\varphi }_k}z\left(t_k\right)={\phi }_k\left(z\left(t_k\right)\right)+x_k,k=1,2,\dots ,m,}\hfill \\ {\displaystyle \sum _{i=0}^m{\kappa }_{i}z\left({\eta }_i\right)=\sum _{j=0}^n{\omega }_j{}_{{\rho }_j}{\mathcal{I}}_{t_j^{+}}^{{\mu }_j;{\varphi }_j}z\left({\xi }_j\right)+A,{\eta }_i\in (t_i,t_{i+1}],{\xi }_j\in (t_j,t_{j+1}],}\hfill \end{array}\right.$$

(44)

then the solution of (44) can be rewritten as

$$\begin{array}{ccc}\hfill z\left(t\right)& =& \frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right){\mathcal{F}}_z\left(s\right){\varphi }_k^{\prime }\left(s\right)ds+[\sum _{r=0}^{k-1}\frac{{\mathcal{P}}_k(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\hfill \\ & & \times {\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_z\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^k{\phi }_r\left(z\left(t_r\right)\right){\mathcal{P}}_k\left(r\right)]\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\hfill \\ & & +\frac{1}{\Xi }(\sum _{j=0}^n\frac{{\omega }_j}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right){\mathcal{F}}_z\left(s\right){\varphi }_j^{\prime }\left(s\right)ds+\sum _{j=0}^n\{[\sum _{r=0}^{j-1}\frac{{\mathcal{P}}_j(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_z\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j{\phi }_r\left(z\left(t_r\right)\right){\mathcal{P}}_j\left(r\right)]\hfill \\ & & \times \frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}{\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\}+A-\sum _{i=0}^m\frac{{\kappa }_i}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\hfill \\ & & \times {\mathcal{F}}_z\left(s\right){\varphi }_r^{\prime }\left(s\right)ds-\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{{\mathcal{P}}_i(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right){\mathcal{F}}_z\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^i{\phi }_r\left(z\left(t_r\right)\right){\mathcal{P}}_i\left(r\right)\left]\frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\right\})\frac{{\mathcal{P}}_k\left(0\right){\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\hfill \\ & & +\frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s){\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},\frac{{\alpha }_k}{{\rho }_k}}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,s)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)x\left(s\right){\varphi }_k^{\prime }\left(s\right)ds+[\sum _{r=0}^{k-1}\frac{{\mathcal{P}}_k(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\hfill \\ & & \times {\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)x\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^k{x}_r{\mathcal{P}}_k\left(r\right)]\frac{{\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right)\hfill \\ & & +\frac{1}{\Xi }(\sum _{j=0}^n\frac{{\omega }_j}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s){\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,s)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)x\left(s\right){\varphi }_j^{\prime }\left(s\right)ds+\sum _{j=0}^n\{[\sum _{r=0}^{j-1}\frac{{\mathcal{P}}_j(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)x\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j{x}_r{\mathcal{P}}_j\left(r\right)]\frac{{\omega }_j{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}}\hfill \\ & & \times {\mathbb{E}}_{\frac{{\alpha }_j}{{\rho }_j},{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}}\left({\lambda }_j{\left({\rho }_j^{-1}{\Phi }_{{\varphi }_j}({\xi }_j,t_j)\right)}^{\frac{{\alpha }_j}{{\rho }_j}}\right)\}-\sum _{i=0}^m\frac{{\kappa }_i}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s){\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},\frac{{\alpha }_i}{{\rho }_i}}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,s)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)x\left(s\right){\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & -\sum _{i=0}^m\{\left[\sum _{r=0}^{i-1}\frac{{\mathcal{P}}_i(r+1)}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s){\mathbb{E}}_{\frac{{\alpha }_r}{{\rho }_r},\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\left({\lambda }_r{\left({\rho }_r^{-1}{\Phi }_{{\varphi }_r}(t_{r+1},s)\right)}^{\frac{{\alpha }_r}{{\rho }_r}}\right)x\left(s\right){\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^i{x}_r{\mathcal{P}}_i\left(r\right)\right]\hfill \\ & & \times \frac{{\kappa }_i{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}}{\mathbb{E}}_{\frac{{\alpha }_i}{{\rho }_i},{\gamma }_i}\left({\lambda }_i{\left({\rho }_i^{-1}{\Phi }_{{\varphi }_i}({\eta }_i,t_i)\right)}^{\frac{{\alpha }_i}{{\rho }_i}}\right)\left\}\right)\frac{{\mathcal{P}}_k\left(0\right){\Phi }_{{\varphi }_k}^{{\gamma }_k-1}(t,t_k)}{{\rho }_k^{{\gamma }_k-1}}{\mathbb{E}}_{\frac{{\alpha }_k}{{\rho }_k},{\gamma }_k}\left({\lambda }_k{\left({\rho }_k^{-1}{\Phi }_{{\varphi }_k}(t,t_k)\right)}^{\frac{{\alpha }_k}{{\rho }_k}}\right).\hfill \end{array}$$

(45)

By applying Lemma 3 with $\left|{\mathcal{P}}_a\left(b\right)\right|\le 1$, for any $t\in \mathcal{J}$, we obtain that

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(z\left(t\right)-u\left(t\right)\right)\right|\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}\Gamma \left(\frac{{\alpha }_k}{{\rho }_k}\right)}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_k^{\prime }\left(s\right)ds+\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}\hfill \\ & & \times [\sum _{r=0}^{k-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^k\left|{\phi }_r\left(z\left(t_r\right)\right)-{\phi }_r\left(z\left(t_r\right)\right)\right|]+\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\Gamma \left(\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}\right)}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s)\hfill \\ & & \times \left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_j^{\prime }\left(s\right)ds+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}\Gamma \left({\gamma }_j+\frac{{\mu }_j}{{\rho }_j}\right)}[\sum _{r=0}^{j-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j\left|{\phi }_r\left(z\left(t_r\right)\right)-{\phi }_r\left(u\left(t_r\right)\right)\right|]\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}\Gamma \left(\frac{{\alpha }_i}{{\rho }_i}\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^i\left|{\phi }_r\left(z\left(t_r\right)\right)-{\phi }_r\left(u\left(t_r\right)\right)\right|\left]\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}\right\})\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}\hfill \\ & & +\frac{1}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}\Gamma \left(\frac{{\alpha }_k}{{\rho }_k}\right)}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s)\left|x\left(s\right)\right|{\varphi }_k^{\prime }\left(s\right)ds+\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}[\sum _{r=0}^{k-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}\hfill \end{array}$$

$$\begin{array}{ccc}& & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^k\left|x_r\right|]+\frac{1}{\left|\Xi \right|{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\Gamma \left(\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}\right)}\hfill \\ & & \times {\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s)\left|x\left(s\right)\right|{\varphi }_j^{\prime }\left(s\right)ds+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}\Gamma \left({\gamma }_j+\frac{{\mu }_j}{{\rho }_j}\right)}\hfill \\ & & \times \left[\sum _{r=0}^{j-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j\left|x_r\right|\right]\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}\Gamma \left(\frac{{\alpha }_i}{{\rho }_i}\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s)\left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}\hfill \\ & & \times \left[\sum _{r=0}^{i-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^i\left|x_r\right|\right]).\hfill \end{array}$$

Thanks to (i) of Remark 4 with (${\mathbb{H}}_1$)–(${\mathbb{H}}_2$), we obtain the following result

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(z\left(t\right)-u\left(t\right)\right)\right|\hfill \\ & \le & \left\{\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]\right(\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right])\hfill \\ & & +\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left[m+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)\right]{\mathbb{N}}_1\}\hfill \\ & & {\times \parallel z-u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+\{\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right]\hfill \\ & & +\frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left[m+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)\right]\}ϵ,\hfill \end{array}$$

which implies that

$${∥z-u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le \left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right){\parallel z-u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+\left\{{\Delta }_1+{\Delta }_2\right\}ϵ.$$

Then ${∥z-u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le {\mathfrak{C}}_{\mathcal{F}}ϵ$, where

$${\mathfrak{C}}_{\mathcal{F}}:=\frac{{\Delta }_1+{\Delta }_2}{1-\left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right)}.$$

(46)

Therefore, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is $\mathcal{UH}$ stable in $\mathcal{B}$. □

Corollary 1.

Under conditions in Theorem 3, if $\mathcal{T}\left(ϵ\right)={\mathfrak{C}}_{\mathcal{F}}ϵ$ so that $\mathcal{T}\left(0\right)=0$, then we have the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) becomes $\mathcal{GUH}$ stable.

4.2. Ulam–Hyers–Rassias Stability Results

To analyze $\mathcal{UHR}$ stability results, we will need the following condition as follows:

(${\mathbb{H}}_7$)

There exists a non-decreasing function $\mathcal{T}\in \mathcal{C}(\mathcal{J},\mathbb{R})$ and there is ${\mathfrak{E}}_{\mathcal{T}}>0$, for each $ϵ>0$, such that the following inequality

$${}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\alpha }_k;{\varphi }_k}\mathcal{T}\left(t\right)\le {\mathfrak{E}}_{\mathcal{T}}\mathcal{T}\left(t\right).$$

(47)

Theorem 4.

Let $f\in (\mathcal{J}\times {\mathbb{R}}^3,\mathbb{R})$ and ${\phi }_k\in \mathcal{C}(\mathbb{R},\mathbb{R})$, ($k=1,2\dots ,m$). If (${\mathbb{H}}_1$), (${\mathbb{H}}_2$), (${\mathbb{H}}_7$), and

$$\left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right)<1$$

are fulfilled. Then the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is $\mathcal{UHR}$ stable with respect to ($\tau ,\mathcal{T}$).

Proof. 

Let $z\in \mathcal{B}$ be any solution of (39) and $u\in \mathcal{B}$ be the solution of the problem (4). By the same process in Theorem 3, we have

$$\begin{array}{ccc}& & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(z\left(t\right)-u\left(t\right)\right)\right|\hfill \\ & \le & \frac{{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}\Gamma \left(\frac{{\alpha }_k}{{\rho }_k}\right)}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_k^{\prime }\left(s\right)ds+\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}\hfill \\ & & \times [\sum _{r=0}^{k-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^k\left|{\phi }_r\left(z\left(t_r\right)\right)-{\phi }_r\left(z\left(t_r\right)\right)\right|]+\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\Gamma \left(\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}\right)}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s)\hfill \\ & & \times \left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_j^{\prime }\left(s\right)ds+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}\Gamma \left({\gamma }_j+\frac{{\mu }_j}{{\rho }_j}\right)}[\sum _{r=0}^{j-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}\hfill \\ & & \times {\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j\left|{\phi }_r\left(z\left(t_r\right)\right)-{\phi }_r\left(u\left(t_r\right)\right)\right|]\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}\Gamma \left(\frac{{\alpha }_i}{{\rho }_i}\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \end{array}$$

$$\begin{array}{ccc}& & +\sum _{i=0}^m\left\{\right[\sum _{r=0}^{i-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|{\mathcal{F}}_z\left(s\right)-{\mathcal{F}}_u\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^i\left|{\phi }_r\left(z\left(t_r\right)\right)-{\phi }_r\left(u\left(t_r\right)\right)\right|\left]\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}\right\})\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}\hfill \\ & & +\frac{{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{{\rho }_k^{\frac{{\alpha }_k}{{\rho }_k}}\Gamma \left(\frac{{\alpha }_k}{{\rho }_k}\right)}{\int }_{t_k}^t{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}-1}(t,s)\left|x\left(s\right)\right|{\varphi }_k^{\prime }\left(s\right)ds+\frac{1}{{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}\hfill \\ & & \times \left[\sum _{r=0}^{k-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^k\left|x_r\right|\right]\hfill \\ & & +\frac{1}{\left|\Xi \right|{\rho }_k^{{\gamma }_k-1}\Gamma \left({\gamma }_k\right)}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|}{{\rho }_j^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}\Gamma \left(\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}\right)}{\int }_{t_j}^{{\xi }_j}{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}-1}({\xi }_j,s)\left|x\left(s\right)\right|{\varphi }_j^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{{\gamma }_j+\frac{{\mu }_j}{{\rho }_j}-1}({\xi }_j,t_j)}{{\rho }_j^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}\Gamma \left({\gamma }_j+\frac{{\mu }_j}{{\rho }_j}\right)}[\sum _{r=0}^{j-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\hfill \\ & & \times \left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds+\sum _{r=1}^j\left|x_r\right|]+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|}{{\rho }_i^{\frac{{\alpha }_i}{{\rho }_i}}\Gamma \left(\frac{{\alpha }_i}{{\rho }_i}\right)}{\int }_{t_i}^{{\eta }_i}{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}-1}({\eta }_i,s)\left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}[\sum _{r=0}^{i-1}\frac{1}{{\rho }_r^{\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r}\Gamma \left(\frac{{\alpha }_r}{{\rho }_r}+1-{\gamma }_r\right)}{\int }_{t_r}^{t_{r+1}}{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r}(t_{r+1},s)\left|x\left(s\right)\right|{\varphi }_r^{\prime }\left(s\right)ds\hfill \\ & & +\sum _{r=1}^i\left|x_r\right|\left]\right).\hfill \end{array}$$

Thanks to (i) of Remark 6 with (${\mathbb{H}}_1$), (${\mathbb{H}}_2$), and (${\mathbb{H}}_7$), we have the following result

$$\begin{array}{ccc}\hfill & & \left|{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left(z\left(t\right)-u\left(t\right)\right)\right|\hfill \\ & \le & \left\{\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]\right(\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}-{\gamma }_m+1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\alpha }_m+{\rho }_m\right)}+\frac{1}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}[\sum _{r=0}^{m-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\frac{1}{\left|\Xi \right|}(\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\alpha }_j+{\mu }_j}{{\rho }_j}}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\alpha }_j+{\mu }_j+{\rho }_j\right)}+\sum _{j=0}^n\frac{\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}\sum _{r=0}^{j-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\hfill \\ & & +\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{\frac{{\alpha }_i}{{\rho }_i}}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\alpha }_i+{\rho }_i\right)}+\sum _{i=0}^m\frac{\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\sum _{r=0}^{i-1}\frac{{\Phi }_{{\varphi }_r}^{\frac{{\alpha }_r}{{\rho }_r}-{\gamma }_r+1}(t_{r+1},t_r)}{{\Gamma }_{{\rho }_r}\left({\alpha }_r+{\rho }_r(2-{\gamma }_r)\right)}\left)\right])\hfill \\ & & +\frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left[m+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\Gamma }_{{\rho }_i}\left({\rho }_i{\gamma }_i\right)}\right)\right]{\mathbb{N}}_1\}\hfill \\ & & {\times \parallel z-u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+\{{\mathfrak{E}}_{\mathcal{T}}\mathcal{T}\left(t\right)\left[1+\frac{1}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left(\sum _{j=0}^n\left|{\omega }_j\right|+\sum _{i=0}^m\left|{\kappa }_i\right|\right)\right]\hfill \end{array}$$

$$\begin{array}{ccc}& & +\frac{{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)}{{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}\left[m+\frac{1}{\left|\Xi \right|}\left(\sum _{j=0}^n\frac{j\left|{\omega }_j\right|{\Phi }_{{\varphi }_j}^{\frac{{\mu }_j}{{\rho }_j}+{\gamma }_j-1}({\xi }_j,t_j)}{{\Gamma }_{{\rho }_j}\left({\rho }_j{\gamma }_j+{\mu }_j\right)}+\sum _{i=0}^m\frac{i\left|{\kappa }_i\right|{\Phi }_{{\varphi }_i}^{{\gamma }_i-1}({\eta }_i,t_i)}{{\rho }_i^{{\gamma }_i-1}\Gamma \left({\gamma }_i\right)}\right)\right]\hfill \\ & & \times \left[{\mathfrak{E}}_{\mathcal{T}}\mathcal{T}\left(t\right)+\tau \right]\}ϵ,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}& & {∥z-u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\hfill \\ & \le & \left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right){\parallel z-u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+\left([{\Delta }_2+{\Delta }_3]{\mathfrak{E}}_{\mathcal{T}}\mathcal{T}\left(t\right)+{\Delta }_2\tau \right)ϵ\hfill \\ & \le & \left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right){\parallel z-u\parallel }_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}+\left([{\Delta }_2+{\Delta }_3]{\mathfrak{E}}_{\mathcal{T}}+{\Delta }_2\right)ϵ(\tau +\mathcal{T}\left(t\right)).\hfill \end{array}$$

Then ${∥z-u∥}_{{\mathcal{PC}}_{{\varphi }_k}^{1-{\gamma }_k}}\le {\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}ϵ(\tau +\mathcal{T}\left(t\right))$, with

$${\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}:=\frac{[{\Delta }_2+{\Delta }_3]{\mathfrak{E}}_{\mathcal{T}}+{\Delta }_2}{1-\left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right)}.$$

(48)

Hence, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) is $\mathcal{UHR}$ stable with respect to $(\tau ,\mathcal{T})$ in $\mathcal{B}$. □

Corollary 2.

Under conditions in Theorem 4, if $ϵ=1$ so that $\mathcal{T}\left(0\right)=0$, then we have the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (4) becomes $\mathcal{GUHR}$ stable.

5. Numerical Examples

This section provides some illustrative examples of the exactness and applicability of our main results.

Example 1.

Consider the following impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s of the form:

$$\left\{\begin{array}{c}{\displaystyle {}_{\frac{k+2}{2}}^H{\mathfrak{D}}_{t_k^{+}}^{\frac{7-3k}{9-3k},\frac{3k+2}{10};{\varphi }_k}u\left(t\right)=-\frac{\sqrt{k^2+1}}{k+1}u\left(t\right)+f(t,u\left(t\right),{}_{\frac{k+2}{2}}^{}{\mathcal{I}}_{t_k}^{\frac{e^{-k}}{3};{\varphi }_k}u\left(t\right),{}_{\frac{k+2}{2}}^{}{\mathcal{I}}_{t_k}^{{sin}^2\frac{\pi }{k+2};{\varphi }_k}u\left(t\right)),}\hfill \\ {\displaystyle \Delta {}_{\frac{k+2}{2}}^{}{\mathcal{I}}_{t_k^{+}}^{\frac{k+2}{2}(1-{\gamma }_k);{\varphi }_k}u\left(t_k\right)={\phi }_k\left(u\left(t_k\right)\right),k=1,2}\hfill \\ {\displaystyle \sum _{i=0}^2\left(\frac{i+1}{3}\right)u\left(\frac{2i+1}{5}\right)=\sum _{j=0}^1\left(\frac{j+2}{2}\right){}_{\frac{j+2}{2}}{\mathcal{I}}_{t_j^{+}}^{\frac{8j+8}{10};{\varphi }_j}u\left(\frac{2j+2}{9}\right)+2.}\hfill \end{array}\right.$$

(49)

Form the problem (49), we obtain that ${\alpha }_k=(7-3k)/(9-3k)$, ${\beta }_k=(3k+2)/10$, ${\rho }_k=(k+2)/2$, ${\lambda }_k=-\sqrt{k^2+1}/(k+1)$, ${\delta }_k=exp(-k)/3$, ${\theta }_k={sin}^2(\pi /(k+2))$, ${\varphi }_k\left(t\right)=ln(t+k+2)/(2k+2)$, $t_k=2k/5$, $k=0,1,2$, $T=6/5$, $k=0,1,2$, ${\kappa }_i=(i+2)/3$, ${\eta }_i=(2i+1)/5$, ${\omega }_j=(j+2)/2$, ${\mu }_j=(8j+8)/10$, ${\xi }_j=(2j+2)/9$, $i=0,1,2$, $j=0,1$, and $A=2$. From the given all data, we can find that $\Xi \approx 3.350618628$, ${\Delta }_1\approx 0.7268674525$, ${\Delta }_2\approx 1.676600970$ and ${\Delta }_3\approx 0.8516099910$. The following functions will be considered for theoretical confirmation:

$\left(i\right)$. Consider the functions

$$\begin{array}{ccc}\hfill f(t,u(t),v(t),w(t\left)\right)& =& \frac{5t^2-3}{2t^2+4}+\frac{(3-t){\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{7-2{sin}^2\pi t}·\frac{|u(t)|}{5+|u(t)|}\hfill \\ & & +\frac{3e^{-2t-1}{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{6-{cos}^2\pi t}·\frac{|v(t)|}{4+|v(t)|}\hfill \\ & & +\frac{4sin2\pi t{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{8-3e^{-t}}·\frac{|w(t)|}{4+|w(t)|},\hfill \\ \hfill {\phi }_k\left(u\left(t_k\right)\right)& =& \frac{4{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t_{k+1},t_k)u\left(t_k\right)}{27-t_k}+3t_k.\hfill \end{array}$$

For $u_i$, $v_i$, $w_i\in \mathbb{R}$, $i=1,2$, and $t\in [0,1]$, we can find that

$$\begin{array}{ccc}\hfill \left|f(t,u_1,v_1,w_1)-f(t,u_2,v_2,w_2)\right|& \le & \frac{3}{25}|u_1-u_2|+\frac{3}{20}|v_1-v_2|+\frac{1}{5}|v_1-v_2|,\hfill \\ \hfill \left|{\phi }_k\left(u_1\right)-{\phi }_k\left(u_2\right)\right|& \le & \frac{4}{25}|u_1-u_2|.\hfill \end{array}$$

The assumption $\left({\mathbb{H}}_1\right)$–$\left({\mathbb{H}}_2\right)$ are satisfied with ${\mathbb{L}}_1=3/5$, ${\mathbb{L}}_2=3/20$, ${\mathbb{L}}_3=1/5$ and ${\mathbb{N}}_1=4/25$. Hence,

$${\Delta }_1\left({\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right)+{\Delta }_2{\mathbb{N}}_1\approx 0.6598729041<1.$$

Since, all the conditions of Theorem 1 are fulfilled. Then, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (49) has a unique solution on $[0,1]$. Moreover, by Theorem 3, we also find that

$${\mathfrak{C}}_{\mathcal{F}}:=\frac{{\Delta }_1+{\Delta }_2}{1-\left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right)}\approx 7.066383275>0$$

Then, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (49) is $\mathcal{UH}$ stable on $[0,1]$. Taking $\mathcal{T}\left(ϵ\right)={\mathfrak{C}}_{\mathcal{F}}ϵ$ via $\mathcal{T}\left(0\right)=0$, then, by Corollary 1, the following impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (49) is $\mathcal{GUH}$ stable on $[0,1]$. Taking $\mathcal{T}\left(t\right)={\Phi }_{{\varphi }_k}(t,t_k)$ with $\tau =1$, we obtain

$${}_{{\rho }_k}^{}{\mathcal{I}}_{t_k^{+}}^{{\alpha }_k;{\varphi }_k}\mathcal{T}\left(t\right)=\frac{{\Phi }_{{\varphi }_k}^{\frac{{\alpha }_k}{{\rho }_k}}(t,t_k)}{{\rho }_k{\Gamma }_{{\rho }_k}\left({\alpha }_k\right)}{\Phi }_{{\varphi }_k}(t,t_k)\le \frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}}(T,t_m)}{{\rho }_m{\Gamma }_{{\rho }_m}\left({\alpha }_m\right)}\mathcal{T}\left(t\right)$$

From (${\mathbb{H}}_7$), we obtain

$${\mathfrak{E}}_{\mathcal{T}}=\frac{{\Phi }_{{\varphi }_m}^{\frac{{\alpha }_m}{{\rho }_m}}(T,t_m)}{{\rho }_m{\Gamma }_{{\rho }_m}\left({\alpha }_m\right)}\approx 0.07794511398>0.$$

Then,

$${\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}:=\frac{[{\Delta }_2+{\Delta }_3]{\mathfrak{E}}_{\mathcal{T}}+{\Delta }_2}{1-\left({\Delta }_1\left[{\mathbb{L}}_1+{\mathbb{L}}_2{\Phi }_1^{\ast }+{\mathbb{L}}_3{\Phi }_2^{\ast }\right]+{\Delta }_2{\mathbb{N}}_1\right)}\approx 5.508713313>0.$$

Hence, by all conditions in Theorem 4, the following impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (49) is $\mathcal{UHR}$ stable on $[0,1]$. Moreover, if $\mathcal{T}\left(ϵ\right)={\mathfrak{C}}_{\mathcal{F},{\mathfrak{E}}_{\mathcal{F}}}ϵ$ with $\mathcal{T}\left(0\right)=0$, then, by Corollary 2, the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$-$\mathbb{NMP}$-$\mathbb{FIBC}$s (49) is $\mathcal{GUHR}$ stable with respect to $(\tau ,\mathcal{T})$.

$\left(ii\right)$. Consider the functions

$$\begin{array}{ccc}\hfill f(t,u(t),v(t),w(t\left)\right)& =& \frac{(2t-1){\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{12-2cos\pi t}\left[sinu\left(t\right)+\frac{2|u(t)|}{3+|u(t)|}\right]\hfill \\ & & +{\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)\left[\frac{3cos\pi t}{4+e^t}·\frac{|v(t)|}{5+|v(t)|}+\frac{6e^{-t}}{10ln(t+e)}·\frac{|w(t)|}{4+|w(t)|}\right],\hfill \\ \hfill {\phi }_k\left(u\left(t_k\right)\right)& =& \frac{(6-t^2){\Phi }_{{\varphi }_k}^{1-{\gamma }_k}(t,t_k)}{5e^t}·\frac{|u(t_k)|}{5+4|u(t_k)|}.\hfill \end{array}$$

For u, v, $w\in \mathbb{R}$, $i=1,2$, and $t\in [0,1]$, we can find that

$$\begin{array}{ccc}\hfill |f(t,u,v,w)|& \le & \frac{2t-1}{12-2cos\pi t}\left(|u|+2\right)+\frac{3cos\pi t}{20+5e^t}|v|+\frac{6e^{-t}}{40ln(t+e)}|w|,\hfill \\ \hfill |{\phi }_k\left(u\right)-{\phi }_k\left(v\right)|& \le & \frac{6}{25}|u-v|,|{\phi }_k\left(u\right)|\le \frac{6}{20}.\hfill \end{array}$$

The assumption $\left({\mathbb{H}}_3\right)$–$\left({\mathbb{H}}_6\right)$ are satisfied with $\psi (|u|)=|u|+2$, $p_1^{\ast }=0.2$, $p_2^{\ast }=0.12$, $p_3^{\ast }=0.15$, ${\mathbb{M}}_1=0.3$ and $b_1=0.24$. Hence, ${\Delta }_2{\Phi }_{{\varphi }_m}^{{\gamma }_m-1}(T,t_m)b_1\approx 0.8262520217<1$, ${(1-\left[{\Delta }_1(p_2^{\ast }{\Phi }_1^{\ast }+p_3^{\ast }{\Phi }_2^{\ast })\right])}^{-1}\approx 1.31318002$, and

$$\underset{{{\rm Y}}_2\in (0,\infty )}{sup}\frac{{{\rm Y}}_2}{{\Delta }_1{p}_1^{\ast }\psi \left({{\rm Y}}_2\right)+{\Delta }_2{\mathbb{M}}_1+\frac{\left|A\right|}{\left|\Xi \right|{\Gamma }_{{\rho }_m}\left({\rho }_m{\gamma }_m\right)}}\approx 6.878833250.$$

Since, all the problem (49) has at least one solution on $[0,1]$.

$\left(iii\right)$. Consider the functions $f(t,u(t),v(t),w(t\left)\right)=0$ and ${\phi }_k\left(u\left(t_k\right)\right)=7-4k$. By using (14), (22), (23), (24), the numerical values of Ξ, ${\Delta }_i$, $i=1,2,3$, for ${\alpha }_k\in \{0.2,0.4,0.6,0.8,1.0\}$, as shown in

Table 1. Numerical values of $\Xi$ and ${\Delta }_i$, $i=1,2,3$, for ${\alpha }_k\in \{0.2,0.4,0.6,0.8,1.0\}$ when ${\varphi }_k\left(t\right)=\frac{ln(t+k+2)}{2k+2}$, ${\beta }_k=\frac{3k+2}{10}$, and ${\rho }_k=\frac{k+2}{2}$ for $k=0,1,2$.

${\mathit{\alpha }}_{\mathit{k}}$	$\mathbf{\Xi }$	${\mathbf{\Delta }}_1$	${\mathbf{\Delta }}_2$	${\mathbf{\Delta }}_3$
0.2	2.892662425	1.3437981140	2.058239791	0.8286028011
0.4	3.213267392	0.8843819924	1.946083874	0.8450705102
0.6	3.328008050	0.6154263942	1.917611370	0.8499260482
0.8	3.199108790	0.4512723574	1.962021647	0.8435009608
1.0	2.903673651	0.3448765974	2.059944134	0.8272976029

. From Lemma 7, we obtain the implicit solutions of the problem (49), as shown in Figure 1, via fixed values of ${\varphi }_k\left(t\right)=\frac{ln(t+k+2)}{2k+2}$, ${\beta }_k=\frac{3k+2}{10}$, and ${\rho }_k=\frac{k+2}{2}$ with vary ${\alpha }_k\in \{0.2,0.4,0.6,0.8,1.0\}$ for $k=0,1,2$. By using (14), (22), (23), (24), the numerical values of Ξ, ${\Delta }_i$, $i=1,2,3$, for ${\alpha }_k\in \{0.2,0.4,0.6$, $0.8,1.0\}$, as shown in

Table 2. Numerical values of $\Xi$ and ${\Delta }_i$, $i=1,2,3$, for ${\alpha }_k\in \{0.2,0.4,0.6,0.8,1.0\}$ when ${\varphi }_k\left(t\right)=t^{\sqrt{{(k+1)}^2+3}}$, ${\beta }_k=\frac{1}{\sqrt{k+2}}$, and ${\rho }_k=3e^{-\sqrt{k+1}}$ for $k=0,1,2$.

${\mathit{\alpha }}_{\mathit{k}}$	$\mathbf{\Xi }$	${\mathbf{\Delta }}_1$	${\mathbf{\Delta }}_2$	${\mathbf{\Delta }}_3$
0.2	0.979991112	4.377072500	3.467001554	0.6667371782
0.4	1.086173111	4.286940343	3.644959141	0.6463248146
0.6	1.192145946	4.039684022	3.713433921	0.6338233539
0.8	1.288330209	3.702297047	3.704677442	0.6253964583
1.0	1.366630096	3.320188403	3.641434555	0.6183465441

. From Lemma 7, we obtain the implicit solutions of the problem (49) as shown in Figure 2 via fixed values of ${\varphi }_k\left(t\right)=\frac{ln(t+k+2)}{2k+2}$, ${\beta }_k=\frac{3k+2}{10}$, and ${\rho }_k=\frac{k+2}{2}$ with vary ${\alpha }_k\in \{0.2,0.4,0.6,0.8,1.0\}$ for $k=0,1,2$. Later, by using (14), (22), (23), (24), the numerical values of Ξ, ${\Delta }_i$, $i=1,2,3$, for ${\alpha }_k\in \{0.2,0.4,0.6$, $0.8,1.0\}$, as shown in

Table 3. Numerical values of $\Xi$ and ${\Delta }_i$, $i=1,2,3$, for ${\alpha }_k\in \{0.2,0.4,0.6,0.8,1.0\}$ when ${\varphi }_k\left(t\right)=sin\left(\frac{\pi t+k+1}{(2+k)t+3k+3}\right)$, ${\beta }_k=cos\left(\frac{\pi }{5-k}\right)$, and ${\rho }_k=(k+2)tan\left(\frac{\pi }{6-k}\right)$ for $k=0,1,2$.

${\mathit{\alpha }}_{\mathit{k}}$	$\mathbf{\Xi }$	${\mathbf{\Delta }}_1$	${\mathbf{\Delta }}_2$	${\mathbf{\Delta }}_3$
0.2	2.706972884	5.693274732	1.322584648	0.7888666540
0.4	3.075302870	3.950732086	1.271388814	0.8122697082
0.6	3.448104601	2.748016737	1.239777125	0.8313127993
0.8	3.800110763	1.924362576	1.227618946	0.8461678966
1.0	4.087413705	1.363408014	1.236212063	0.8565872889

. From Lemma 7, we obtain the implicit solutions of the problem (49) as shown in Figure 3 via fixed values of ${\varphi }_k\left(t\right)=sin\left(\frac{\pi t+k+1}{(2+k)t+3k+3}\right)$, ${\beta }_k=cos\left(\frac{\pi }{5-k}\right)$, and ${\rho }_k=(k+2)tan\left(\frac{\pi }{6-k}\right)$ with vary ${\alpha }_k\in \{0.2,0.4,0.6,0.8,1.0\}$ for $k=0,1,2$. In addition, we will show the implicit solutions of the problem (49) as shown in Figure 4 for each values of ${\varphi }_k\left(t\right)$, ${\alpha }_k$, ${\beta }_k$, ${\rho }_k$ are given as in

Table 4. The values of ${\varphi }_k\left(t\right)$, ${\alpha }_k$, ${\beta }_k$, ${\rho }_k$, and $\Xi$.

Case	${\mathit{\varphi }}_{\mathit{k}}\left(\mathit{t}\right)$	${\mathit{\alpha }}_{\mathit{k}}$	${\mathit{\beta }}_{\mathit{k}}$	${\mathit{\rho }}_{\mathit{k}}$	$\mathbf{\Xi }$
I	$\frac{ln(t+k+2)}{2k+2}$	$\frac{7-3k}{9-3k}$	$\frac{3k+2}{10}$	$\frac{k+2}{2}$	3.350618629
II	$t^{\sqrt{{(k+1)}^2+3}}$	$\frac{\sqrt{k+1}}{2}$	$\frac{1}{\sqrt{k+2}}$	3 $e^{-\sqrt{k+1}}$	1.221677115
III	$sin\left(\frac{\pi t+k+1}{(2+k)t+3k+3}\right)$	$sin\left(\frac{\pi }{4-k}\right)$	$cos\left(\frac{\pi }{5-k}\right)$	$(k+2)tan\left(\frac{\pi }{6-k}\right)$	3.593850012

.

6. Conclusions

In this paper, we studied the impulsive $({\rho }_k,{\varphi }_k)$-Hilfer $\mathbb{FIDE}$s with a constant coefficient involving $\mathbb{NMP}$-$\mathbb{FIBC}$s. Firstly, we created some essential properties to apply to our main results. The formula of the solution to the linear ($\rho ,\varphi$)-Hilfer fractional Cauchy problem was constructed in the form of the Mittag–Leffler kernel. The non-linear impulsive $({\rho }_k,{\varphi }_k)$-Hilfer fractional Cauchy $\mathbb{BVP}$ was converted into a fixed-point problem via an auxiliary lemma regarding a linear variant of the problem. The uniqueness result was investigated by Banach’s fixed point theorem, while the existence result was proved by a fixed point theorem due to O’Regan. In addition, by applying non-linear functional analysis methods and qualitative theory, a variety of $\mathcal{UH}$ stability, $\mathcal{UHR}$ stability, and their generalization are also examined. To confirm all the achieved theoretical results, numerical examples were given to present the application of our main results in the recent past. Apart from that, our main results are not only novel in the context of the impulsive problem at hand, but they also show some new special situations by adjusting the parameters involved. They have enriched the qualitative theory literature on non-linear impulsive (${\rho }_k,{\varphi }_k$)-Hilfer $\mathbb{FIDE}$s of order in $(0,1]$ equipped with $\mathbb{NMP}$-$\mathbb{FIBC}$s. In future work areas, we recommend working on the qualitative theory literature on non-linear fractional integro-differential equations/inclusions involving a special function, such as the linear Cauchy-type problem with variable coefficients, stability, or the algorithms to solve the ($\rho ,\varphi$)-Hilfer fractional differential equations in mathematical software.