Nonlocal Nonlinear Fractional-Order Sequential Hilfer–Caputo Multivalued Boundary-Value Problems

Abstract

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study.

1. Introduction

Fractional differential equations frequently arise in various research disciplines, including engineering, biology, robotics, physics, ecology, control theory, chemistry, economics, and viscoelasticity, among others. A comprehensive exploration of fractional calculus and fractional differential equations can be found in the monographs by Miller and Ross [1], Kilbas et al. [2], Lakshmikantham et al. [3], Diethelm [4], and Zhou [5], etc.

Many studies have focused on Caputo- and Riemann–Liouville-type fractional differential equations. However, several new definitions of fractional derivatives, such as Hadamard, Hilfer, Caputo–Hadamard, etc., have been introduced to enhance the scope of fractional calculus and meet the requirements of innovative modeling techniques. The Hilfer fractional derivative [6] generalizes both the Riemann–Liouville and Caputo fractional derivatives and finds its applications in science and engineering [7,8,9], mathematics [10,11], etc. The concept of the Hilfer generalized proportional fractional derivative was offered in [12]. For details and applications of differential inclusions, see, for instance, [13,14,15].

Numerous researchers have investigated boundary-value problems for fractional differential equations, addressing different types of equations and boundary conditions. For comprehensive treatment of fractional boundary-value problems involving nonlocal multi-point, multistrip integral boundary conditions, we refer the reader to [16,17,18,19,20]. Hilfer fractional differential inclusions with nonlocal boundary conditions were explored in [21]. In [22], the authors studied a sequential ${\psi }_1$-Hilfer and ${\psi }_2$-Caputo fractional boundary-value problem with integro-differential boundary conditions. In a recently published text [23], one can find useful results on nonlinear systems of fractional differential equations involving different fractional derivative operators and nonlocal boundary conditions.

In the present research, our aim is to examine the existence of solutions for the following sequential $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional differential inclusion with $(k_2,{\psi }_2)$-Riemann–Liouville-type multistrip boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{k_1,H}D_{a^{+}}^{{\alpha }_1,\beta ;{\psi }_1}\left({}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(t\right)-g(t,x\left(t\right))\right)\in F(t,x\left(t\right)),t\in [a,b],\hfill \\ {\displaystyle {}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(a\right)=0,x\left(b\right)={\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i,{\psi }_2}x\left({\xi }_i\right)},}\hfill \end{array}\right.\end{array}$$

(1)

where ${}^{k_1,H}D_{a^{+}}^{{\alpha }_1,\beta ;{\psi }_1}$ and ${}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}$, respectively, denote the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator of order $0<{\alpha }_1\le 1$ with the parameter $0\le \beta \le 1$ and the $(k_2,{\psi }_2)$-Caputo fractional derivative operator of order $0<{\alpha }_2\le 1$; ${}^{k_2}I_{a^{+}}^{{\phi }_i,{\psi }_2}$ represents the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator of order ${\phi }_i>0$; ${\eta }_i\in \mathbb{R}$, $i=1,2,\dots ,\ell$, $a<b$; $a<{\xi }_1<{\xi }_2<\dots <{\xi }_{\ell }<b$; $g:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a given continuous function; and $F:[a,b]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ represents a multivalued map ($\mathcal{P}\left(\mathbb{R}\right)$ defines the family of all nonempty subsets of $\mathbb{R}$).

Here it is worthwhile to mention that the present study is motivated by the fact that the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator is of more general form and specializes to several other forms of fractional derivative operators by fixing the values of ${\psi }_1$ and $\beta .$ For example, the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator takes the form of (i) the $(k_1,{\psi }_1)$-Riemann–Liouville fractional derivative operator for $\beta =0$ and k-Riemann–Liouville fractional derivative operator for $\beta =0$ and ${\psi }_1\left(t\right)=t$; (ii) the $(k_1,{\psi }_1)$-Caputo fractional derivative operator for $\beta =1$ and k-Caputo fractional derivative operator for ${\psi }_1\left(t\right)=t$ and $\beta =1$; (iii) the k-Hilfer–Katugampola fractional derivative operator for ${\psi }_1\left(t\right)=t^{\rho }$; (iv) the k-Katugampola fractional derivative operator for ${\psi }_1\left(t\right)=t^{\rho }$ and $\beta =0$; (v) the $k_1$-Caputo–Katugampola fractional derivative operator when ${\psi }_1\left(t\right)=t^{\rho }$ and $\beta =1$; (vi) the $k_1$-Hilfer–Hadamard fractional derivative operator for ${\psi }_1\left(t\right)=logt$ and $k_1$-Hadamard fractional derivative operator for ${\psi }_1\left(t\right)=logt$ and $\beta =0$; and (vii) the $k_1$-Caputo–Hadamard fractional derivative operator for ${\psi }_1\left(t\right)=logt$ and $\beta =1.$ Also $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operators appearing in the multistrip boundary conditions correspond to the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators for $k_2=1$, ${\psi }_2\left(t\right)=t$ and $k_2=1,{\psi }_2\left(t\right)=t$, respectively. For more details, see [24].

Nonlocal conditions are considered to be more realistic than traditional ones, as such conditions effectively model certain aspects of physical phenomena, where classical conditions do not. These conditions play a vital role in characterizing the internal properties of physical and chemical processes, where conventional initial and boundary conditions are inadequate. Nonlocal boundary conditions appear in various fields like petroleum extraction, thermodynamics, elasticity, and wave propagation [25].

The main objective of the present article is to explore the existence criteria for solutions to problem (1) as described below.

(i)

The upper semicontinuous case: Assuming the multivalued map to be convex-valued and $L^1$-Carathéodory, we prove four existence results. The first one relies on the nonlinear alternative of the Leray–Schauder type for Kakutani maps, while the second one is based on a practical version of Martelli’s fixed-point theorem for bounded multifunctions. The third and fourth results, respectively, depend upon the nonlinear alternative for contractive maps and Krasnoselskii’s fixed-point theorem for multivalued maps.

(ii)

The lower semicontinuous case: In this case, the multifunction does not necessarily have convex values. Our approach relies on the nonlinear alternative of the Leray–Schauder type for single-valued maps, combined with Bressan and Colombo’s selection theorem for lower semicontinuous maps having decomposable values.

(iii)

The Lipschitz case: An existence result for the non-convex-valued right-hand side of the fractional differential inclusion is obtained with the aid of a fixed-point theorem for multivalued maps due to Covitz and Nadler.

Our findings make a significant contribution to the evolving field of sequential Hilfer and Caputo fractional differential inclusions. The results obtained in this paper mark a novel advancement for fractional boundary-value problems involving sequential $(k,\psi )$-type Hilfer and Caputo fractional differential inclusions with nonlocal multistrip boundary conditions. Though the methodologies used for deriving the desired abstract results are well established, their application to problem (1) is quite lengthy and original.

We arrange the rest of the article as follows. Essential preliminaries related to our work are outlined in Section 2. Section 3 presents the existence results for the upper semicontinuous case, followed by Section 4 addressing the lower semicontinuous case. In Section 5, an existence result for the Lipschitz case is obtained. Examples illustrating the main results are offered in Section 6.

2. Preliminaries

In the first part of this section, we recall the necessary concepts of fractional calculus, while the second part deals with multivalued analysis.

2.1. Fractional Calculus

In the following, we suppose that $\psi \in C^1([a,b],\mathbb{R})$ is a positive, continuous, and increasing function with ${\psi }^{\prime }\left(a\right)\ne 0$ for each $t\in [a,b]$.

Definition 1 

([26]). For $z\in \mathbb{C}$ with a positive real part and $k\in \mathbb{R}$, the k-gamma function defined by

$${\mathsf{\Gamma }}_k\left(z\right)={\int }_0^{\infty }{t}^{z-1}{e}^{-{\displaystyle \frac{t^k}{k}}}dt,k>0,$$

satisfies the properties

$$\begin{array}{c}\hfill \mathsf{\Gamma }\left(z\right)=\underset{k\to 1}{lim}{\mathsf{\Gamma }}_k\left(z\right),{\mathsf{\Gamma }}_k\left(z\right)=k^{{\displaystyle \frac{z}{k}}-1}\mathsf{\Gamma }\left({\displaystyle \frac{z}{k}}\right)\mathit{and}{\mathsf{\Gamma }}_k(z+k)=z{\mathsf{\Gamma }}_k\left(z\right).\end{array}$$

Definition 2 

([27]). The $(k,\psi )$-Riemann–Liouville fractional integral of order α for a function $f\in L^1([a,b],\mathbb{R})$ is given by

$$\begin{array}{c}\hfill {}^{k}I_{a^{+}}^{\alpha ;\psi }f\left(t\right)={\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k\left(\alpha \right)}}{\int }_a^t{\psi }^{\prime }\left(s\right){\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{{\displaystyle \frac{\alpha }{k}}-1}f\left(s\right)ds,\alpha ,k\in {\mathbb{R}}^{+}.\end{array}$$

Definition 3 

([24]). We define the $(k,\psi )$-Caputo fractional derivative of order $\alpha \in {\mathbb{R}}^{+}$ for a function $f\in C\left(\right[a,b],\mathbb{R})$ as

$$\begin{array}{cc}\hfill {}^{k,C}D_{a^{+}}^{\alpha ;\psi }f\left(t\right)& ={}^{k}I_{a^{+}}^{nk-\alpha ;\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n}f\left(t\right)\hfill \\ & ={\displaystyle \frac{1}{k{\mathsf{\Gamma }}_k(nk-\alpha )}}{\int }_a^t{\psi }^{\prime }\left(s\right){(\psi \left(t\right)-\psi \left(s\right))}^{n-{\displaystyle \frac{\alpha }{k}}-1}{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(s\right)}}{\displaystyle \frac{d}{ds}}\right)}^{n}f\left(s\right)ds,\hfill \end{array}$$

where $k\in {\mathbb{R}}^{+}$ and $n=⌈{\displaystyle \frac{\alpha }{k}}⌉$ is the ceiling function of ${\displaystyle \frac{\alpha }{k}}$.

Definition 4 

([24]). The $(k,\psi )$-Hilfer fractional derivative of order $\alpha \in {\mathbb{R}}^{+}$ and type $\beta \in (0,1]$ for a function $f\in C\left(\right[a,b],\mathbb{R})$ is defined by

$$\begin{array}{c}\hfill {}^{k,H}D_{a^{+}}^{\alpha ,\beta ;\psi }f\left(t\right)={}^{k}I_{a^{+}}^{\beta (nk-\alpha );\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{}^{k}I_{a^{+}}^{(1-\beta )(nk-\alpha );\psi }f\left(t\right),n=⌈{\displaystyle \frac{\alpha }{k}}⌉,k\in {\mathbb{R}}^{+}.\end{array}$$

Lemma 1. 

Let $f\in C^n\left([a,b]\mathbb{R}\right)$, $\alpha ,k\in {\mathbb{R}}^{+}$ and $n=⌈{\displaystyle \frac{\alpha }{k}}⌉$. Then, the following relation holds:

$$\begin{array}{c}\hfill {}^{k}I_{a^{+}}^{\alpha ;\psi }\left({}^{k,C}D_{a^{+}}^{\alpha ;\psi }f\left(t\right)\right)=f\left(t\right)-\sum _{j=0}^{n-1}{\displaystyle \frac{{\left(\psi \left(t\right)-\psi \left(a\right)\right)}^j}{{\mathsf{\Gamma }}_k(jk+k)}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{j}f\left(t\right)\right]}_{t=a}.\end{array}$$

Lemma 2 

([24]). Let $f\in C^n([a,b],\mathbb{R})$ and ${}^{k}I_{a^{+}}^{nk-\mu ;\psi }f\in C^n([a,b],\mathbb{R})$ with $\mu ,k\in {\mathbb{R}}^{+}$ and $n=⌈{\displaystyle \frac{\mu }{k}}⌉$. Then

$$\begin{array}{c}\hfill {}^{k}I_{a^{+}}^{\mu ;\psi }\left({}^{k,H}D_{a^{+}}^{\mu ;\psi }f\left(t\right)\right)=f\left(t\right)-\sum _{j=1}^n{\displaystyle \frac{{\left(\psi \left(t\right)-\psi \left(0\right)\right)}^{\frac{\mu }{k}-j}}{{\mathsf{\Gamma }}_k(\mu -jk+k)}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n-j}{}^{k}I_{a^{+}}^{nk-\mu }f\left(t\right)\right]}_{t=a}.\end{array}$$

Lemma 3 

([24]). Let $\alpha ,k\in {\mathbb{R}}^{+}$ with $\alpha <k$, $0\le \beta \le 1$ and ${\epsilon }_k=\alpha +\beta (nk-\alpha )$ and $f\in C^n([a,b],\mathbb{R})$. Then

$$\begin{array}{c}\hfill {}^{k}I_{a^{+}}^{{\epsilon }_k;\psi }\left({}^{k,RL}D_{a^{+}}^{{\epsilon }_k;\psi }f\left(t\right)\right)={}^{k}I_{a^{+}}^{\alpha ;\psi }\left({}^{k,H}D_{a^{+}}^{\alpha ,\beta ;\psi }f\left(t\right)\right).\end{array}$$

Lemma 4 

([24]). Let ${\rho }_1,{\rho }_2\in {\mathbb{R}}^{+}$. Then,

$$\begin{array}{c}\hfill {}^{k}I_{a^{+}}^{{\rho }_1;\psi }{}^{k}I_{a^{+}}^{{\rho }_2;\psi }f\left(t\right)={}^{k}I_{a^{+}}^{{\rho }_1+{\rho }_2;\psi }f\left(t\right).\end{array}$$

Lemma 5 

([24]). Let $\alpha ,k\in {\mathbb{R}}^{+}$ and let $\mu \in \mathbb{R}$, ${\displaystyle \frac{\mu }{k}}>-1$. Then,

$$\begin{array}{cc}\hfill {}^{k}I_{a^{+}}^{\alpha ;\psi }{\left(\psi \left(t\right)-\psi \left(a\right)\right)}^{{\displaystyle \frac{\mu }{k}}}& {\displaystyle =\frac{{\mathsf{\Gamma }}_k(\mu +k)}{{\mathsf{\Gamma }}_k(\mu +k+\alpha )}{\left(\psi \left(t\right)-\psi \left(a\right)\right)}^{\frac{\mu +\alpha }{k}},}\hfill \\ \hfill {}^{k,H}D_{a^{+}}^{\alpha ;\psi }{\left(\psi \left(t\right)-\psi \left(a\right)\right)}^{{\displaystyle \frac{\mu }{k}}}& {\displaystyle =\frac{{\mathsf{\Gamma }}_k(\mu +k)}{{\mathsf{\Gamma }}_k(\mu +k-\alpha )}{\left(\psi \left(t\right)-\psi \left(a\right)\right)}^{\frac{\mu -\alpha }{k}}.}\hfill \end{array}$$

Remark 1 

([24]). The relationship between the $(k,\psi )$-Hilfer fractional derivative and $(k,\psi )$-Riemann–Liouville fractional integral is given by

$$\begin{array}{cc}\hfill {}^{k,H}D_{a^{+}}^{\alpha ,\beta ;\psi }f\left(t\right)& ={}^{k}I_{a^{+}}^{{\epsilon }_k-\alpha ;\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{}^{k}I_{a^{+}}^{(nk-{\epsilon }_k);\psi }\varkappa \left(t\right)\hfill \\ \hfill & ={{}^{k}I}_{a^{+}}^{{\epsilon }_k-\alpha ;\psi }\left({}^{k,RL}D_{a^{+}}^{{\epsilon }_k;\psi }\right)f\left(t\right).\hfill \end{array}$$

Letting ${\epsilon }_k=\alpha +\beta (nk-\alpha )$ or $(1-\beta )(nk-\alpha )=nk-{\epsilon }_k$, we have $n-1<{\displaystyle \frac{{\epsilon }_k}{k}}\le n$ for $n-1<{\displaystyle \frac{\alpha }{k}}\le n$ and $0\le \beta \le 1$.

2.2. Set-Valued Analysis

We begin this subsection with some related concepts of multivalued analysis [13,15].

Let $\mathcal{P}\left(\mathbb{R}\right)$ denote the family of all nonempty subsets of $\mathbb{R}$ and $(X,\parallel ·\parallel )$ be a normed space. Then we define ${\mathcal{P}}_{\mathfrak{q}}\left(X\right)=\{\mathfrak{Z}\in \mathcal{P}\left(X\right):\mathfrak{Z}\ne \varnothing ,\mathrm{which} \mathrm{has} \mathrm{the} \mathrm{property}\mathfrak{q}\}.$ In particular, we have the following:

• ${\mathcal{P}}_{cl}\left(X\right)=\{\mathfrak{Z}\in \mathcal{P}\left(X\right):\mathfrak{Z} \mathrm{is} \mathrm{closed}\}$;
• ${\mathcal{P}}_{cp}\left(X\right)=\{\mathfrak{Z}\in \mathcal{P}\left(X\right):\mathfrak{Z} \mathrm{is} \mathrm{compact}\}$;
• ${\mathcal{P}}_{cp,c}\left(X\right)=\{\mathfrak{Z}\in \mathcal{P}\left(X\right):\mathfrak{Z} \mathrm{is} \mathrm{compact} \mathrm{and} \mathrm{convex}\}$;
• ${\mathcal{P}}_{b,cl,c}\left(X\right)=\{\mathfrak{Z}\in \mathcal{P}\left(X\right):\mathfrak{Z} \mathrm{is} \mathrm{bounded}, \mathrm{closed}, \mathrm{and}$ $\mathrm{convex}\}$, etc.

Define the set of selections of a multifunction F for each $x\in C\left(\right[a,b],\mathbb{R})$ as

$$S_{F,x}:=\{z\in L^1([a,b],\mathbb{R}):z\left(t\right)\in F(t,x\left(t\right))\mathrm{for} \mathrm{a.e.}t\in [a,b]\},$$

and the graph of F by $\mathit{Gr}\left(F\right)=\left\{\right(x,y)\in X\times Y,y\in F(x\left)\right\}.$

A multivalued map $F:X\to \mathcal{P}\left(X\right)$ is said to be

(a)

Convex (closed) valued if, for every $a\in X,$ the set $F\left(a\right)$ is convex (closed);

(b)

Upper semicontinuous (u.s.c.) on X if, for every $a_0\in X,$ the set $F\left(a_0\right)$ is a nonempty closed subset of X, and for any open set $\mathcal{N}$ in X that contains $F\left(a_0\right),$ there exists an open neighborhood ${\mathcal{N}}_0$ of $a_0$ such that $F\left({\mathcal{N}}_0\right)\subseteq N;$

(c)

Bounded on bounded sets if the union $F\left(\mathbb{B}\right)={\cup }_{x\in \mathbb{B}}F\left(x\right)$ remains bounded in X for every $\mathbb{B}\in {\mathcal{P}}_b\left(X\right),$ meaning that ${sup}_{x\in \mathbb{B}}\{sup\{\left|y\right|:y\in F\left(x\right)\left\}\right\}<\infty ;$

(d)

Completely continuous if $F\left(\mathbb{B}\right)$ is relatively compact for every $\mathbb{B}\in {\mathcal{P}}_b\left(X\right)$.

A fixed point of F exists if there is some $a\in X$ such that $a\in F\left(a\right).$ The set of fixed points of the multivalued operator F is denoted by $\mathit{Fix}F$.

A function $F:J\to {\mathcal{P}}_{cl}\left(\mathbb{R}\right)$ is called measurable if the function $t⟼d(b,F(t\left)\right)=inf\left\{\right|b-c|:c\in F(t\left)\right\}$ is measurable for every $b\in \mathbb{R}$.

In the following remark, we describe the connection between closed graphs and upper semicontinuity [15].

Remark 2. 

If $F:X\to Y$ is upper semicontinuous, then its graph $\mathit{Gr}\left(F\right)$ is a closed subset of $X\times Y.$ In other words, if ${\left\{x_n\right\}}_{n\in \mathbb{N}}\subset X$ and ${\left\{y_n\right\}}_{n\in \mathbb{N}}\subset X$ are sequences such that $x_n\to x_{*}$, $y_n\to y_{*}$ as $n\to \infty$, and $y_n\in F\left(x_n\right)$, then $y_{*}\in F\left(x_{*}\right)$. Conversely, if F is completely continuous and has a closed graph, then it is upper semicontinuous.

For further details on multivalued maps, we refer the reader to the books by Deimling [15], Hu and Papageorgiou [13], Gorniewicz [28], and Castaing and Valadier [29].

3. Existence Results—The Upper Semicontinuous Case

Denote by $C\left(\right[a,b],\mathbb{R})$ the Banach space of all continuous functions mapping $[a,b]$ into $\mathbb{R},$ endowed with the sup-norm $\parallel x\parallel :=sup\left\{\right|x\left(t\right)|:t\in [a,b\left]\right\}.$ Similarly, the space $L^1([a,b],\mathbb{R})$ consisting of functions $x:[a,b]\to \mathbb{R}$ for which the norm ${\parallel x\parallel }_{L^1}={\int }_a^b\left|x\left(s\right)\right|ds$ is finite.

In the following lemma, we solve a linear variant of the single-valued version of the boundary-value problem (1), which is used to transform problem (1) into a fixed-point problem.

Lemma 6. 

Let $h,\omega \in C\left(\right[a,b],\mathbb{R})$ be given functions and

$$\begin{array}{c}\hfill \mathsf{\Delta }:=1-\sum _{i=1}^{\ell }{\eta }_i{\displaystyle \frac{{({\psi }_2\left({\xi }_i\right)-{\psi }_2\left(a\right))}^{\frac{{\phi }_i}{k_2}}}{{\mathsf{\Gamma }}_{k_2}({\phi }_i+k_2)}}\ne 0.\end{array}$$

Then, the sequential $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional integro-differential linear boundary-value problem

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{k_1,H}D_{a^{+}}^{{\alpha }_1,\beta ;{\psi }_1}\left({}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(t\right)-h\left(t\right)\right)=\omega \left(t\right),t\in [a,b],\hfill \\ {\displaystyle {}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(a\right)=0,x\left(b\right)={\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i,{\psi }_2}x\left({\xi }_i\right)},}\hfill \end{array}\right.\end{array}$$

(2)

is equivalent to the integral equation

$$\begin{array}{cc}\hfill x\left(t\right)& ={}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left(t\right)+{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(t\right)}\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\mathsf{\Delta }}}[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}h\left({\xi }_i\right)+{\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i+{\alpha }_2,{\psi }_2}}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left({\xi }_i\right)\right)\hfill \\ \hfill & -{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left(b\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(b\right)\right)].\hfill \end{array}$$

(3)

Proof. 

Operating the integral operator ${}^{k_1}I_{a^{+}}^{a_1;{\psi }_1}$ on both sides of the fractional differential equation in (2), and using Lemma 2, we obtain

$$\begin{array}{c}\hfill {}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(t\right)=h\left(t\right)+{{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(t\right)+{\displaystyle \frac{{\left({\psi }_1\left(t\right)-{\psi }_1\left(a\right)\right)}^{\frac{{\theta }_{k_1}}{k_1}-1}}{\mathsf{\Gamma }\left({\theta }_{k_1}\right)}}{c}_1,\end{array}$$

(4)

where

$$\begin{array}{c}\hfill c_1={\left[\left({\displaystyle \frac{k_1}{{\psi }_1^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right){{}^{k_1}I}_{a^{+}}^{k_1-{\theta }_{k_1};{\psi }_1}\omega \left(t\right)\right]}_{t=a}.\end{array}$$

Using the condition ${}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(a\right)=0$ in (4), we find that $c_1=0$ since ${\displaystyle \frac{{\theta }_{k_1}}{k_1}}<1.$ Therefore, (4) becomes

$$\begin{array}{c}\hfill {}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(t\right)=h\left(t\right)+{}^{k_1}I_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(t\right).\end{array}$$

(5)

Operating the fractional integral operator ${}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}$ on both sides of (5) yields

$$\begin{array}{cc}\hfill x\left(t\right)& ={}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left(t\right)+{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(t\right)\right)+d_1,\hfill \end{array}$$

(6)

where $d_1$ is an arbitrary real constant. Inserting (6) in the second boundary condition of (2), we obtain

$$\begin{array}{ccc}& & {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left(b\right)+{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(b\right)\right)+d_1\hfill \\ & =& {\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i,{\psi }_2}}\left[{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left({\xi }_i\right)+{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(t\right)\right)+d_1\right]\hfill \\ & =& \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}h\left({\xi }_i\right)+{\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i+{\alpha }_2,{\psi }_2}}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left({\xi }_i\right)\right)\hfill \\ & & +d_1\sum _{i=1}^{\ell }{\eta }_i{\displaystyle \frac{{({\psi }_2\left({\xi }_i\right)-{\psi }_2\left(a\right))}^{\frac{{\phi }_i}{k_2}}}{{\mathsf{\Gamma }}_{k_2}({\phi }_i+k_2)}},\hfill \end{array}$$

which, on solving for $d_1$, yields

$$\begin{array}{ccc}\hfill d_1& =& {\displaystyle \frac{1}{\mathsf{\Delta }}}[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}h\left({\xi }_i\right)+{\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i+{\alpha }_2,{\psi }_2}}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left({\xi }_i\right)\right)\hfill \\ & & -{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left(b\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(b\right)\right)].\hfill \end{array}$$

Replacing $d_1$ with its value above in (6), we obtain (3). By direct computation, one can prove the converse of the lemma. □

For estimating the bounds, the solution (3) can be rewritten as

$$\begin{array}{cc}\hfill x\left(t\right)& =\left\{{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left(t\right)+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}h\left({\xi }_i\right)-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}h\left(b\right)\right]\right\}\hfill \\ \hfill & +\{{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(t\right)}\right)\hfill \\ & +{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[{\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i+{\alpha }_2,{\psi }_2}}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\omega \left(b\right)\right)\right]\}\hfill \\ \hfill & =x_h\left(t\right)+x_{\omega }\left(t\right).\hfill \end{array}$$

(7)

Lemma 7. 

Let $h,\omega \in C\left(\right[a,b],\mathbb{R}).$ Then, we have

(i) 

${\displaystyle |x_h\left(t\right)|\le \parallel h\parallel \left\{\left(1+\frac{1}{\left|\mathsf{\Delta }\right|}\right)A_0+\frac{1}{\left|\mathsf{\Delta }\right|}\widehat{A_0}\right\}:=\parallel h\parallel {\mathsf{\Theta }}_0,}$

where

$${\mathsf{\Theta }}_0=\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_0+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\widehat{A_0},$$

(8)

$$A_0={\displaystyle \frac{{({\psi }_2\left(b\right)-{\psi }_2\left(a\right))}^{\frac{{\alpha }_2}{k_2}}}{{\mathsf{\Gamma }}_{k_2}({\alpha }_2+k_2)}}and\widehat{A_0}=\sum _{i=1}^{\ell }{\eta }_i{\displaystyle \frac{{({\psi }_2\left(b\right)-{\psi }_2\left(a\right))}^{\frac{{\varphi }_i+{\alpha }_2}{k_2}}}{{\mathsf{\Gamma }}_{k_2}({\varphi }_i+{\alpha }_2+k_2)}}.$$

(ii) 

${\displaystyle |x_{\omega }|\le \parallel \omega \parallel \left\{\left(1+\frac{1}{\left|\mathsf{\Delta }\right|}\right)A_1+\frac{1}{\left|\mathsf{\Delta }\right|}\sum _{i=1}^{\ell }\left|{\eta }_i\right|A_{i1}\right\}:=\parallel \omega \parallel {\mathsf{\Theta }}_1,}$

where

$${\mathsf{\Theta }}_1=\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\sum _{i=1}^{\ell }\left|{\eta }_i\right|A_{i1},$$

(9)

$$\begin{array}{c}\hfill A_1={\displaystyle \frac{1}{k_2{\mathsf{\Gamma }}_{k_1}({\alpha }_1+k_1){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^b{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(b\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}{\left({\psi }_1\left(s\right)-{\psi }_1\left(a\right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}}ds\end{array}$$

and

$$\begin{array}{c}\hfill A_{i1}={\displaystyle \frac{1}{k_2{\mathsf{\Gamma }}_{k_1}({\alpha }_1+k_1){\mathsf{\Gamma }}_{k_2}({\alpha }_2+{\phi }_i)}}{\int }_a^{{\xi }_i}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left({\xi }_i\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2+{\phi }_i}{k_2}}-1}{\left({\psi }_1\left(s\right)-{\psi }_1\left(a\right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}}ds.\end{array}$$

Proof. 

(i) From (7), we have

$$\begin{array}{ccc}\hfill |x_h\left(t\right)|& \le & {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}\left|h\left(t\right)\right|+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}|h\left({\xi }_i\right)|+{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}\left|h\left(b\right)\right|\right]\hfill \\ & \le & \parallel h\parallel \left\{{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}\left(1\right)+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}\left(1\right)+{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}\left(1\right)\right]\right\}\hfill \\ & =& \parallel h\parallel \{{\displaystyle \frac{1}{{\alpha }_2{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{({\psi }_2\left(t\right)-{\psi }_2\left(a\right))}^{{\displaystyle \frac{{\alpha }_2}{k_2}}}\hfill \\ & & +{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}[\sum _{i=1}^{\ell }{\eta }_i{\displaystyle \frac{1}{({\varphi }_i+{\alpha }_2){\mathsf{\Gamma }}_{k_2}({\varphi }_i+{\alpha }_2)}}{({\psi }_2\left(t\right)-{\psi }_2\left(a\right))}^{{\displaystyle \frac{{\varphi }_i+{\alpha }_2}{k_2}}}\hfill \\ & & +{\displaystyle \frac{1}{{\alpha }_2{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{({\psi }_2\left(t\right)-{\psi }_2\left(a\right))}^{{\displaystyle \frac{{\alpha }_2}{k_2}}}\left]\right\}\hfill \\ & =& \parallel h\parallel \left\{\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_0+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\widehat{A_0}\right\}\hfill \\ & =& \parallel h\parallel {\mathsf{\Theta }}_0.\hfill \end{array}$$

(ii) Similarly, we can find that

$$\begin{array}{ccc}\hfill |x_{\omega }\left(t\right)|& \le & {{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left|\omega \left(t\right)\right|}\right)\hfill \\ & & +{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[{\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i+{\alpha }_2,{\psi }_2}}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left|\omega \left({\xi }_i\right)\right|\right)+{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left|\omega \left(b\right)\right|\right)\right]\hfill \\ & \le & {\displaystyle \frac{\parallel \omega \parallel }{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^b{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(b\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}d\tau ds\hfill \\ & & +{\displaystyle \frac{\parallel \omega \parallel }{\left|\mathsf{\Delta }\right|k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right)}}[\sum _{i=1}^{\ell }\left|{\eta }_i\right|{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{k_2}({\alpha }_2+{\phi }_i)}}{\int }_a^{{\xi }_i}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left({\xi }_i\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2+{\phi }_i}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}d\tau ds\hfill \\ & & +{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^b{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(b\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}{\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}d\tau ds]\hfill \\ & =& {\displaystyle \frac{\parallel \omega \parallel }{k_2{\mathsf{\Gamma }}_{k_1}({\alpha }_1+k_1){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^b{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(b\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}{\left({\psi }_1\left(s\right)-{\psi }_1\left(a\right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}}ds\hfill \\ & & +{\displaystyle \frac{\parallel \omega \parallel }{\left|\mathsf{\Delta }\right|k_2{\mathsf{\Gamma }}_{k_1}({\alpha }_1+k_1)}}[\sum _{i=1}^{\ell }\left|{\eta }_i\right|{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{k_2}({\alpha }_2+{\phi }_i)}}{\int }_a^{{\xi }_i}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left({\xi }_i\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2+{\phi }_i}{k_2}}-1}\hfill \\ & & \times {\left({\psi }_1\left(s\right)-{\psi }_1\left(a\right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}}ds\hfill \\ & & +{\displaystyle \frac{1}{{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^b{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(b\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}{\left({\psi }_1\left(s\right)-{\psi }_1\left(a\right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}}ds]\hfill \\ & =& \parallel \omega \parallel \left\{\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\sum _{i=1}^{\ell }\left|{\eta }_i\right|A_{i1}\right\}\hfill \\ & =& \parallel \omega \parallel {\mathsf{\Theta }}_1.\hfill \end{array}$$

□

Definition 5. 

A function $x\in C\left(\right[a,b],\mathbb{R})$ is referred to as a solution of the sequential boundary-value problem (1) if there exists a function $u\in L^1([a,b],\mathbb{R})$ such that $u\left(t\right)\in F(t,x(t\left)\right)$ almost everywhere on $[a,b]$ and it satisfies the equations

$${\displaystyle {}^{k_1,H}D_{a^{+}}^{{\alpha }_1,\beta ;{\psi }_1}\left({}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(t\right)-g(t,x\left(t\right))\right)=u\left(t\right),{}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(a\right)=0,x\left(b\right)={\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i,{\psi }_2}x\left({\xi }_i\right)}.}$$

Definition 6 

([13]). A set-valued function $F:[a,b]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}\left(\mathbb{R}\right)$ is defined as Carathéodory if

(i) 

The mapping $t\to F(t,x)$ is measurable for every $x\in \mathbb{R}$;

(ii) 

The function $x\to F(t,x)$ is upper semicontinuous for almost every $t\in [a,b]$.

Moreover, a Carathéodory function F is referred to as an $L^1$-Carathéodory if

(iii) 

There exists a function ${\phi }_{\rho }\in L^1([a,b],{\mathbb{R}}^{+})$ such that $\parallel F(t,x)\parallel =sup\left\{\right|u|:u\in F(t,x)\}\le {\phi }_{\rho }\left(t\right)$ for all $x\in \mathbb{R}$ with $\parallel x\parallel \le \rho$ for each $\rho >0$ and for almost every $t\in [a,b]$.

The following lemma is used in the sequel.

Lemma 8 

([30]). Let $F:[a,b]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}\left(\mathbb{R}\right)$ be an $L^1$-Carathéodory multivalued map and let $\mathcal{Z}$ be a linear continuous mapping from $L^1([a,b],\mathbb{R})$ to $C\left(\right[a,b],\mathbb{R})$. Then, the operator

$$\mathcal{Z}\circ S_F:C([a,b],\mathbb{R})\to {\mathcal{P}}_{cp,c}\left(C([a,b],\mathbb{R})\right),x\mapsto (\mathcal{Z}\circ S_F)\left(x\right)=\mathcal{Z}\left(S_{F,x}\right)$$

is a closed graph operator in $C\left(\right[a,b],\mathbb{R})\times C\left(\right[a,b],\mathbb{R}).$

3.1. Existence Result via Leray–Schauder Nonlinear Alternative

Our first result, dealing with a convex-valued F, is based on the Leray–Schauder nonlinear alternative for multivalued maps (stated below).

Theorem 1 

(Nonlinear alternative for Kakutani maps [31]). Let $\mathcal{N}$ be a Banach space, ${\mathcal{N}}_1$ a closed convex subset of $\mathcal{N},$ U an open subset of ${\mathcal{N}}_1$, and $0\in U.$ Suppose that $F:\overline{U}\to {\mathcal{P}}_{cp,c}\left({\mathcal{N}}_1\right)$ is an upper semicontinuous and compact map. Then,

(i) 

There exists a fixed point of F in $\overline{U},$ or

(ii) 

There exists some $u\in \partial U$ and $\lambda \in (0,1)$ such that $u\in \lambda F\left(u\right).$

Theorem 2. 

Suppose that the following conditions hold:

(H₁)

The set-valued function $F:[a,b]\times \mathbb{R}\to {\mathcal{P}}_{cp,c}\left(\mathbb{R}\right)$ is of $L^1$-Carathéodory type;

(H₂)

There exists a strictly increasing function $\mathsf{\Psi }\in C\left(\right[0,\infty ),(0,\infty \left)\right)$ and a positive, continuous function Q such that

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\left\{\right|z|:z\in F(t,x)\}\le Q\left(t\right)\mathsf{\Psi }(\parallel x\parallel )for each(t,x)\in [a,b]\times \mathbb{R};$$

(H₃)

There exists a constant $M>0$ such that

$${\displaystyle \frac{M}{{\displaystyle \parallel p\parallel {\mathsf{\Theta }}_0+\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }\left(M\right){\mathsf{\Theta }}_1}}}>1,$$

where ${\mathsf{\Theta }}_0,{\mathsf{\Theta }}_1$ are given in Lemma 7.

(H₄)

There exists a function $p\in C([a,b],{\mathbb{R}}^{+})$ such that $\left|g\right(t,x\left)\right|\le p\left(t\right),$ for all $(t,x)\in [a,b]\times \mathbb{R}.$

Then, the sequential fractional multivalued boundary value problem involving the $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo operators, given in (1), admits at least one solution on the interval $[a,b]$.

Proof. 

Introduce an operator $\mathbb{F}:C\left(\right[a,b],\mathbb{R})⟶\mathcal{P}\left(C\right([a,b],\mathbb{R}\left)\right)$ as

$$\left(\mathbb{F}x\right)\left(t\right)=\left\{\begin{array}{cc}& h\in C\left(\right[a,b],\mathbb{R}):\hfill \\ & h\left(t\right)=\left\{\begin{array}{c}{\displaystyle {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))}\hfill \\ {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]}\hfill \\ {\displaystyle +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)}\hfill \\ {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)}\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)}\right)\right],}\hfill \end{array}\right.\hfill \end{array}\right\}$$

for $v\in S_{F,x}$ and $t\in [a,b],$ and transform the nonlocal boundary-value problem (1) into a fixed-point problem $\mathbb{F}x=x$. The fixed points of the operator $\mathbb{F}$ are obviously solutions to the boundary-value problem (1).

We verify the hypotheses of the Leray–Schauder multivalued nonlinear alternative (Theorem 1) through several steps.

Step 1. The set $\mathbb{F}x$ is convex for each $x\in C\left(\right[a,b],\mathbb{R})$.

This property is obvious as F is convex-valued.

Step 2. The operator $\mathbb{F}$ maps bounded sets of $C\left(\right[a,b],\mathbb{R})$ to its bounded subsets.

Consider the bounded set ${\mathbb{B}}_{r_0}=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel \le r_0\}$ in $C\left(\right[a,b],\mathbb{R})$. Then, for each $h\in \mathbb{F}x,$ with $x\in {\mathbb{B}}_{r_0}$, there exists an element $v\in S_{F,x}$ such that

$$\begin{array}{ccc}\hfill h\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)\right)\right].}\hfill \end{array}$$

Then, for $t\in [a,b],$ it follows by Lemma 7 that

$$\begin{array}{ccc}\hfill \left|h\right(t\left)\right|& \le & {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}\left|g(t,x\left(t\right))\right|+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}|g({\xi }_i,x\left({\xi }_i\right))|+{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}\left|g(b,x\left(b\right))\right|\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left|v\left(t\right)\right|}\right)\hfill \\ & & {\displaystyle +\frac{1}{\left|\mathsf{\Delta }\right|}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left|v\left({\xi }_i\right)\right|\right)+{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left|v\left(b\right)\right|\right)\right]}\hfill \\ & \le & \parallel p\parallel \left\{\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_0+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\widehat{A_0}\right\}\hfill \\ & & +\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }(\parallel x\parallel )\left\{\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\sum _{i=1}^{\ell }{\eta }_i{A}_{i1}\right\}\hfill \\ & \le & \parallel p\parallel {\mathsf{\Theta }}_0+\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }\left(r_0\right){\mathsf{\Theta }}_1,\hfill \end{array}$$

and consequently, we get

$$\parallel h\parallel \le \parallel p\parallel {\mathsf{\Theta }}_0+\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }\left(r_0\right){\mathsf{\Theta }}_1.$$

Step 3. The mapping $\mathbb{F}$ sends bounded sets of $C\left(\right[a,b],\mathbb{R})$ into equicontinuous sets.

Let $t_1,t_2\in [a,b]$ with $t_1<t_2$ and $x\in {\mathbb{B}}_{r_0}.$ Then, for each $h\in \mathbb{F}x,$ we obtain

$$\begin{array}{ccc}& & |h\left(t_2\right)-h\left(t_1\right)|\hfill \\ & \le & |{\displaystyle \frac{1}{k_2{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_2}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}g(s,x\left(s\right))ds\hfill \\ & & -{\displaystyle \frac{1}{k_2{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_1}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_1\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}g(s,x\left(s\right))ds|\hfill \\ & & +|{\displaystyle \frac{1}{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_2}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}v\left(\tau \right)d\tau ds\hfill \\ & & -{\displaystyle \frac{1}{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_1}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_1\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}v\left(\tau \right)d\tau ds|\hfill \\ & \le & {\displaystyle \frac{\parallel p\parallel }{k_2{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_2}{\psi }_2^{\prime }\left(s\right)\left|{\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}-{\left({\psi }_2\left(t_1\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\right|ds\hfill \\ & & +{\displaystyle \frac{\parallel p\parallel }{k_2{\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_{t_1}^{t_2}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}ds\hfill \\ & & +{\displaystyle \frac{\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }\left(r_0\right)}{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_1}{\psi }_2^{\prime }\left(s\right)\left|{\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}-{\left({\psi }_2\left(t_1\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\right|\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}d\tau ds\hfill \\ & & +{\displaystyle \frac{\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }\left(r_0\right)}{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_{t_1}^{t_2}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}d\tau ds,\hfill \end{array}$$

which tends to zero as $t_2-t_1\to 0,$ independently of $x\in {\mathbb{B}}_{r_0}.$ So, the operator $\mathbb{F}:C\left(\right[a,b],\mathbb{R})\to \mathcal{P}\left(C\right([a,b],\mathbb{R}\left)\right)$ is completely continuous by the application of the Arzelá–Ascoli theorem.

Next, we show that the operator $\mathbb{F}$ is upper semicontinuous (u.s.c.). It suffices to demonstrate that the graph of $\mathbb{F}$ is closed according to Remark 2.

Step 4. The graph of $\mathbb{F}$ is closed.

Assume that $x_n\to x_{*},h_n\in \mathbb{F}{x}_n$ and that $h_n\to h_{*}.$ Our goal is to establish that $h_{*}\in \mathbb{F}{x}_{*}.$ Since $h_n\in \mathbb{F}{x}_n,$ there exists $v_n\in S_{F,x_n}$ such that, for every $t\in [a,b],$

$$\begin{array}{ccc}\hfill h_n\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_n\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_n\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_n\left(b\right)\right)\right].}\hfill \end{array}$$

We will show that there exists $v_{*}\in S_{F,x_{*}}$ such that, for each $t\in [a,b],$ we have

$$\begin{array}{ccc}\hfill h_{*}\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_{*}\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_{*}\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_{*}\left(b\right)\right)\right].}\hfill \end{array}$$

Let the linear operator $\mathcal{Z}:L^1([a,b],\mathbb{R})\to C([a,b],\mathbb{R})$ be given by

$$\begin{array}{ccc}\hfill v\mapsto \left(\mathcal{Z}v\right)\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)\right)\right].}\hfill \end{array}$$

Observe that $\parallel h_n-h_{*}\parallel \to 0,$ as $n\to \infty .$ Then, by Lemma 8, the operator $\mathcal{Z}\circ S_F$ has a closed graph. Moreover, we have $h_n\left(t\right)\in \mathcal{Z}\left(S_{F,x_n}\right).$ Since $x_n\to x_{*},$ therefore

$$\begin{array}{ccc}\hfill h_{*}\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_{*}\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_{*}\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_{*}\left(b\right)\right)\right],}\hfill \end{array}$$

for some $v_{*}\in S_{F,x_{*}}$.

Step 5. We aim to demonstrate the existence of an open set $U\subseteq C\left(\right[a,b],\mathbb{R}),$ such that, for every $x\in \partial U$ and any $\lambda \in (0,1)$, it holds that $x\notin \lambda \mathbb{F}x.$

Suppose $\lambda \in (0,1)$ and $x\in \lambda \mathbb{F}\left(x\right).$ Then, there exists some $v\in L^1([a,b],\mathbb{R})$ with $v\in S_{F,x}$ such that, for every $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill x\left(t\right)& =& \lambda {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+\lambda {\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +\lambda {{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)\hfill \\ & & {\displaystyle +\lambda \frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)\right)\right].}\hfill \end{array}$$

As in step 2, we have

$$\begin{array}{c}\hfill \left|x\left(t\right)\right|\le \parallel p\parallel {\mathsf{\Theta }}_0+(\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }(\parallel x\parallel \left)\right){\mathsf{\Theta }}_1,\end{array}$$

which implies that

$${\displaystyle \frac{\parallel x\parallel }{\parallel p\parallel {\mathsf{\Theta }}_0+\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }(\parallel x\parallel ){\mathsf{\Theta }}_1}}\le 1.$$

By $\left(H_3\right)$, $\parallel x\parallel \ne M$. Consider the set

$$U=\{x\in C([a,b],\mathbb{R}):\parallel x\parallel <M\},$$

and observe that $\mathbb{F}:\overline{U}\to \mathcal{P}\left(C([a,b],\mathbb{R})\right)$ is an upper semicontinuous, compact multivalued map with closed and convex values. By the definition of $U,$ there is no $x\in \partial U$ for some $\lambda \in (0,1),$ such that $x\in \lambda \mathbb{F}x.$ Applying the Leray–Schauder nonlinear alternative (Theorem 1), we conclude that the operator $\mathbb{F}$ has a fixed point $x\in \overline{U}$. Consequently, at least one solution to the sequential $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional multi-point boundary-value problem (1) exists on $[a,b],$ completing the proof. □

3.2. Existence Result via Martelli’s Fixed-Point Theorem

In this subsection we apply Martelli’s fixed-point theorem to establish our second existence result.

Theorem 3 

([32]). Let X be a Banach space and $T:X\to X$ be a condensing upper semicontinuous and acyclic-valued mapping. If the equation $x\in T\left(x\right)$ does not have any solution, then the set $D=\{x\in X:\lambda x\in T(x)for some\lambda >1\}$ is unbounded.

Remark 3. 

(i) An upper semicontinuous map $F:X⟶X$ is said to be condensing if for any subset $B\subseteq X$ with $\alpha \left(B\right)\ne 0$, we have $\alpha \left(F\right(B\left)\right)<\alpha \left(B\right)$, where α denotes the Kuratowski measure of noncompactness. For properties of the Kuratowski measure, we refer to Banas and Goebel [33].

(ii) 

A completely continuous operator is condensing [34].

(iii) 

A convex set satisfies the acyclicity property [35].

Hence, Martelli’s fixed-point theorem has the following easily applicable form.

Lemma 9 

([32]). Let X be a Banach space and $N:X⟶{\mathcal{P}}_{b,cl,c}\left(X\right)$ be a completely continuous multivalued map. If the set

$$\mathsf{\Omega }:=\{y\in X:\lambda y\in N(y)\mathit{for}\mathit{some}\lambda >1\}$$

is bounded, then N has a fixed point.

Theorem 4. 

Assume that $\left(H_1\right)$, $\left(H_4\right)$, and the following condition hold:

(H₅)

There exists a function $q\in C\left(\right[a,b],\mathbb{R})$ such that

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\left\{\right|z|:z\in F(t,x)\}\le q\left(t\right)for a.e.t\in [a,b]\mathit{and}\mathit{each}x\in \mathbb{R}.$$

Then, at least one solution to the problem (1) exists on $[a,b]$.

Proof. 

Employing the arguments used in the proof of Theorem 2, one can find that the operator $\mathbb{F}:C\left(\right[a,b],\mathbb{R})\to \mathcal{P}\left(C\right([a,b],$ $\mathbb{R}\left)\right)$, defined in the beginning of the proof of Theorem 2, is a compact, upper semicontinuous, multivalued mapping with closed and convex images. We now establish the boundedness of the set

$$\mathcal{V}=\{x\in C([a,b],\mathbb{R}):\lambda x\in \mathbb{F}(x),\lambda >1\}.$$

Let $x\in \mathcal{V}.$ Then there exists some $\lambda >1$ such that $\lambda x\in \mathbb{F}\left(x\right),$ and consequently, we can find a function $v\in S_{F,x}$ such that

$$\begin{array}{ccc}\hfill x\left(t\right)& =& {\displaystyle \frac{1}{\lambda }}({}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)\right)\right]).}\hfill \end{array}$$

As in step 2 of Theorem 2, it can be shown that $\parallel x\parallel \le \parallel p\parallel {\mathsf{\Theta }}_0+\parallel q\parallel {\mathsf{\Theta }}_1$ for each $t\in [a,b]$. Therefore, the set $\mathcal{V}$ is bounded. Hence, we deduce by Lemma 9 that the operator $\mathbb{F}$ has at least one fixed point. In consequence, there exists a solution to the problem (1) on $[a,b]$. □

3.3. Existence Results via Nonlinear Alternative for Contractive Maps

The following fixed-point theorem pertains to multivalued mappings and is commonly referred to as the nonlinear alternative for contractive maps ([36], Corollary 3.8).

Theorem 5 

([36]). Let $\mathcal{E}$ be a be bounded neighborhood of $0\in X,$ where X is a Banach space. Suppose that the multivalued operators $Z_1:X\to {\mathcal{P}}_{cp,c}\left(X\right)$ and $Z_2:\overline{\mathcal{E}}\to {\mathcal{P}}_{cp,c}\left(X\right)$ are such that the following hold:

(a) 

$Z_1$ is contraction;

(b) 

$Z_2$ is upper semicontinuous and compact.

Then, for the multivalued operator $G=Z_1+Z_2,$ either

(i) 

G has a fixed point in $\overline{\mathcal{E}}$, or

(ii) 

There exists a point $u\in \partial \mathcal{E}$ and a scalar $\lambda \in (0,1)$ such that $u\in \lambda G\left(u\right)$.

Theorem 6. 

Assume that $\left(H_1\right)\text{–}\left(H_5\right)$ and the following condition hold:

(H₆)

There exists a constant $L_g<{\mathsf{\Theta }}_0^{-1}$ such that

$$|g(t,x\left(t\right))-g(t,y\left(t\right))|\le L_g|x\left(t\right)-y\left(t\right)|,t\in [a,b],x,y\in \mathbb{R}.$$

Then, the problem (1) has at least one solution on $[a,b].$

Proof. 

Define an operator $\mathcal{A}:C\left(\right[a,b],\mathbb{R})⟶C\left(\right[a,b],\mathbb{R})$ by

$${\displaystyle \mathcal{A}x\left(t\right)={}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right],}$$

(10)

and a multivalued operator $\mathcal{B}:C\left(\right[a,b],\mathbb{R})⟶\mathcal{P}\left(C\right([a,b],\mathbb{R}\left)\right)$ by

$$\mathcal{B}\left(x\right)=\left\{\begin{array}{cc}\hfill & h\in C\left(\right[a,b],\mathbb{R}):\hfill \\ \hfill & h\left(t\right)=\left\{\begin{array}{c}{\displaystyle {{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)}\hfill \\ {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)}\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)}\right)\right].}\hfill \end{array}\right.\hfill \end{array}\right\}$$

(11)

Observe that $\mathbb{F}=\mathcal{A}+\mathcal{B}$, where $\mathbb{F}:C\left(\right[a,b],\mathbb{R})⟶\mathcal{P}\left(C\right([a,b],\mathbb{R}\left)\right)$ is defined in the beginning of the proof of Theorem 2. We verify that the operators $\mathcal{A}$ and $\mathcal{B}$ satisfy the hypothesis of Theorem 5 on $[a,b]$.

Step 1: We verify that $\mathcal{A}$ is a contraction on $C\left(\right[a,b],\mathbb{R})$. For $x,y\in C\left(\right[a,b],\mathbb{R}),$ we have

$$\begin{array}{ccc}\hfill \left|\mathcal{A}x\right(t)-\mathcal{A}y(t\left)\right|& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}|g(t,x\left(t\right))-g(t,y\left(t\right))|\hfill \\ & & +{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}|g({\xi }_i,x\left({\xi }_i\right))-g(\xi ,y\left(\xi \right))|\hfill \\ & & +{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}|g(b,x\left(b\right))-g(b,y\left(b\right))|]\hfill \\ & \le & L_g|x\left(t\right)-y\left(t\right)|\left\{\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_0+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\widehat{A_0}\right\}\hfill \\ & =& L_g{\mathsf{\Theta }}_0|x\left(t\right)-y\left(t\right)|,\hfill \end{array}$$

which, on taking the supremum for $t\in [a,b]$, yields

$$\parallel \mathcal{A}x-\mathcal{A}y\parallel \le L_g{\mathsf{\Theta }}_0\parallel x-y\parallel .$$

This shows that $\mathcal{A}$ is a contraction according to the given condition $L_g{\mathsf{\Theta }}_0<1$.

Step 2: As argued in Theorem 2, it can be established that the operator $\mathcal{B}$ is compact, convex-valued, and completely continuous.

In consequence, the operators $\mathcal{A}$ and $\mathcal{B}$ fulfill all the requirements of Theorem 5. Thus, by the conclusion of this theorem, either condition (i) or condition (ii) must hold. We will now prove that conclusion (ii) is not feasible. If $x\in \lambda \mathcal{A}\left(x\right)+\lambda \mathcal{B}\left(x\right)$ for some $\lambda \in (0,1),$ then there exists $v\in S_{F,x}$ such that

$$\begin{array}{ccc}\hfill x\left(t\right)& =& \lambda {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+\lambda {\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +\lambda {{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)\hfill \\ & & {\displaystyle +\lambda \frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)\right)\right],}\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel x\parallel \le \parallel p\parallel {\mathsf{\Theta }}_0+\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }(\parallel x\parallel ){\mathsf{\Theta }}_1.\end{array}$$

(12)

If condition (ii) of Theorem 5 is satisfied, then we can find $\lambda \in (0,1)$ and $x\in \partial B_{\rho },$ where $B_{\rho }=\{x\in C([0,1],\mathbb{R}):\parallel x\parallel \le \rho \}$ with $x=\lambda \mathbb{F}\left(x\right).$ Then, x is a solution to problem (1) with $\parallel x\parallel =M.$ (We can take a large enough ball so that (ii) does not occur.) Now, by inequality (12), we have

$${\displaystyle \frac{M}{\parallel p\parallel {\mathsf{\Theta }}_0+\parallel \mathsf{\Lambda }\parallel \mathsf{\Psi }\left(M\right){\mathsf{\Theta }}_1}}\le 1,$$

which contradicts $\left(H_3\right).$ Hence, it follows by Theorem 5 that the operator $\mathbb{F}$ has a fixed point in $[a,b]$. Therefore, there exists a solution to the problem (1) on $[a,b]$. This completes the proof. □

3.4. Existence Results via Krasnoselskii’s Multivalued Fixed-Point Theorem

Here, we prove our last existence result by applying the multivalued version of Krasnoselskii’s fixed-point theorem [37].

Lemma 10 

(Krasnoselskii’s multivalued fixed-point theorem [37]). Let $\mathcal{X}$ be a Banach space, $\mathcal{Y}\in {\mathcal{P}}_{b,cl,c}\left(\mathcal{X}\right)$ and $A,B:\mathcal{Y}\to {\mathcal{P}}_{cp,c}\left(\mathcal{X}\right)$ be two multivalued operators. If (i) $Ay+By\subset \mathcal{Y}$ for all $y\in \mathcal{Y}$, (ii) A is contraction, and (iii) B is u.s.c and compact, then there exists $y\in \mathcal{Y}$ such that $y\in Ay+By$.

Theorem 7. 

Assume that $\left(H_1\right),\mathrm{ }$$\left(H_4\right),\mathrm{ }$$\left(H_5\right)$, and $\left(H_6\right)$ hold. Then, the sequential $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional multi-point boundary-value problem (1) has at least one solution on $[a,b].$

Proof. 

We will demonstrate that the operators $\mathcal{A}$ and $\mathcal{B}$ defined in (10) and (11), respectively, satisfy the hypotheses of Theorem 10 on the interval $[a,b]$ through several steps. Let us first show that $\mathcal{A},\mathcal{B}:B_r\to {\mathcal{P}}_{cp,c}\left(C([a,b],\mathbb{R})\right)$ determine the multivalued mappings, where $B_r=\{x\in C([a,b],\mathbb{R}):|x|\le r\}$ represents a bounded subset of $C\left(\right[a,b],\mathbb{R})$. We will focus on establishing that the operator $\mathcal{B}$ has compact values on $B_r$ and it is convex for every $x\in C\left(\right[a,b],\mathbb{R})$. Observe that the operator $\mathcal{B}$ can be expressed as the composition $\mathcal{L}\circ S_F$, where $\mathcal{L}$ denotes the continuous linear operator mapping $L^1([a,b],\mathbb{R})$ into $C\left(\right[a,b],\mathbb{R})$, defined by

$$\begin{array}{ccc}\hfill \mathcal{L}\left(v\right)\left(t\right)& =& {{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)}\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)}\right)\right].}\hfill \end{array}$$

Assume that $x\in B_r$ is arbitrary, and consider a sequence $\left\{v_n\right\}$ in $S_{F,x}$. By the definition of $S_{F,x}$, it follows that $v_n\left(t\right)\in F(t,x\left(t\right))$ for almost every $t\in [a,b]$. Since $F(t,x(t\left)\right)$ is compact for each $t\in [a,b]$, we can extract a subsequence from $\left\{v_n\left(t\right)\right\}$ (which we still denote by $\left\{v_n\left(t\right)\right\}$ for simplicity) that converges in measure to some $v\left(t\right)\in S_{F,x}$ for almost every $t\in [a,b]$. Moreover, continuity of the operator $\mathcal{L}$ implies that $\mathcal{L}\left(v_n\right)\left(t\right)\to \mathcal{L}\left(v\right)\left(t\right)$ pointwise on $[a,b]$.

To establish that the convergence is uniform, it suffices to prove that the sequence $\{\mathcal{L}\left(v_n\right)\}$ is equicontinuous. Take any $t_1,t_2\in [a,b]$ with $t_1<t_2$. Then, we have

$$\begin{array}{ccc}& & |\mathcal{L}\left(v_n\right)\left(t_2\right)-\mathcal{L}\left(v_n\right)\left(t_1\right)|\hfill \\ & \le & |{\displaystyle \frac{1}{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_2}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}{v}_n\left(\tau \right)d\tau ds\hfill \\ & & -{\displaystyle \frac{1}{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_1}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_1\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}{v}_n\left(\tau \right)d\tau ds|\hfill \\ & \le & {\displaystyle \frac{\parallel q\parallel }{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_a^{t_1}{\psi }_2^{\prime }\left(s\right)\left|{\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}-{\left({\psi }_2\left(t_1\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\right|\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}d\tau ds\hfill \\ & & +{\displaystyle \frac{\parallel q\parallel }{k_1{k}_2{\mathsf{\Gamma }}_{k_1}\left({\alpha }_1\right){\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)}}{\int }_{t_1}^{t_2}{\psi }_2^{\prime }\left(s\right){\left({\psi }_2\left(t_2\right)-{\psi }_2\left(s\right)\right)}^{{\displaystyle \frac{{\alpha }_2}{k_2}}-1}\hfill \\ & & \times {\int }_a^s{\psi }_1^{\prime }\left(\tau \right){\left({\psi }_1\left(s\right)-{\psi }_1\left(\tau \right)\right)}^{{\displaystyle \frac{{\alpha }_1}{k_1}}-1}d\tau ds.\hfill \end{array}$$

We observe that the right-hand side of the above inequality approaches zero as $t_2\to t_1$. Therefore, the sequence $\left\{\mathcal{L}\left(v_n\right)\right\}$ is equicontinuous. By the Arzelà–Ascoli theorem, it follows that there exists a uniformly convergent subsequence, which we continue to denote by $\left\{v_n\right\}$, such that $\mathcal{L}\left(v_n\right)\to \mathcal{L}\left(v\right)$. Note that $\mathcal{L}\left(v\right)$ belongs to $\mathcal{L}\left(S_{F,x}\right)$. Consequently, we conclude that $\mathcal{B}\left(x\right)=\mathcal{L}\left(S_{F,x}\right)$ is compact for every $x\in B_r$. Thus, $\mathcal{B}\left(x\right)$ is compact.

To prove that $\mathcal{B}\left(x\right)$ is convex for all $x\in C\left(\right[a,b],\mathbb{R})$, let us take $z_1,z_2\in \mathcal{B}\left(x\right)$ and choose $v_1,v_2\in S_{F,x}$ such that

$$\begin{array}{ccc}\hfill z_j\left(t\right)& =& {{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_j\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_j\left({\xi }_i\right)}\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_j\left(b\right)}\right)\right],j=1,2,}\hfill \end{array}$$

for almost all $t\in [a,b]$. Let $0\le \omega \le 1$. Then, we have

$$\begin{array}{ccc}\hfill [\omega z_1+(1-\omega )z_2]\left(t\right)& =& {{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}[\omega v_1\left(s\right)+(1-\omega )v_2\left(s\right)]\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}[\omega v_1\left(s\right)+(1-\omega )v_2\left(s\right)]\left({\xi }_i\right)}\right)}\hfill \\ & & -{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}[\omega v_1\left(s\right)+(1-\omega )v_2\left(s\right)]\left(b\right)}\right)].\hfill \end{array}$$

Since F has convex values, so $S_{F,x}$ is convex and $\omega v_1\left(s\right)+(1-\omega )v_2\left(s\right)\in S_{F,x}$. Thus, $\omega z_1+(1-\omega )z_2\in \mathcal{B}\left(x\right)$. Consequently, $\mathcal{B}$ is convex-valued.

Next, we verify that $\mathcal{A}\left(y\right)+\mathcal{B}\left(y\right)\subset {\mathbb{B}}_{r_0}$ for all $y\in {\mathbb{B}}_{r_0}$ with $r_0>\parallel p\parallel {\mathsf{\Theta }}_0+\parallel q\parallel {\mathsf{\Theta }}_1.$ Letting $y\in {\mathbb{B}}_{r_0}$ and $h\in \mathcal{B}$, we choose $v\in S_{F,y}$ such that

$$\begin{array}{ccc}\hfill h\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right)\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)\right)\right].}\hfill \end{array}$$

As in step 2 of Theorem 2, for each $t\in [a,b]$, it can be established that

$$\parallel h\parallel \le \parallel p\parallel {\mathsf{\Theta }}_0+\parallel q\parallel {\mathsf{\Theta }}_1<r_0,$$

which implies that $\parallel h\parallel \le r_0.$ Therefore, $\mathcal{A}\left(y\right)+\mathcal{B}\left(y\right)\subset {\mathbb{B}}_{r_0}$ for all $y\in {\mathbb{B}}_{r_0}.$ By using the condition $\left(H_6\right)$, it is easy to verify that the operator $\mathcal{A}$ is a contraction. As in steps 3 and 4 of Theorem 2, we can obtain that $\mathcal{B}$ is u.s.c. and compact. Thus, the operators $\mathcal{A}$ and $\mathcal{B}$ satisfy the hypothesis of Lemma 10 and hence we deduce by its conclusion that $y\in \mathcal{A}\left(y\right)+\mathcal{B}\left(y\right)$ in ${\mathbb{B}}_{r_0}.$ In consequence, there exists a solution to the problem (1) in ${\mathbb{B}}_{r_0}$ and the proof is complete. □

4. Existence Results—The Lower Semicontinuous Case

Let $F:X\to \mathcal{P}\left(E\right)$ be a multivalued operator with nonempty closed values, where X is a nonempty closed subset of a Banach space E. The operator F is lower semicontinuous (l.s.c.) if the set $\{y\in X:F(y)\cap M\ne \varnothing \}$ is open for any open set M in E.

Consider a subset A of $J\times \mathbb{R}$. The set A is said to be $\mathcal{L}\otimes \mathcal{W}$-measurable if it belongs to the $\sigma$–algebra generated by all sets of the form $\mathcal{J}\times \mathcal{D}$, where $\mathcal{J}$ is a Lebesgue measurable subset of J and $\mathcal{D}$ is a Borel-measurable subset of $\mathbb{R}$. A subset $\mathcal{A}$ of $L^1(J,\mathbb{R})$ is said to be decomposable if, for any $x,y\in \mathcal{A}$ and any measurable subset $\mathcal{J}\subset J$, the function $x{\chi }_{\mathcal{J}}+y{\chi }_{J-\mathcal{J}}$ is also contained in $\mathcal{A}$, where ${\chi }_{\mathcal{J}}$ represents the characteristic function of $\mathcal{J}$.

Now, let us consider a set-valued mapping $F:J\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ with nonempty compact images. We define the corresponding Nemytskii operator $\mathcal{F}:C(J\times \mathbb{R})\to \mathcal{P}\left(L^1(J,\mathbb{R})\right)$ by

$$\mathcal{F}\left(x\right)=\{w\in L^1(J,\mathbb{R}):w\left(t\right)\in F(t,x\left(t\right))\mathrm{for}\mathrm{ }\mathrm{a.e.}t\in J\}.$$

Definition 7 

([38]). Let $F:J\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ be a multivalued function taking nonempty compact values. We say that F is of lower semicontinuous type (l.s.c. type) if its associated Nemytskii operator $\mathcal{F}$ is lower semicontinuous and its values are nonempty, closed, and decomposable.

Theorem 8 

([38]). Let Y be a separable metric space, and let $N:Y\to \mathcal{P}\left(L^1(J,\mathbb{R})\right)$ be a multivalued mapping that is lower semicontinuous, with nonempty, closed, and decomposable values. Then, N admits a continuous selection, that is, there exists a continuous (single-valued) function $\varphi :Y\to L^1(J,\mathbb{R})$ such that $\varphi \left(x\right)\in N\left(x\right)$ for every $x\in Y.$

In the following result, we combine the nonlinear alternative of the Leray–Schauder type with Bressan and Colombo’s fixed-point theorem (Theorem 8) to study the case when the mapping F has non-convex values.

Theorem 9. 

Suppose that $\left(H_2\right)$, $\left(H_3\right),$ $\left(H_4\right)$, and the following condition hold:

(H₇)

$F:[a,b]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is a nonempty compact-valued multivalued map such that

(a) 

$(t,x)⟼F(t,x)$ is $\mathcal{L}\otimes \mathcal{W}$-measurable;

(b) 

$x⟼F(t,x)$ is lower semicontinuous for each $t\in [a,b].$

Then, at least one solution to the sequential $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional multi-point boundary-value problem (1) exists on $[a,b].$

Proof. 

From $\left(H_2\right)$ and $\left(B_1\right)$, it is easy to see that F is of l.s.c. type ([39]). Then, by Theorem 8, there exists a continuous function $f:C([a,b],\mathbb{R})\to L^1([a,b],\mathbb{R})$ such that $f\left(x\right)\in \mathcal{F}\left(x\right)$ for all $x\in C\left(\right[a,b],\mathbb{R})$.

Consider the problem

$$\left\{\begin{array}{c}{\displaystyle {}^{k_1,H}D_{a^{+}}^{{\alpha }_1,\beta ;{\psi }_1}\left({}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(t\right)-g(t,x\left(t\right)\right)=f\left(x\left(t\right)\right),t\in (a,b),}\hfill \\ {\displaystyle {}^{k_2,C}D_{a^{+}}^{{\alpha }_2;{\psi }_2}x\left(a\right)=0,x\left(b\right)={\displaystyle \sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\phi }_i,{\psi }_2}x\left({\xi }_i\right)}.}\hfill \end{array}\right.$$

(13)

Note that $x\in C^2([a,b],\mathbb{R})$ satisfying problem (13) also solves problem (1). We reformulate (13) as a fixed-point problem by introducing an operator $\overline{\mathcal{F}}$ as

$$\begin{array}{ccc}\hfill \overline{\mathcal{F}}x\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}f\left(x\left({\xi }_i\right)\right)}\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}f(x\left(\left(b\right)\right)}\right)\right]}\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}f\left(x\left(t\right)\right)}\right),t\in [a,b].\hfill \end{array}$$

It can then be readily verified that $\overline{\mathcal{F}}$ is both continuous and completely continuous. The rest of the proof follows the same reasoning as Theorem 2, so we omit the details. This concludes the proof. □

5. Existence Results—The Lipschitz Case

In this section, we apply Covitz and Nadler’s fixed-point theorem for multivalued operators [40] to prove an existence result for problem (1) when the right-hand side of the inclusion is not necessarily convex-valued.

Definition 8 

([14]). Let $(X,d)$ be a metric space derived from a normed space $(X;\parallel ·\parallel )$ and define the Hausdorff distance $H_d:\mathcal{P}\left(X\right)\times \mathcal{P}\left(X\right)\to \mathbb{R}\cup \{\infty \}$ by

$$H_d(U,V)=max\{\underset{u\in U}{sup}d(u,V),\underset{v\in V}{sup}d(U,v)\},$$

where $d(U,v)={inf}_{u\in U}d(u,v)$ and $d(u,V)={inf}_{v\in V}d(u,v)$. Then, $({\mathcal{P}}_{b,cl}\left(X\right),H_d)$ forms a metric space, while $({\mathcal{P}}_{cl}\left(X\right),H_d)$ constitutes a generalized metric space (see [14]).

Definition 9 

([40]). A multivalued mapping $N:X\to {\mathcal{P}}_{cl}\left(X\right)$ is said to be

(a) 

$\gamma$–Lipschitz if there exists a constant $\gamma >0$ such that

$$H_d(N\left(x\right),N\left(y\right))\le \gamma d(x,y)for allx,y\in X;$$

(b) 

A contraction if it is $\gamma$–Lipschitz with $\gamma <1$.

Lemma 11 

(Covitz and Nadler’s fixed-point theorem [40]). Let $(X,d)$ be a complete metric space. If $N:X\to {\mathcal{P}}_{cl}\left(X\right)$ is a multivalued contraction, then the set of fixed points of N, denoted by $FixN$, is nonempty.

Theorem 10. 

Suppose that $\left(H_6\right)$ and the following conditions are satisfied:

(H₈)

For the multifunction $F:[a,b]\times \mathbb{R}\to {\mathcal{P}}_{cp}\left(\mathbb{R}\right)$, the mapping $t\to F(t,x)$ is measurable on $[a,b]$ for every $x\in \mathbb{R};$

(H₉)

There exists a continuous, nonnegative function $\mu \in C([a,b],{\mathbb{R}}^{+})$ such that, for almost every $t\in [a,b]$ and $x,\overline{x}\in \mathbb{R},$ the Hausdorff distance satisfies

$$H_d(F(t,x),F(t,\overline{x}))\le \mu \left(t\right)|x-\overline{x}|,$$

with $d(0,F(t,0\left)\right)\le \mu \left(t\right)$ almost everywhere on $[a,b]$.

Then, at least one solution to the sequential fractional multi-point boundary-value problem involving the $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo operators (1) exists on $[a,b],$ provided that

$$L_g{\mathsf{\Theta }}_0+\parallel \mu \parallel {\mathsf{\Theta }}_1<1,$$

(14)

where the constants ${\mathsf{\Theta }}_i,i=0,1$ are given in Lemma 7.

Proof. 

We will demonstrate that the operator $\mathbb{F}:C\left(\right[a,b],\mathbb{R})⟶\mathcal{P}\left(C\right([a,b],\mathbb{R}\left)\right),$ defined at the beginning of the proof of Theorem 2, satisfies the hypothesis of Lemma 11.

By the measurable selection theorem (see Theorem III.6 in [29]), the multivalued map F is measurable, implying the existence of a measurable selection. Using condition $\left(H_9\right),$ we obtain $\left|v\right(t\left)\right|\le \mu \left(t\right)(1+|x\left(t\right)\left|\right)$, which ensures that $v\in L^1([a,b],\mathbb{R}),$ meaning that F is integrably bounded. Consequently, $S_{F,x}\ne \varnothing .$

Let ${\left\{u_n\right\}}_{n\ge 0}\in \mathbb{F}\left(x\right)$ with $u_n\to u(n\to \infty )$ in $C\left(\right[a,b],\mathbb{R}).$ Then $u\in C\left(\right[a,b],\mathbb{R})$ and we can find $v_n\in S_{F,x_n}$ such that, for each $t\in [a,b]$,

$$\begin{array}{ccc}\hfill u_n\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_n\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_n\left(b\right)\right)\right]}\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_n\left(t\right)}\right).\hfill \end{array}$$

Since F has compact values, we can find a subsequence (if necessary) $v_n$ converging to v in $L^1([a,b],\mathbb{R}).$ Thus, $v\in S_{F,x}$ and for each $t\in [a,b]$, we have

$$\begin{array}{ccc}\hfill u_n\left(t\right)\to u\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(b\right)\right)\right]}\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}v\left(t\right)}\right).\hfill \end{array}$$

Thus, $u\in \mathbb{F}\left(x\right).$

We now aim to show that there exists a constant $\delta$ (specifically, $\delta :=L_g{\mathsf{\Theta }}_0+\parallel \mu \parallel {\mathsf{\Theta }}_1<1)$ satisfying

$$H_d(\mathbb{F}\left(x\right),\mathbb{F}\left(\widehat{x}\right))\le \delta \parallel x-\widehat{x}\parallel \mathrm{for} \mathrm{all}x,\widehat{x}\in C([a,b],\mathbb{R}).$$

Let $x,\widehat{x}\in C([a,b],\mathbb{R})$ and $h_1\in \mathbb{F}\left(x\right)$. Then, there exists a function $v_1\left(t\right)\in F(t,x\left(t\right))$ such that, for each $t\in [a,b]$,

$$\begin{array}{ccc}\hfill h_1\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,x\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,x\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,x\left(b\right))\right]\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_1\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_1\left(b\right)\right)\right]}\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_1\left(t\right)}\right).\hfill \end{array}$$

By $\left(H_9\right)$, we have

$$H_d(F(t,x),F(t,\widehat{x}))\le \mu \left(t\right)|x\left(t\right)-\widehat{x}\left(t\right)|.$$

Hence, there exists $z\in F(t,\widehat{x}\left(t\right))$ such that

$$|v_1\left(t\right)-z|\le \mu \left(t\right)|x\left(t\right)-\widehat{x}\left(t\right)|,t\in [a,b].$$

Define $W:[a,b]\to \mathcal{P}\left(\mathbb{R}\right)$ by

$$W\left(t\right)=\{t\in \mathbb{R}:|v_1\left(t\right)-z|\le \mu \left(t\right)|x\left(t\right)-\widehat{x}\left(t\right)\left|\right\}.$$

Hence, there exists a function $v_2\left(t\right)$ that serves as a measurable selection for $W,$ due to the fact that the multivalued mapping $W\left(t\right)\cap F(t,\widehat{x}\left(t\right))$ is measurable (see Proposition III.4 [29]). So, $v_2\left(t\right)\in F(t,\widehat{x}\left(t\right)),$ and the inequality $|v_1\left(t\right)-v_2\left(t\right)|\le \mu \left(t\right)|x\left(t\right)-\widehat{x}\left(t\right)|$ holds for every $t\in [a,b]$. Now, for each $t\in [a,b]$, we define

$$\begin{array}{ccc}\hfill h_2\left(t\right)& =& {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(t,\widehat{x}\left(t\right))+{\displaystyle \frac{1}{\mathsf{\Delta }}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}g({\xi }_i,\widehat{x}\left({\xi }_i\right))-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}g(b,\widehat{x}\left(b\right))\right]\hfill \\ & & {\displaystyle +\frac{1}{\mathsf{\Delta }}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_2\left({\xi }_i\right)\right)-{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_2\left(b\right)\right)\right]}\hfill \\ & & +{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}{v}_2\left(t\right)}\right).\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{ccc}& & |h_1\left(t\right)-h_2\left(t\right)|\hfill \\ & \le & {}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}\left|g\right(t,x\left(t\right)-g(t,\widehat{x}\left(t\right))|\hfill \\ & & +{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\left[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\varphi }_i+{\alpha }_2;{\psi }_2}|g({\xi }_i,x\left({\xi }_i\right))-g({\xi }_i,\widehat{x}\left({\xi }_i\right))|-{}^{k_2}I_{a^{+}}^{{\alpha }_2;{\psi }_2}|g(b,x\left(b\right))-g(b,\widehat{x}\left(b\right))|\right]\hfill \\ & & {\displaystyle +\frac{1}{\left|\mathsf{\Delta }\right|}[\sum _{i=1}^{\ell }{\eta }_i{}^{k_2}I_{a^{+}}^{{\alpha }_2+{\phi }_i,{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left(\right|v_1\left(s\right)-v_2\left(s\right)\left|\right)\left({\xi }_i\right)\right)}\hfill \\ & & -{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left(\right|v_1\left(s\right)-v_2\left(s\right)\left|\right)\left(b\right)\right)]+{{}^{k_2}I}_{a^{+}}^{{\alpha }_2;{\psi }_2}\left({\displaystyle {{}^{k_1}I}_{a^{+}}^{{\alpha }_1;{\psi }_1}\left(\right|v_1\left(s\right)-v_2\left(s\right)\left|\right)\left(t\right)}\right)\hfill \\ & \le & L_g|x\left(t\right)-\widehat{x}\left(t\right)|\left\{\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_0+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\widehat{A_0}\right\}\hfill \\ & & +\parallel \mu \parallel |x\left(t\right)-y\left(t\right)|\left\{\left(1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\right)A_1+{\displaystyle \frac{1}{\left|\mathsf{\Delta }\right|}}\sum _{i=1}^{\ell }{\eta }_i{A}_{1i}\right\}\hfill \\ & \le & \left(L_g{\mathsf{\Theta }}_0+\parallel \mu \parallel {\mathsf{\Theta }}_1)\right)\parallel x-\widehat{x}\parallel .\hfill \end{array}$$

Therefore,

$$\parallel h_1-h_2\parallel \le \left(L_g{\mathsf{\Theta }}_0+\parallel \mu \parallel {\mathsf{\Theta }}_1)\right)\parallel x-\widehat{x}\parallel .$$

Reversing the roles of x and $\widehat{x}$, we get

$$H_d(\mathbb{F}\left(x\right),\mathbb{F}\left(\overline{x}\right))\le \left(L_g{\mathsf{\Theta }}_0+\parallel \mu \parallel {\mathsf{\Theta }}_1)\right)\parallel x-\widehat{x}\parallel .$$

Thus, $\mathbb{F}$ is a contraction by (14). Consequently, there exists a fixed point x for $\mathbb{F}$ by Lemma 11. Therefore, a solution to problem (1) exists and the proof is finished. □

6. Examples

Example 1. 

Consider the following sequential $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional differential inclusions with multi-point boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle {}^{\frac{2}{3},H}D_{\frac{1}{5}}^{\frac{7}{13},\frac{1}{2};t^2+1}\left({}^{\frac{8}{11},C}D_{\frac{1}{5}}^{\frac{9}{13};{log}_e(t+1)}x\left(t\right)-g(t,x\left(t\right))\right)\in F(t,x\left(t\right)),t\in \left[\frac{1}{5},\frac{12}{5}\right],}\hfill \\ {\displaystyle {}^{\frac{8}{11},C}D_{\frac{1}{5}}^{\frac{9}{13};{log}_e(t+1)}x\left(\frac{1}{5}\right)=0,x\left(\frac{12}{5}\right)=\frac{1}{\sqrt{23}}{}^{\frac{8}{11}}I_{\frac{1}{5}}^{\frac{1}{3},{log}_e(t+1)}x\left(\frac{4}{5}\right)+\frac{1}{\sqrt{27}}{}^{\frac{8}{11}}I_{\frac{1}{5}}^{\frac{2}{3},{log}_e(t+1)}x\left(\frac{7}{5}\right)}\hfill \\ {\displaystyle +\frac{1}{\sqrt{29}}{}^{\frac{8}{11}}I_{\frac{1}{5}}^{\frac{5}{3},{log}_e(t+1)}x\left(\frac{11}{5}\right).}\hfill \end{array}\right.\end{array}$$

(15)

Comparing (15) with (1), we have $k_1=2/3$, ${\alpha }_1=7/13$, $\beta =1/2$, ${\psi }_1=t^2+1$, $k_2=8/11$, ${\alpha }_2=9/13$, ${\psi }_2={log}_e(t+1)$, $a=1/5$, $b=12/5$, ${\eta }_1=1/\sqrt{23}$, ${\eta }_2=1/\sqrt{27}$, ${\eta }_3=1/\sqrt{29}$, ${\phi }_1=1/3$, ${\phi }_2=2/3$, ${\phi }_3=5/3$, ${\xi }_1=4/5$, ${\xi }_2=7/5$, ${\xi }_3=11/5$, $\ell =3$. The relation ${\mathsf{\Gamma }}_k\left(z\right)=k^{{\displaystyle \frac{z}{k}}-1}\mathsf{\Gamma }(z/k)$ yields ${\mathsf{\Gamma }}_{k_1}({\alpha }_1+k_1)\approx 2.814004879$, ${\mathsf{\Gamma }}_{k_2}\left({\alpha }_2\right)\approx 1.046036330$, ${\mathsf{\Gamma }}_{k_2}({\alpha }_2+k_2)\approx 0.7241789981$, ${\mathsf{\Gamma }}_{k_2}({\phi }_1+k_2)\approx 0.7653370305$, ${\mathsf{\Gamma }}_{k_2}({\phi }_2+k_2)\approx 0.7226223046$, ${\mathsf{\Gamma }}_{k_2}({\phi }_3+k_2)\approx 1.282354117$, ${\mathsf{\Gamma }}_{k_2}({\alpha }_2+{\phi }_1)\approx 0.7781496433$, ${\mathsf{\Gamma }}_{k_2}({\alpha }_2+{\phi }_2)\approx 0.7214905589$, ${\mathsf{\Gamma }}_{k_2}({\alpha }_2+{\phi }_3)\approx 1.239627128$, ${\mathsf{\Gamma }}_{k_2}({\phi }_1+{\alpha }_2+k_2)\approx 0.7981021981$, ${\mathsf{\Gamma }}_{k_2}({\phi }_2+{\alpha }_2+k_2)\approx 0.9804871698$, and ${\mathsf{\Gamma }}_{k_2}({\phi }_3+{\alpha }_2+k_2)\approx 2.924248609$. Using the foregoing values, we find that $\mathsf{\Delta }\approx 0.4910158029$, $A_0\approx 1.435311074$, $\widehat{A_0}\approx 0.5608666437$, ${\mathsf{\Theta }}_0\approx 5.500715295$. $A_1\approx 0.7117780817$, $A_{11}\approx 0.02770136184$, $A_{21}\approx 0.06958600105$, $A_{31}\approx 0.04306515146$, and ${\mathsf{\Theta }}_1\approx 2.216705297$.

$\left(i\right)$ Let us consider the map

$$F(t,x)=\left[{\displaystyle \frac{1}{9t+8}}\left(e^{-x^4}{sin}^{2}x\right),{\displaystyle \frac{1}{5t+7}}\left({\displaystyle \frac{{\left|x\right|}^{2025}}{1+{\left|x\right|}^{2023}}}+e^{-x^2}\right)\right]$$

(16)

and

$$g(t,x)={\displaystyle \frac{{cos}^{2}x}{25t^2+8}}.$$

(17)

Now, for $f\in F$, we obtain

$$\begin{array}{ccc}\hfill \left|f\right|& \le & max\left({\displaystyle \frac{1}{9t+8}}\left(e^{-x^4}{sin}^{2}x\right),{\displaystyle \frac{1}{5t+7}}\left({\displaystyle \frac{{\left|x\right|}^{2025}}{1+{\left|x\right|}^{2023}}}+e^{-x^2}\right)\right)\hfill \\ & \le & \left({\displaystyle \frac{1}{5t+7}}\right)\left({\parallel x\parallel }^2+1\right),t\in \left[{\displaystyle \frac{1}{5}},{\displaystyle \frac{12}{5}}\right],x\in \mathbb{R}.\hfill \end{array}$$

Thus, we obtain

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\left\{\right|x|:x\in F(t,x)\}\le \left({\displaystyle \frac{1}{5t+7}}\right)\left({\parallel x\parallel }^2+1\right):=\mathsf{\Lambda }\left(t\right)\mathsf{\Psi }(\parallel x\parallel ),$$

where $\mathsf{\Lambda }\left(t\right)=1/(5t+7)$ and $\mathsf{\Psi }\left(x\right)=x^2+1$. Furthermore, the bound of g is given by

$$\left|g(t,x)\right|\le {\displaystyle \frac{1}{25t^2+8}}:=p\left(t\right).$$

Then, we get $\parallel \mathsf{\Lambda }\parallel =1/8$ and $\parallel p\parallel =1/9$. Therefore, there exists a constant

$$M\in (1.580013170,2.028946492)$$

satisfying inequality in $\left(H_3\right)$. Hence, from Theorem 2, it follows that problem (15) with F and g given in (16) and (17), respectively, admits at least one solution on the interval $[1/5,12/5]$.

$\left(ii\right)$ If the multi-map $F:[1/5,12/5]\times \mathbb{R}\to \mathcal{P}\left(\mathbb{R}\right)$ is defined by

$$x\to F(t,x)=\left[{\displaystyle \frac{e^{-x^6}{sin}^{8}x}{1+t^2}}+{\displaystyle \frac{1}{2}},{\displaystyle \frac{x^{16}}{1+x^{16}}}+{\displaystyle \frac{3}{2}}(t^2+1)\right]$$

(18)

and

$$g(t,x)={\displaystyle \frac{{tan}^{-1}{x}^{12}}{1+\sqrt{t}}}+{\displaystyle \frac{1}{3}}.$$

(19)

Then, we obtain

$${\parallel F(t,x)\parallel }_{\mathcal{P}}:=sup\left\{\right|x|:x\in F(t,x)\}\le {\displaystyle \frac{3}{2}}{t}^2+{\displaystyle \frac{5}{2}}:=q\left(t\right)$$

and

$$\left|g(t,x)\right|\le {\displaystyle \frac{\pi }{2(1+\sqrt{t})}}:=p\left(t\right).$$

Therefore, all the assumptions of Theorem 4 are satisfied. Consequently, we can conclude that the boundary-value problem (15) with functions F and g, given by (18) and (19), respectively, has at least one solution on the interval $[1/5,12/5]$.

$\left(iii\right)$ Let $F(t,x)$ and $g(t,x)$ be given by

$$F(t,x)=\left[0,{\displaystyle \frac{(5t+3)|sinx|+t}{60(10t+3)}}\right],$$

(20)

$$g(t,x)={\displaystyle \frac{4}{{(5t+4)}^2}}\left({\displaystyle \frac{\left|x\right|}{1+\left|x\right|}}\right)+{\displaystyle \frac{1}{4}}.$$

(21)

Then, we have

$$H_d\left(F(t,x),F(t,\widehat{x})\right)\le {\displaystyle \frac{5t+3}{60(10t+3)}}|x-\widehat{x}|.$$

We choose $\mu \left(t\right)=(5t+3)/\left(60\right(10t+3\left)\right)$, and observe that

$${\displaystyle d(0,F(t,0))=\frac{t}{60(10t+3)}<\frac{5t+3}{60(10t+3)}=\mu \left(t\right)}$$

for almost all $t\in [1/5,12/5]$ and $\parallel \mu \parallel =1/75.$ For the given function g, we note that

$$|g(t,x)-g(t,y)|\le {\displaystyle \frac{4}{{(5t+4)}^2}}|x-y|\le {\displaystyle \frac{4}{25}}|x-y|,$$

with $L_g=4/25$. Since $L_g{\mathsf{\Theta }}_0+\parallel \mu \parallel {\mathsf{\Theta }}_1\approx 0.909670517<1$, we deduce that there exists at least one solution to the problem (15), with F and g given in (20) and (21), respectively, on the interval $[1/5,12/5]$, according to Theorem 10.

7. Conclusions

A boundary-value problem for fractional differential inclusions involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators subject to $(k_2,{\psi }_2)$-Riemann–Liouville-type integral multistrip boundary conditions is analyzed in this article. Applying various fixed-point theorems for convex and non-convex multivalued maps, we presented different criteria for the existence of solutions for the problem at hand. For the upper semicontinuous case, we utilized the Leray–Schauder nonlinear alternative for multivalued maps, Martell’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem, assuming that the multivalued map had convex values and satisfies the $L^1$-Carathéodory conditions. In the lower semicontinuous case, our approach was based on the Leray–Schauder-type nonlinear alternative for single-valued maps, combined with the Bressan–Colombo selection theorem for lower semicontinuous maps with decomposable values. In the Lipschitz case, we applied the Covitz–Nadler fixed-point theorem for contractive multivalued maps. The application of the obtained abstract results has been demonstrated through illustrative examples. Our results are novel, significant, and useful in the given configuration as several new results for a variety of fractional boundary-value problems involving different fractional derivative and integral operators appear as special cases, as explained in the paragraph following the formulation of the problem (1). In future, we plan to study the present fractional differential inclusions supplemented with nonlocal multi-point, fractional-integral multistrip and fractional integro-differential boundary conditions. Moreover, the present investigation will also be extended to a coupled-system variant of the problem at hand.