*Article* **On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions**

**Muath Awadalla 1,\*, Muthaiah Subramanian 2, Kinda Abuasbeh <sup>1</sup> and Murugesan Manigandan <sup>3</sup>**


**Abstract:** In this study, we examine the existence and Hyers–Ulam stability of a coupled system of generalized Liouville–Caputo fractional order differential equations with integral boundary conditions and a connection to Katugampola integrals. In the first and third theorems, the Leray–Schauder alternative and Krasnoselskii's fixed point theorem are used to demonstrate the existence of a solution. The Banach fixed point theorem's concept of contraction mapping is used in the second theorem to emphasise the analysis of uniqueness, and the results for Hyers–Ulam stability are established in the next theorem. We establish the stability of Ulam–Hyers using conventional functional analysis. Finally, examples are used to support the results. When a generalized Liouville–Caputo (*ρ*) parameter is modified, asymmetric results are obtained. This study presents novel results that significantly contribute to the literature on this topic.

**Keywords:** generalized fractional derivatives; generalized fractional integrals; coupled system; existence; fixed point

**MSC:** 34A08; 34B10; 34D10

#### **1. Introduction**

We consider the nonlinear coupled fractional differential equations with generalized Liouville–Caputo derivatives

$$\begin{cases} \,^{\rho}\_{\mathbb{C}} \mathcal{D}^{\sharp}\_{0^{+}} p(\tau) = f(\tau, p(\tau), q(\tau)), \tau \in \mathcal{G} := [0, \mathcal{T}],\\ \,^{\rho}\_{\mathbb{C}} \mathcal{D}^{\sharp}\_{0^{+}} q(\tau) = g(\tau, p(\tau), q(\tau)), \tau \in \mathcal{G} := [0, \mathcal{T}], \end{cases} \tag{1}$$

enhanced with boundary conditions which are defined by:

$$\begin{cases} p(0) = 0, \quad q(0) = 0, \\ p(T) = \varepsilon^{\rho} \mathcal{I}\_{0^{+}}^{\xi} q(\varpi) = \frac{\varepsilon \rho^{1-\zeta}}{\Gamma(\xi)} \int\_{0}^{\mathcal{Q}} \frac{\theta^{\rho-1}}{(\varpi^{\rho} - \theta^{\rho})^{1-\zeta}} q(\theta) d\theta, \\ q(T) = \pi^{\rho} \mathcal{I}\_{0+}^{\varrho} p(\sigma) = \frac{\tau \rho^{1-\varrho}}{\Gamma(\varrho)} \int\_{0}^{\mathcal{O}} \frac{\theta^{\rho-1}}{(\sigma^{\rho} - \theta^{\rho})^{1-\varrho}} p(\theta) d\theta, \\ 0 < \sigma < \mathcal{O} < \mathcal{T}, \end{cases} \tag{2}$$

where *<sup>ρ</sup> C*D*<sup>ξ</sup>* <sup>0</sup><sup>+</sup> , *<sup>ρ</sup> C*D*<sup>ζ</sup>* <sup>0</sup><sup>+</sup> are the Liouville–Caputo-type generalized fractional derivative of order <sup>1</sup> <sup>&</sup>lt; *<sup>ξ</sup>*, *<sup>ζ</sup>* <sup>≤</sup> 2, *<sup>ρ</sup> C*I*ς* <sup>0</sup><sup>+</sup> , *<sup>ρ</sup> C*I <sup>0</sup><sup>+</sup> are the generalized fractional integral of order (Katugampola type) , *<sup>ς</sup>* <sup>&</sup>gt; 0, *<sup>ρ</sup>* <sup>&</sup>gt; 0, *<sup>f</sup>* , *<sup>g</sup>* : G × <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> are continuous functions, , *<sup>π</sup>* <sup>∈</sup> <sup>R</sup>. The strip

**Citation:** Awadalla, M.; Subramanian, M.; Abuasbeh, K.; Manigandan, M. On the Generalized Liouville–Caputo Type Fractional Differential Equations Supplemented with Katugampola Integral Boundary Conditions. *Symmetry* **2022**, *14*, 2273. https://doi.org/10.3390/sym14112273

Academic Editor: Anton V. Krysko

Received: 10 October 2022 Accepted: 26 October 2022 Published: 29 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

conditions states that the value of the unknown function at the right end point *τ* = *T* of the given interval is proportional to the values of the unknown function on the strips of varying lengths. When *ρ* = 1, the generalized Liouville–Caputo equation is changed to the Caputo sense, which leads to asymmetric results. In a similar way, when *ρ* = 1, the Katugampola integrals are changed to Riemann-Liouville integrals, which leads to cases that are not symmetric. To the best of our knowledge, the stability analysis of boundary value problems (BVPs) is still in its early stages. This paper's primary contribution is to study existence and Ulam-Hyers stability analysis. In addition, we demonstrate the problem (1)–(2) employed by Leray–Schauder, Banach and Krasnoselskii's fixed point theorems to prove the existence and uniqueness of solutions. The system (1) is the well-known fractional-order coupled logistic system [1]:

$$\begin{cases} \mathcal{D}^{\mathbb{A}}u(\tau) = r\_1 u(\tau) - \frac{r\_1}{k\_1} u(\tau)(u(\tau) + v(\tau)), \tau \in I\_{\tau} \\\mathcal{D}^{\mathbb{A}}v(\tau) = r\_2 v(\tau) - \frac{r\_2}{k\_2} v(\tau)(v(\tau) + u(\tau)), \end{cases}$$

and the Lotka–Volterra prey-predator system [1]:

$$\begin{cases} \mathcal{D}^{\alpha}u(\tau) = u(\tau)(a - u(\tau)E - \gamma v(\tau)), \tau \in I, \\\mathcal{D}^{\beta}v(\tau) = v(\tau)(-b + \gamma Ev(\tau) - \beta E). \end{cases}$$

We now provide some recent results related to our problem (1)–(2). In [2], the authors discussed the existence results for coupled system of fractional differential equations Riemann–Liouville derivatives

$$\begin{cases} \mathcal{D}\_{0^{+}}^{\kappa\_{1}}(\mathcal{D}\_{0^{+}}^{\beta\_{1}}\mathbf{x}(t)) + f(t, \mathbf{x}(t), y(t)), t \in [0, 1],\\ \mathcal{D}\_{0^{+}}^{\kappa\_{2}}(\mathcal{D}\_{0^{+}}^{\beta\_{2}}y(t)) + f(t, \mathbf{x}(t), y(t)), t \in [0, 1], \end{cases} \tag{3}$$

with the Riemann–Stieltjes integral boundary conditions:

$$\begin{cases} \mathcal{D}\_{0^{+}}^{\delta\_{1}}\mathbf{x}(0) = \mathbf{0}, \; \mathbf{x}(0) = \mathbf{0}, \; \mathcal{D}\_{0^{+}}^{\delta\_{2}}\mathbf{y}(0) = \mathbf{0}, \; \mathbf{y}(0) = \mathbf{0} \\ \mathbf{x}(1) = \gamma\_{1}\mathcal{L}\_{0^{+}}^{\delta\_{1}}\mathbf{y}(\xi) + \sum\_{i=1}^{p} \int\_{0}^{1} \mathbf{y}(\tau) d\mathcal{H}\_{i}(\tau), \\ \mathbf{y}(1) = \gamma\_{2}\mathcal{L}\_{0^{+}}^{\delta\_{2}}\mathbf{x}(\eta) + \sum\_{j=1}^{q} \int\_{0}^{1} \mathbf{x}(\tau) d\mathcal{K}\_{i}(\tau), \end{cases} \tag{4}$$

where *α*<sup>1</sup> is in the interval (0, 1), *β*<sup>1</sup> is in the interval (1, 2), *α*<sup>2</sup> is in the interval (0, 1], *β*<sup>2</sup> is in the interval (1, 2], *p*, *q* ∈ *N*, and *γ*1, *γ*2, *δ*1, *δ*<sup>2</sup> > 0, 0 < *ξ*, *η* < 1 K*j*(*t*), *j* = 1, ... , *q*, H*i*(*t*), *i* = 1, ... , *p* are bounded variation functions. Both function f and g are nonlinear. They used several theorems from fixed point index theory to prove the main results. In [3], the authors investigated existence of solutions for coupled system of fractional differential equations with Hilfer derivatives

$$\begin{cases} (^{H}\mathcal{D}\_{0^{+}}^{\mathfrak{a}\_{1}\mathfrak{f}\_{1}}\mathbf{x})(t) + \lambda\_{1}(^{H}\mathcal{D}\_{0^{+}}^{\mathfrak{a}\_{1}-1,\mathfrak{f}\_{1}}\mathbf{x})(t) = f(t,\mathbf{x}(t),\mathbf{R}^{\{\delta\_{q},\ldots,\delta\_{1}\}}\mathbf{x}(t),\mathbf{y}(t)), t \in [0,T],\\ (^{H}\mathcal{D}\_{0^{+}}^{\mathfrak{a}\_{2}\mathfrak{f}\_{2}}\mathbf{y})(t) + \lambda\_{2}(^{H}\mathcal{D}\_{0^{+}}^{\mathfrak{a}\_{2}-1,\mathfrak{f}\_{2}}\mathbf{y})(t) = f(t,\mathbf{x}(t),\mathbf{y}(t),\mathbf{R}^{\{\underline{\zeta}\_{q},\ldots,\zeta\_{1}\}}\mathbf{y}(t)), t \in [0,T],\end{cases} \tag{5}$$

with Riemann–Liouville and Hadamard-type iterated integral boundary conditions:

$$\begin{cases} \boldsymbol{\mathfrak{x}}(0) = \boldsymbol{0}, \,\boldsymbol{y}(0) = \boldsymbol{0},\\ \boldsymbol{\mathfrak{x}}(T) = \sum\_{i=1}^{m} \boldsymbol{\varepsilon}\_{i} \boldsymbol{R}^{(\mu\_{\rho}, \dots, \mu\_{1})} \boldsymbol{y}(\eta\_{i}) \,\eta\_{i} \in (\mathbf{0}, T),\\ \boldsymbol{y}(T) = \sum\_{j=1}^{n} \theta\_{j} \boldsymbol{R}^{(\nu\_{\rho}, \dots, \nu\_{1})} \boldsymbol{x}(\boldsymbol{\xi}\_{j}) \,\boldsymbol{\xi}\_{i} \in (\mathbf{0}, T), \end{cases} \tag{6}$$

where *<sup>H</sup>*D*αl*,*β<sup>l</sup>* is the Hilfer fractional derivative operator of order *<sup>α</sup><sup>l</sup>* with parameters *<sup>β</sup>l*, *l* ∈ 1, 2, 1 < *α<sup>l</sup>* < 2, 0 ≤ *β<sup>l</sup>* ≤ 1, *λ*1, *λ*2, *i*, *θ<sup>j</sup>* ∈ R\{0}, *i* = 1, 2, ... , *m*, *j* = 1, 2, ... , *n*, *<sup>f</sup>* , *<sup>g</sup>* : [0, *<sup>T</sup>*] × R × R × R → ×R are nonlinear continuous functions and *<sup>R</sup>*(*φτ*,...,*φ*1), *φ<sup>r</sup>* ∈ {*δ*, *ζ*, *μ*, *ν*}, *r* ∈ {*q*, *p*, *ρ*|*q*, *p*, *ρ* ∈ N}, involves the iterated Riemann–Liouville and

Hadamard fractional integral operators. They used several theorems from fixed point index theory to prove the main results. Numerous scientific and engineering phenomena are mathematically modelled using fractional order differential and integral operators. The main benefit of adopting these operators is their nonlocality, which enables the description of the materials and processes involved in the history of the phenomenon. As a result, compared to their integer-order counterparts, fractional-order models are more precise and informative. As a result of the extensive use of fractional calculus techniques in a range of real-world occurrences, such as those described in the texts cited [4–8] numerous researchers developed this significant branch of mathematical study. In recent years, a lot of research has been done on fractional differential equations with different boundary conditions. Nonlocal nonlinear fractional-order boundary value problems, in particular, have attracted a lot of attention (BVPs). The idea of nonlocal circumstances, which help to describe physical processes occurring inside the confines of a specific domain, was originally introduced in the work of Bitsadze and Samarski [9]. It is challenging to defend the assumption of a circular cross-section in computational fluid dynamics investigations of blood flow problems because to the changing shape of a blood vessel throughout the vessel. To solve that problem, integral boundary conditions have been developed. In addition, the ill-posed parabolic backward problems are solved under integral boundary conditions. Integral boundary conditions are also essential in mathematical models of bacterial self-regularization, as shown in [10]. Fractional order differential equations, as well as inclusions including Riemann–Liouville, Liouville–Caputo (Caputo), and Hadamard-type derivatives, among others, have all been included in the literature on the topic recently. For some recent works on the topic, we point the reader to several papers [11–15] and the references listed therein. The use of fractional differential systems in mathematical representations of physical and engineering processes has drawn considerable interest. See [16–22] for additional details on the theoretical evolution of such systems. The following is the remainder of the article: Section 2 introduces some fundamental definitions, lemmas, and theorems that support our main results. For the existence and uniqueness of solutions to the given system (1) and (2), we use various conditions and some standard fixed-point theorems in Section 3. Section 4 discusses the Ulam–Hyers stability of the given system (1) and (2) under certain conditions. In Section 6, examples are provided to demonstrate the main results. Finally, the consequences of existence, uniqueness, and stability for the problem (1) and (75) are provided.

#### **2. Preliminaries**

For our research, we recall some preliminary definitions of generalized Liouville– Caputo fractional derivatives and Katugampola fractional integrals.

The space of all complex-valued Lebesgue measurable functions *φ* on (*c*, *d*) equipped with the norm is denoted by <sup>Z</sup>*<sup>q</sup> <sup>b</sup>* (*c*, *d*) :

$$||\phi||\_{\mathcal{Z}\_b^q} = \left(\int\_c^d |z^b \phi(z)|^q \frac{dz}{z}\right)^{\frac{1}{q}} < \infty, b \in \mathbb{R}, 1 \le q \le \infty.$$

Let <sup>L</sup>1(*c*, *<sup>d</sup>*) represent the space of all Lebesgue measurable functions *<sup>ϕ</sup>* on (*c*, *<sup>d</sup>*) endowed with the norm:

$$\|\|\boldsymbol{\varrho}\|\|\_{\mathcal{L}^1} = \int\_{\mathcal{L}}^d |\boldsymbol{\varrho}(\boldsymbol{z})| d\boldsymbol{z} < \infty.$$

We further recall that AC*n*(E, <sup>R</sup>) = {*<sup>p</sup>* : E → <sup>R</sup> : *<sup>p</sup>*, *<sup>p</sup>* , ... , *<sup>p</sup>*(*n*−1) ∈ C(E, <sup>R</sup>) and *<sup>p</sup>*(*n*−1) is absolutely continuous. For 0 <sup>≤</sup> <sup>&</sup>lt; 1, we define <sup>C</sup>,*ρ*(E, <sup>R</sup>) = { *<sup>f</sup>* : E → <sup>R</sup> : (*τ<sup>ρ</sup>* <sup>−</sup> *<sup>a</sup>ρ*) *<sup>f</sup>*(*τ*) ∈ C(E, <sup>R</sup>) endowed with the norm *<sup>f</sup>* C,*<sup>ρ</sup>* <sup>=</sup> (*τ<sup>ρ</sup>* <sup>−</sup> *<sup>a</sup>ρ*) *<sup>f</sup>*(*τ*)C . Moreover, we define the class of functions *f* that have absolute continuous *δn*−<sup>1</sup> derivative, denoted by AC*<sup>n</sup> <sup>γ</sup>*(E, <sup>R</sup>), as follows: AC*<sup>n</sup> <sup>γ</sup>*(E, <sup>R</sup>) = { *<sup>f</sup>* : E → <sup>R</sup> : *<sup>γ</sup>n*−<sup>1</sup> *<sup>f</sup>* ∈ AC(E, <sup>R</sup>), *<sup>γ</sup>* <sup>=</sup> *<sup>τ</sup>*1−*<sup>ρ</sup> <sup>d</sup> <sup>d</sup><sup>τ</sup>* }, which is equipped with the norm *f* C*<sup>n</sup> <sup>γ</sup>*, = <sup>∑</sup>*n*−<sup>1</sup> *<sup>k</sup>*=<sup>0</sup> *γ<sup>k</sup> <sup>f</sup>* C <sup>+</sup> *γ<sup>n</sup> <sup>f</sup>* C,*<sup>ρ</sup>* is defined by

$$\mathcal{C}^{n}\_{\gamma,\varepsilon}(\mathcal{E}, \mathbb{R}) = \left\{ f : \mathcal{E} \to \mathbb{R} : \gamma^{n-1} f \in \mathcal{C}(\mathcal{E}, \mathbb{R}), \gamma^{n} f \in \mathcal{C}\_{\varepsilon,\rho}(\mathcal{E}, \mathbb{R}), \gamma = \tau^{1-\rho} \frac{d}{d\tau} \right\}.$$

Notice that <sup>C</sup>*<sup>n</sup> <sup>γ</sup>*,0 <sup>=</sup> <sup>C</sup>*<sup>n</sup> <sup>γ</sup>*. We define space <sup>P</sup> <sup>=</sup> {*p*(*τ*) : *<sup>p</sup>*(*τ*) ∈ C(E, <sup>R</sup>)} equipped with the norm ||*p*|| = sup{|*p*(*τ*)|, *τ* ∈ E}- this is a Banach space. Furthermore Q = {*q*(*τ*) : *<sup>q</sup>*(*τ*) ∈ C(E, <sup>R</sup>)} equipped with the norm is ||*q*|| <sup>=</sup> sup{|*q*(*τ*)|, *<sup>τ</sup>* ∈ E} is a Banach space. Then the product space (P×Q, ||(*p*, *q*)||) is also a Banach space with norm ||(*p*, *q*)|| = ||*p*|| + ||*q*||.

**Definition 1** ([23])**.** *The left and right-sided generalized fractional integrals (GFIs) of <sup>f</sup>* ∈ Z*<sup>q</sup> <sup>b</sup>* (*c*, *d*) *of order ξ* > 0 *and ρ* > 0 *for* −∞ < *c* < *τ* < *d* < ∞*, are defined as follows:*

$$(\,^\rho T\_{\mathfrak{c}^+}^\sharp f)(\tau) = \frac{\rho^{1-\mathfrak{f}}}{\Gamma(\mathfrak{f})} \int\_{\mathfrak{c}}^\tau \frac{\theta^{\rho-1}}{(\tau^\rho - \theta^\rho)^{1-\mathfrak{f}}} f(\theta) d\theta \,\tag{7}$$

$$(\prescript{\rho}{}{\mathcal{T}}\_{d^{-}}^{\tilde{\xi}}f)(\tau) = \frac{\rho^{1-\tilde{\xi}}}{\Gamma(\tilde{\xi})} \int\_{\tau}^{d} \frac{\theta^{\rho-1}}{(\theta^{\rho} - \tau^{\rho})^{1-\tilde{\xi}}} f(\theta) d\theta. \tag{8}$$

**Definition 2** ([24])**.** *The generalized fractional derivatives (GFDs) which are associated with GFIs (7) and (8) for* 0 ≤ *c* < *τ* < *d* < ∞*, are defined as follows:*

$$\begin{split} (\prescript{\rho}{}{\mathcal{D}}\_{\mathcal{C}^{+}}^{\mathfrak{f}} f)(\tau) &= \left(\tau^{1-\rho} \frac{d}{d\tau}\right)^{n} (\prescript{\rho}{}{\mathcal{D}}\_{\mathcal{C}^{+}}^{n-\mathfrak{f}} f)(\tau) \\ &= \frac{\rho^{\mathfrak{f}-n+1}}{\Gamma(n-\mathfrak{f})} \left(\tau^{1-\rho} \frac{d}{d\tau}\right)^{n} \int\_{\mathscr{C}}^{\tau} \frac{\theta^{\rho-1}}{(\tau^{\rho}-\theta^{\rho})^{\mathfrak{f}-n+1}} f(\theta) d\theta,\end{split} \tag{9}$$

$$\begin{split} \left( \ell^{\rho} \mathcal{D}\_{d^{-}}^{\tilde{\xi}} f \right)(\tau) &= \left( -\tau^{1-\rho} \frac{d}{d\tau} \right)^{n} (\ell^{\rho} \mathcal{T}\_{d^{-}}^{n-\tilde{\xi}} f)(\tau) \\ &= \frac{\rho^{\tilde{\xi}-n+1}}{\Gamma(n-\tilde{\xi})} \left( -\tau^{1-\rho} \frac{d}{d\tau} \right)^{n} \int\_{\tau}^{d} \frac{\theta^{\rho-1}}{(\tau^{\rho}-\theta^{\rho})^{\tilde{\xi}-n+1}} f(\theta) d\theta, \end{split} \tag{10}$$

*if the integrals exist.*

**Definition 3** ([25])**.** *The above GFDs define the left and right-sided generalized Liouville–Caputo type fractional derivatives of f* ∈ AC*<sup>n</sup> <sup>γ</sup>*[*c*, *d*] *of order ξ* ≥ 0

$$\mathcal{D}\_{\mathbb{C}}^{\rho} \mathcal{D}\_{\mathfrak{c}^+}^{\mathbb{E}} f(z) = ^{\rho} \mathcal{D}\_{\mathfrak{c}^+}^{\mathbb{E}} \left[ f(\tau) - \sum\_{k=0}^{n-1} \frac{\gamma^k f(\mathfrak{c})}{k!} \left( \frac{\tau^{\rho} - \mathfrak{c}^{\rho}}{\rho} \right)^k \right] (z) , \gamma = z^{1-\rho} \frac{d}{dz} \prime \tag{11}$$

$$\mathcal{D}\_{\mathbb{C}}^{\rho} \mathcal{D}\_{d^{-}}^{\mathbb{E}} f(z) =^{\rho} \mathcal{D}\_{d^{-}}^{\mathbb{E}} \left[ f(\tau) - \sum\_{k=0}^{n-1} \frac{(-1)^{k} \gamma^{k} f(d)}{k!} \left( \frac{d^{\rho} - \tau^{\rho}}{\rho} \right)^{k} \right] (z), \gamma = z^{1-\rho} \frac{d}{dz}, \tag{12}$$

*when n* = [*ξ*] + 1.

**Lemma 1** ([25])**.** *Let <sup>ξ</sup>* <sup>≥</sup> 0, *<sup>n</sup>* = [*ξ*] + <sup>1</sup> *and f* ∈ AC*<sup>n</sup> <sup>γ</sup>*[*c*, *d*], *where* 0 < *c* < *d* < ∞*. Then,* *1. if <sup>ξ</sup>* <sup>∈</sup>/ <sup>N</sup>

$$\prescript{\rho}{}{\mathcal{D}}\_{\varepsilon^{+}}^{\mathbb{E}} f(\tau) = \frac{1}{\Gamma(n - \mathbb{E})} \int\_{\varepsilon}^{\tau} \left( \frac{\tau^{\rho} - \theta^{\rho}}{\rho} \right)^{n - \mathbb{E} - 1} \frac{(\gamma^{n} f)(\theta) d\theta}{\theta^{1 - \rho}} = \prescript{\rho}{}{\mathcal{Z}}\_{\varepsilon^{+}}^{n - \mathbb{E}} (\gamma^{n} f)(\tau), \tag{13}$$

$$\mathcal{I}\_{\mathbb{C}}^{\rho} \mathcal{D}\_{d^{-}}^{\mathbb{E}} f(\tau) = \frac{1}{\Gamma(n - \underline{\zeta})} \int\_{\tau}^{d} \left( \frac{\theta^{\rho} - \tau^{\rho}}{\rho} \right)^{n - \overline{\zeta} - 1} \frac{(-1)^{n} (\gamma^{n} f)(\theta) d\theta}{\theta^{1 - \rho}} = \mathcal{I}\_{\overline{d^{-}}}^{\overline{n - \overline{\zeta}}} (\gamma^{n} f)(\tau). \tag{14}$$
  $2. \quad \dot{f} \, \overline{\zeta} \in \mathbb{N}$ 

$$\prescript{0}{}{\mathcal{D}}\_{c^{+}}^{\sharp}f = \gamma^{n}f, \qquad \prescript{\rho}{}{\mathcal{D}}\_{d^{-}}^{\sharp}f = (-1)^{n}\gamma^{n}f. \tag{15}$$

**Lemma 2** ([25])**.** *Let f* ∈ AC*<sup>n</sup> <sup>γ</sup>*[*c*, *<sup>d</sup>*] *or* <sup>C</sup>*<sup>n</sup> <sup>γ</sup>*[*c*, *<sup>d</sup>*] *and <sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*. Then,*

$$\begin{aligned} \,^\rho \mathcal{Z}\_{c^+ \gets}^\sharp \mathcal{D}\_{c^+}^\sharp f(z) &= f(z) - \sum\_{k=0}^{n-1} \frac{\gamma^k f(c)}{k!} \left( \frac{z^\rho - c^\rho}{\rho} \right)^k, \\\\ \,^\rho \mathcal{Z}\_{d^- \gets}^\sharp \mathcal{D}\_{d^-}^\sharp f(z) &= f(z) - \sum\_{k=0}^{n-1} \frac{(-1)^k \gamma^k f(d)}{k!} \left( \frac{d^\rho - z^\rho}{\rho} \right)^k. \end{aligned}$$

In particular, for 0 < *ξ* ≤ 1, we have

$$\prescript{\rho}{}{\mathcal{T}}\_{\mathcal{C}^{+}\mathbb{C}}^{\mathbb{F}}\prescript{\rho}{}{\mathcal{D}}\_{\mathcal{C}+}^{\mathbb{F}}f(z) = f(z) - f(c), \qquad \prescript{\rho}{}{\mathcal{T}}\_{d^{-}\mathbb{C}}^{\mathbb{F}}\prescript{\rho}{}{\mathcal{D}}\_{d^{-}}^{\mathbb{F}}f(z) = f(z) - f(d).$$

We introduce the following notations for computational ease:

$$\mathcal{E}\_1 = \varepsilon \frac{\mathcal{O}^{\rho(\emptyset + 1)}}{\rho^{\emptyset + 1} \Gamma(\emptyset + 2)}, \; \mathcal{E}\_2 = \pi \frac{\mathcal{O}^{\rho(\emptyset + 1)}}{\rho^{\emptyset + 1} \Gamma(\emptyset + 2)}, \; \hat{\mathcal{E}} = \frac{\mathcal{T}^{\rho}}{\rho}, \tag{16}$$

$$\mathcal{G} = \hat{\mathcal{E}}^2 - \mathcal{E}\_1 \mathcal{E}\_2 \neq 0,\tag{17}$$

$$
\delta(\tau) = \left(\frac{\tau^{\rho}}{\rho \mathcal{G}}\right). \tag{18}
$$

Next, we are proving a lemma, which is vital in converting the given problem to a fixed-point problem.

**Lemma 3.** *Given the functions* ˆ *<sup>f</sup>* , *<sup>g</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>C</sup>*(0, <sup>T</sup> ) ∪ L(0, <sup>T</sup> ), *<sup>p</sup>*, *<sup>q</sup>* ∈ AC<sup>2</sup> *<sup>γ</sup>*(E) *and* Λ = 0. *Then the solution of the coupled BVP:*

$$\begin{cases} \,^{\rho}\_{\mathcal{C}} \mathcal{D}\_{0+}^{\mathcal{E}} p(\tau) = f(\tau), \,\tau \in \mathcal{E} := [0, \mathcal{T}],\\ \,^{\rho}\_{\mathcal{C}} \mathcal{D}\_{0+}^{\mathcal{E}} q(\tau) = \mathcal{\xi}(\tau), \,\tau \in \mathcal{E} := [0, \mathcal{T}],\\ p(0) = 0, \quad q(0) = 0, \quad p(\mathcal{T}) = \varepsilon^{\rho} \mathcal{T}\_{0+}^{\mathcal{E}} q(\mathcal{o}), \quad q(\mathcal{T}) = \pi^{\rho} \mathcal{T}\_{0+}^{\mathcal{E}} p(\sigma) \quad 0 < \sigma < \mathcal{o} < \mathcal{T}\_{\text{s}} \end{cases} \tag{19}$$

*is given by*

$$p(\mathbf{r}) = ^\rho \mathcal{Z}\_{0+}^\xi \hat{f}(\mathbf{r}) + \delta(\mathbf{r}) \left[ \hat{\mathcal{E}} \Big( \boldsymbol{\varepsilon}^\rho \mathcal{Z}\_{0+}^{\mathsf{F}+\mathsf{c}} \circ \mathcal{Y}(\boldsymbol{\sigma}) - ^\rho \mathcal{Z}\_{0+}^\mathsf{F} \hat{f}(\mathcal{T}) \Big) + \mathcal{E}\_1 \Big( \boldsymbol{\pi}^\rho \mathcal{Z}\_{0+}^{\mathsf{F}+\mathsf{c}} \hat{f}(\boldsymbol{\sigma}) - ^\rho \mathcal{Z}\_{0+}^\mathsf{F} \mathcal{G}(\mathcal{T}) \Big) \right] \tag{20}$$

*and*

$$q(\tau) = ^\rho \mathcal{Z}\_{0^+}^\sharp \hat{\mathbf{g}}(\tau) + \delta(\tau) \left[ \hat{\mathcal{E}} \left( \pi^\rho \mathcal{Z}\_{0^+}^{\mathbb{S}+\varrho} \hat{f}(\sigma) - {}^\rho \mathcal{Z}\_{0^+}^\sharp \hat{\mathbf{g}}(T) \right) + \mathcal{E}\_2 \left( \epsilon^\rho \mathcal{Z}\_{0^+}^{\mathbb{S}+\varsigma} \hat{\mathbf{g}}(\sigma) - {}^\rho \mathcal{Z}\_{0+}^\sharp \hat{f}(T) \right) \right]. \tag{21}$$

**Proof.** When *<sup>ρ</sup>*I*<sup>ξ</sup>* 0+ , *ρ* I*ζ* <sup>0</sup><sup>+</sup> are applied to the FDEs in (19) and Lemma 2 is used, the solution of the FDEs in (19) for *τ* ∈ E is

$$p(\tau) = ^\rho \mathcal{Z}\_{0^+}^\sharp f(\tau) + a\_1 + a\_2 \frac{\tau^\rho}{\rho} = \frac{\rho^{1-\tilde{\zeta}}}{\Gamma(\tilde{\zeta})} \int\_0^\tau \theta^{\rho-1} (\tau^\rho - \theta^\rho)^{\tilde{\zeta}-1} \tilde{f}(\theta) d\theta + a\_1 + a\_2 \frac{\tau^\rho}{\rho}, \tag{22}$$

$$q(\tau) = ^\rho \mathcal{Z}\_{0^+}^\zeta \xi(\tau) + b\_1 + b\_2 \frac{\tau^\rho}{\rho} = \frac{\rho^{1-\zeta}}{\Gamma(\zeta)} \int\_0^\tau \theta^{\rho-1} (\tau^\rho - \theta^\rho)^{\zeta-1} \xi(\theta) d\theta + b\_1 + b\_2 \frac{\tau^\rho}{\rho}, \tag{23}$$

respectively, for some *a*1, *a*2, *b*1, *b*<sup>2</sup> ∈ R. Making use of the boundary conditions *p*(0) = *q*(0) = 0 in (22) and (23) respectively, we get *a*<sup>1</sup> = *b*<sup>1</sup> = 0. Next, we obtain by using the generalized integral operators *<sup>ρ</sup>*I*<sup>ξ</sup>* <sup>0</sup>+, *ρ* I*ζ* <sup>0</sup><sup>+</sup> (22) and (23) respectively,

$$\prescript{}{\rho}{\mathcal{I}}\_{0+}^{\varrho} p(\tau) = ^{\rho} \prescript{}{\mathcal{I}}\_{0^{+}}^{\xi+\varrho} f(\tau) + a\_1 \frac{\tau^{\rho\varrho}}{\rho^{\varrho} \Gamma(\varrho+1)} + a\_2 \frac{\tau^{\rho(\varrho+1)}}{\rho^{\varrho+1} \Gamma(\varrho+2)},\tag{24}$$

$$\prescript{}{0}{\tau}\mathcal{Z}\_{0^{+}}^{\xi}q(\tau) = ^{\rho}\mathcal{Z}\_{0^{+}}^{\zeta+\zeta}\mathcal{G}(\tau) + b\_{1}\frac{\tau^{\rho\zeta}}{\rho^{\varsigma}\Gamma(\zeta+1)} + b\_{2}\frac{\tau^{\rho(\zeta+1)}}{\rho^{\varsigma+1}\Gamma(\zeta+2)},\tag{25}$$

which, when combined with the boundary conditions *<sup>p</sup>*(<sup>T</sup> ) = *<sup>ρ</sup>*I*<sup>ς</sup>* <sup>0</sup><sup>+</sup> *q*(), *<sup>q</sup>*(<sup>T</sup> ) = *<sup>π</sup>ρ*I <sup>0</sup><sup>+</sup> *p*(*σ*), gives the following results:

$${}^{\rho}\mathcal{T}\_{0+}^{\tilde{\varsigma}}f(\mathcal{T}) + a\_1 + a\_2 \frac{\mathcal{T}^{\rho}}{\rho} = \varepsilon^{\rho}\mathcal{T}\_{0+}^{\tilde{\varsigma}+\varsigma}\mathfrak{g}(\mathcal{a}) + b\_1 \frac{\varepsilon\mathcal{a}^{\rho\varsigma}}{\rho^{\varsigma}\Gamma(\varsigma+1)} + b\_2 \frac{\varepsilon\mathcal{a}^{\rho(\varsigma+1)}}{\rho^{\varsigma+1}\Gamma(\varsigma+2)},\tag{26}$$

$$\pi^{\rho} \mathcal{Z}\_{0^{+}}^{\tilde{\zeta}} \xi(\mathcal{T}) + b\_1 + b\_2 \frac{\mathcal{T}^{\rho}}{\rho} = \pi^{\rho} \mathcal{Z}\_{0^{+}}^{\tilde{\zeta} + \varrho} \hat{f}(\sigma) + a\_1 \frac{\pi \sigma^{\rho \varrho}}{\rho^{\varrho} \Gamma(\varrho + 1)} + a\_2 \frac{\pi \sigma^{\rho(\varrho + 1)}}{\rho^{\varrho + 1} \Gamma(\varrho + 2)}.\tag{27}$$

Next, we obtain

$$a\_2 \hat{\mathcal{E}} - b\_2 \mathcal{E}\_1 = \epsilon^{\rho} \mathcal{I}\_{0^+}^{\tilde{\varsigma} + \zeta} \mathcal{g}(\mathcal{a}) - ^{\rho} \mathcal{I}\_{0+}^{\tilde{\varsigma}} f(\mathcal{T}),\tag{28}$$

$$b\_2 \widehat{\mathcal{E}} - a\_2 \mathcal{E}\_2 = \pi^\rho \mathcal{Z}\_{0^+}^{\tilde{\ast} + \varrho} \widehat{f}(\sigma) - ^\rho \mathcal{Z}\_{0 +}^{\tilde{\ast}} \widehat{g}(\mathcal{T}).\tag{29}$$

by employing the notations (16) in (26) and (27) respectively. We find that when we solve the system of Equations (28) and (29) for *a*<sup>2</sup> and *b*2,

$$a\_2 = \frac{1}{\mathcal{G}} \left[ \hat{\mathcal{E}} \left( \varepsilon^{\rho} \mathcal{Z}\_{0^+}^{\mathbb{S}+\zeta} \mathfrak{z}(\boldsymbol{\sigma}) - {}^{\rho} \mathcal{Z}\_{0+}^{\mathbb{S}} f(\boldsymbol{\mathcal{T}}) \right) + \mathcal{E}\_1 \left( \pi^{\rho} \mathcal{Z}\_{0^+}^{\mathbb{S}+\varrho} f(\boldsymbol{\sigma}) - {}^{\rho} \mathcal{Z}\_{0+}^{\mathbb{S}} \mathfrak{z}(\boldsymbol{\mathcal{T}}) \right) \right],\tag{30}$$

$$b\_2 = \frac{1}{\mathcal{G}} \left[ \mathcal{E}\_2 \left( \epsilon^{\rho} \mathcal{Z}\_{0^+}^{\mathbb{S}+\xi} \mathfrak{z}(\mathcal{o}) - ^{\rho} \mathcal{Z}\_{0+}^{\mathbb{S}} f(\mathcal{T}) \right) + \hat{\mathcal{E}} \left( \pi^{\rho} \mathcal{Z}\_{0^+}^{\mathbb{S}+\varrho} f(\sigma) - ^{\rho} \mathcal{Z}\_{0^+}^{\mathbb{S}} \mathfrak{z}(\mathcal{T}) \right) \right]. \tag{31}$$

Substituting the values of *a*1, *a*2, *b*1, *b*<sup>2</sup> in (22) and (23) respectively, we get the solution for the BVP (19).

#### **3. Existence Results for the Problem (1) and (2)**

As a result of Lemma 3, we define an operator Δ : P×Q→P×Q by

$$
\Delta(p,q)(\tau) = (\Delta\_1(p,q)(\tau), \Delta\_2(p,q)(\tau)),
\tag{32}
$$

where

$$\Delta\_1(p,q)(\tau) = \prescript{\rho}{}{T}\_{0+}^{\tilde{\tau}} f(\tau, p(\tau), q(\tau)) + \delta(\tau) \left[ \hat{\mathcal{E}} \left( \varepsilon^{\rho} \mathcal{T}\_{0^+}^{\tilde{\tau}+\xi} g(\sigma, p(\sigma), q(\sigma)) - \prescript{\rho}{}{T}\_{0+}^{\tilde{\tau}} f(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T})) \right) \right]$$

$$+ \mathcal{E}\_1 \Big( \pi^{\rho} \mathcal{T}\_{0^+}^{\tilde{\tau}+\theta} f(\sigma, p(\sigma), q(\sigma)) - \prescript{\rho}{}{T}\_{0+}^{\tilde{\tau}} g(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T})) \Big) \Big]. \tag{33}$$

$$\Delta\_2(p,q)(\mathbf{r}) = {}^{\rho}\mathcal{Z}\_{0+}^{\tilde{\xi}}(\mathbf{r},p(\tau),q(\tau)) + \delta(\tau) \left[ \hat{\mathcal{E}}\left(\pi^{\rho}\mathcal{Z}\_{0+}^{\tilde{\xi}+\varrho}f(\sigma,p(\sigma),q(\sigma)) - {}^{\rho}\mathcal{Z}\_{0+}^{\tilde{\xi}}(\mathcal{T},p(\mathcal{T}),q(\mathcal{T}))\right) \right]$$

$$+ \mathcal{E}\_2\Big(\varepsilon^{\rho}\mathcal{Z}\_{0+}^{\tilde{\xi}+\varepsilon}g(\sigma,p(\sigma),q(\sigma)) - {}^{\rho}\mathcal{Z}\_{0+}^{\tilde{\xi}}f(\mathcal{T},p(\mathcal{T}),q(\mathcal{T})) \Big) \Big].\tag{34}$$

For brevity's sake, we'll use the following notations:

$$\mathcal{J}\_1 = \frac{\left(\mathcal{T}^{\rho\_5^z} (1 + |\delta| |\hat{\mathcal{E}}|) \right)}{\rho^\sharp \Gamma(\tilde{\xi} + 1)} + \frac{|\delta| |\pi| |\mathcal{E}\_1| \sigma^{\rho(\tilde{\xi} + \varrho)}}{\rho^{\tilde{\xi} + \varrho} \Gamma(\tilde{\xi} + \varrho + 1)},\tag{35}$$

$$\mathcal{K}\_1 = |\delta| \left( \frac{|\mathcal{E}\_1| \mathcal{T}^{\rho \mathbb{\tilde{V}}}}{\rho^{\mathbb{\tilde{V}}} \Gamma(\mathbb{\tilde{J}} + 1)} + \frac{|\hat{\mathcal{E}}| |\varepsilon| \mathcal{a}^{\rho(\mathbb{\tilde{J}} + \emptyset)}}{\rho^{\mathbb{\tilde{V}} + \emptyset} \Gamma(\mathbb{\tilde{J}} + \emptyset + 1)} \right), \tag{36}$$

$$\mathcal{J}\_2 = |\delta| \left( \frac{\mathcal{T}^{\rho \tilde{\varsigma}} |\mathcal{E}\_2|}{\rho^{\sharp} \Gamma(\tilde{\varsigma} + 1)} + \frac{|\pi||\hat{\mathcal{E}}| \sigma^{\rho(\tilde{\varsigma} + \varrho)}}{\rho^{\sharp + \varrho} \Gamma(\tilde{\varsigma} + \varrho + 1)} \right), \tag{37}$$

$$\mathcal{K}\_2 = \frac{\left(\mathcal{T}^{\rho\tilde{\varsigma}} (1 + |\delta| |\hat{\mathcal{E}}|) \right)}{\rho^{\zeta} \Gamma(\zeta + 1)} + \frac{|\delta| |\varepsilon| |\mathcal{E}\_2| \mathcal{a}^{\rho(\zeta + \zeta)}}{\rho^{\zeta + \zeta} \Gamma(\zeta + \zeta + 1)} \tag{38}$$

$$\Phi = \min \{ 1 - \left[ \psi\_1 (\mathcal{J}\_1 + \mathcal{J}\_2) + \psi\_1 (\mathcal{K}\_1 + \mathcal{K}\_2) \right] , 1 - \left[ \psi\_2 (\mathcal{J}\_1 + \mathcal{J}\_2) + \psi\_2 (\mathcal{K}\_1 + \mathcal{K}\_2) \right] \}. \tag{39}$$

**Theorem 1.** *Assume that <sup>f</sup>* , *<sup>g</sup>* : E × <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> *are continuous functions satisfying the condition:* (A1) *there exists constants <sup>ψ</sup>m*, *<sup>ψ</sup>*<sup>ˆ</sup> *<sup>m</sup>* ≥ <sup>0</sup>(*<sup>m</sup>* = 1, 2) *and <sup>ψ</sup>*0, *<sup>ψ</sup>*<sup>ˆ</sup> <sup>0</sup> > 0 *such that*

$$\begin{aligned} |f(\mathfrak{r}, o\_1, o\_2)| &\leq \psi\_0 + \psi\_1 |o\_1| + \psi\_2 |o\_2|, \\ |g(\mathfrak{r}, o\_1, o\_2)| &\leq \hat{\psi}\_0 + \hat{\psi}\_1 |o\_1| + \hat{\psi}\_2 |o\_2|, \forall o\_m \in \mathbb{R}, m = 1, 2. \end{aligned}$$

*If <sup>ψ</sup>*1(J<sup>1</sup> + J2) + *<sup>ψ</sup>*<sup>ˆ</sup> <sup>1</sup>(K<sup>1</sup> + K2) < 1, *<sup>ψ</sup>*2(J<sup>1</sup> + J2) + *<sup>ψ</sup>*<sup>ˆ</sup> <sup>2</sup>(K<sup>1</sup> + K2) < 1*. Then* ∃ *at least one solution for the BVP (1) and (2) on* E*, where* J1, K1, J2, K<sup>2</sup> *are given by (35)–(38) respectively.*

**Proof.** We define operator Δ : P×Q→ P×Q as being completely continuous in the first step. The continuity of the functions *f* and *g* implies that the operators Δ<sup>1</sup> and Δ<sup>2</sup> are continuous. As a result, the operator Δ is continuous. Let Ψ ⊂P×Q be a bounded set to demonstrate the uniformly bounded operator <sup>Δ</sup>. Then <sup>N</sup><sup>ˆ</sup> <sup>1</sup> and <sup>N</sup><sup>ˆ</sup> <sup>2</sup> are positive constants such that <sup>|</sup> *<sup>f</sup>*(*τ*, *<sup>p</sup>*(*τ*), *<sup>q</sup>*(*τ*))| ≤ <sup>N</sup><sup>ˆ</sup> 1, <sup>|</sup>*g*(*τ*, *<sup>p</sup>*(*τ*), *<sup>q</sup>*(*τ*))| ≤ <sup>N</sup><sup>ˆ</sup> 2, ∀(*p*, *q*) ∈ Ψ. Then we have

$$\begin{split} |\Delta\_{1}(p,q)(\tau)| &\leq \rho^{\mathsf{T}}\mathbb{I}\_{0+}^{\mathsf{T}}|f(\tau,p(\tau),q(\tau))|+|\delta(\tau)|\left[|\hat{\mathcal{E}}[(|\sigma|\ {}^{\mathsf{p}}T^{\mathsf{T}+\mathsf{e}}\_{0^{+}}|g(\sigma,p(\sigma),q(\sigma))|+{}^{\mathsf{p}}\mathcal{I}\_{0+}^{\mathsf{T}}|f(\mathcal{T},p(\mathcal{T}),q(\mathcal{T}))|\right] \\ &\quad+|\mathcal{E}\_{1}[(|\pi|\ {}^{\mathsf{p}}T^{\mathsf{T}+\mathsf{e}}\_{0^{+}}|f(\sigma,p(\sigma),q(\sigma))|+{}^{\mathsf{p}}\mathcal{I}\_{0+}^{\mathsf{T}}|g(\mathcal{T},p(\mathcal{T}),q(\mathcal{T}))| \; \atop &\leq \mathcal{N}\_{1}\left\{\frac{|\boldsymbol{\beta}|\left|\boldsymbol{\pi}|\mathcal{E}\_{1}|\sigma^{\mathsf{p}(\xi+\varepsilon)}}{\rho^{\mathsf{p}+\mathsf{e}}\Gamma(\xi+\mathsf{e}+1)}+\frac{(\mathcal{T}^{\mathsf{p}\tilde{\xi}}(1+|\boldsymbol{\beta}|\{\hat{\mathcal{E}}|) )}{\rho^{\mathsf{p}}\Gamma(\xi+1)}\right) \right\} \\ &\quad+\mathcal{N}\_{2}\left\{\left(\frac{|\boldsymbol{\beta}|\left|\boldsymbol{\pi}|\boldsymbol{\beta}|\sigma^{\mathsf{p}(\xi+\mathsf{e})}}{\rho^{\mathsf{p}+\mathsf{e}}\Gamma(\xi+\mathsf{e}+1)}+\frac{|\mathcal{E}\_{1}|\mathcal{T}^{\mathsf{p}\tilde{\xi}}}{\rho^{\mathsf{p}}\Gamma(\xi+1)}\right|\boldsymbol{\beta}|\right) .\end{split}$$

when taking the norm and using (35) and (36), that yields for (*p*, *q*) ∈ Ψ,

$$||\Delta\_1(p,q)|| \le \mathcal{J}\_1 \hat{\mathcal{N}}\_1 + \mathcal{K}\_1 \hat{\mathcal{N}}\_2.\tag{40}$$

Likewise, we obtain

$$\begin{split} ||\Lambda\_{2}(p,q)|| \leq & \mathcal{N}\_{2} \left\{ \frac{|\delta||\varepsilon||\mathcal{E}\_{2}|\varpi^{\rho(\zeta+\varepsilon)}}{\rho^{\zeta+\varepsilon}\Gamma(\zeta+\varrho+1)} + \frac{\left(\mathcal{T}^{\rho\tilde{\zeta}}(1+|\delta||\hat{\mathcal{E}}|)\right)}{\rho^{\zeta}\Gamma(\zeta+1)} \right\} \\ &+ \mathcal{N}\_{1} \left\{ |\delta| \left(\frac{|\pi||\hat{\mathcal{E}}|\sigma^{\rho(\zeta+\varrho)}}{\rho^{\zeta+\varrho}\Gamma(\zeta+\varrho+1)} + \frac{\mathcal{T}^{\rho\tilde{\varepsilon}}|\mathcal{E}\_{2}|}{\rho^{\zeta}\Gamma(\zeta+1)}\right) \right\} \\ \leq & \mathcal{I}\_{2}\mathcal{N}\_{1} + \mathcal{K}\_{2}\mathcal{N}\_{2} \end{split} \tag{41}$$

using (37) and (38). Based on the inequalities (40) and (41), we can conclude that Δ<sup>1</sup> and Δ<sup>2</sup> are uniformly bounded, which indicates that the operator Δ is uniformly bounded. Next, we show that Δ is equicontinuous. Let *τ*1, *τ*<sup>2</sup> ∈ E with *τ*<sup>1</sup> < *τ*2. Then we have


independent of (*p*, *<sup>q</sup>*) with respect to <sup>|</sup> *<sup>f</sup>*(*τ*, *<sup>p</sup>*(*τ*1), *<sup>q</sup>*(*τ*1))| ≤ <sup>N</sup><sup>ˆ</sup> <sup>1</sup> and <sup>|</sup>*g*(*τ*, *<sup>p</sup>*(*τ*1), *<sup>q</sup>*(*τ*1))| ≤ <sup>N</sup><sup>ˆ</sup> 2. Similarly, we can express |Δ2(*p*, *q*)(*τ*2) − Δ2(*p*, *q*)(*τ*1)| → 0 as *τ*<sup>2</sup> → *τ*<sup>1</sup> independent of (*p*, *q*) in terms of the boundedness of *f* and *g*. As a result of the equicontinuity of Δ<sup>1</sup> and Δ2, operator Δ is equicontinuous. As a result of the Arzela–Ascoli theorem, the operator is compact. Finally, we demonstrate that the set Π(Δ) = {(*p*, *q*) ∈ P×Q : *λ*Δ(*p*, *q*); 0 < *λ* < 1} is bounded. Let (*p*, *q*) ∈ Π(Δ).Then (*p*, *q*) = *λ*Δ(*p*, *q*) . For any *τ* ∈ E, we have *p*(*τ*) = *λ*Δ1(*p*, *q*)(*τ*), *q*(*τ*) = *λ*Δ2(*p*, *q*)(*τ*). By utilizing (A1) in (33), we obtain

$$\begin{split} |p(\tau)| &\leq \ell^{\rho} \mathcal{Z}\_{0+}^{\tilde{\tau}}(\psi\_{0}, \Psi\_{1} | p(\tau)|, \psi\_{2} | q(\tau)|) \\ &\quad + |\delta(\tau)| \Big( |\hat{\mathcal{E}}| \Big( |\varepsilon|^{\rho} \mathcal{Z}\_{0+}^{\tilde{\tau}+\zeta}(\psi\_{0} + \psi\_{1} |p(\sigma)| + \psi\_{2} |q(\sigma)|) + \ell^{\rho} \mathcal{Z}\_{0+}^{\tilde{\tau}}(\psi\_{0} + \psi\_{0} |p(\mathcal{T})| + \psi\_{2} |q(\mathcal{T})|) \Big) \\ &\quad + |\mathcal{E}\_{1}| \Big( |\pi|^{\rho} \mathcal{Z}\_{0+}^{\tilde{\tau}+\varrho}(\psi\_{0} + \psi\_{1} |p(\sigma)| + \psi\_{2} |q(\sigma)|) + \ell^{\rho} \mathcal{Z}\_{0+}^{\tilde{\tau}}(\psi\_{0} + \psi\_{1} |p(\mathcal{T})| + \psi\_{2} |q(\mathcal{T})|) \Big) \Big). \end{split}$$

which results when taking the norm for *τ* ∈ E,

$$||p|| \le (\psi\_0 + \psi\_1 ||p|| + \psi\_2 ||q||) \mathcal{J}\_1 + (\psi\_0 + \psi\_1 ||p|| + \psi\_2 ||q||) \mathcal{K}\_1. \tag{43}$$

Similarly, we are capable of obtaining that

$$||q|| \le (\psi\_0 + \psi\_1||p|| + \psi\_2||q||)\mathcal{K}\_2 + (\psi\_0 + \psi\_1||p|| + \psi\_2||q||)\mathcal{J}\_2. \tag{44}$$

From (43) and (44), we get

$$\begin{aligned} |||p|||+||q||| &= \up{\psi}\_{0}(\mathcal{J}\_{1}+\mathcal{J}\_{2}) + \hat{\psi}\_{0}(\mathcal{K}\_{1}+\mathcal{K}\_{2}) + ||p|| \big{\big} \left[\up{\psi}\_{1}(\mathcal{J}\_{1}+\mathcal{J}\_{2}) + \hat{\psi}\_{1}(\mathcal{K}\_{1}+\mathcal{K}\_{2})\right] \\ &+ ||q|| \big{\big} \big{\big} \big{\psi}\_{1}(\mathcal{J}\_{1}+\mathcal{J}\_{2}) + \hat{\psi}\_{1}(\mathcal{K}\_{1}+\mathcal{K}\_{2})\big{\big} .\end{aligned}$$

which results, with ||(*p*, *q*)|| = ||*p*|| + ||*q*||,

$$||(p\_\prime q)|| \le \frac{\psi\_0(\mathcal{I}\_1 + \mathcal{I}\_2) + \psi\_0(\mathcal{K}\_1 + \mathcal{K}\_2)}{\Phi}.$$

As a result, Π(Δ) is bounded. Thus, the nonlinear alternative of Leray–Schauder [26] is valid and the operator Δ has at least one fixed point. It implies that the BVP (1) and (2) contain at least one solution on E.

**Theorem 2.** *Assume that <sup>f</sup>* , *<sup>g</sup>* : E × <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> *are continuous functions satisfying the condition: (*A2*) there exists constants <sup>φ</sup>m*, *<sup>φ</sup>*<sup>ˆ</sup> *<sup>m</sup>* ≥ 0(*m* = 1, 2) *such that*

$$\begin{aligned} |f(\boldsymbol{\tau},o\_1,o\_2) - f(\boldsymbol{\tau},\boldsymbol{\delta}\_1,\boldsymbol{\delta}\_2)| &\leq \boldsymbol{\phi}\_1|o\_1 - \boldsymbol{\delta}\_1| + \boldsymbol{\phi}\_2|o\_2 - \boldsymbol{\delta}\_2|,\\ |g(\boldsymbol{\tau},o\_1,o\_2) - g(\boldsymbol{\tau},\boldsymbol{\delta}\_1,\boldsymbol{\delta}\_2)| &\leq \boldsymbol{\phi}\_1|o\_1 - \boldsymbol{\delta}\_1| + \boldsymbol{\phi}\_2|o\_2 - \boldsymbol{\delta}\_2|, \forall o\_m,\boldsymbol{\delta}\_m \in \mathbb{R}, m = 1,2. \end{aligned}$$

*Furthermore, there exist* S1, S<sup>2</sup> > 0 *such that* | *f*(*τ*, 0, 0)|≤S1, |*g*(*τ*, 0, 0)|≤S2, *Then, given that*

$$(\mathcal{J}\_1 + \mathcal{J}\_2)(\phi\_1 + \phi\_2) + (\mathcal{K}\_1 + \mathcal{K}\_2)(\hat{\phi}\_1 + \hat{\phi}\_2) < 1,\tag{45}$$

*the BVP (1) and (2) has a unique solution on* E*, where* J1, K1, J2, K<sup>2</sup> *are given by (35)–(38) respectively.*

**Proof.** Let us fix *<sup>ϕ</sup>* <sup>≤</sup> (J1+J2)S1+(K1+K2)S<sup>2</sup> <sup>1</sup>−((J1+J2)(*φ*1+*φ*2)+(K1+K2)(*φ*<sup>ˆ</sup> <sup>1</sup>+*φ*ˆ <sup>2</sup>)) and demonstrate that <sup>Δ</sup>B*<sup>ϕ</sup>* ⊂ B*<sup>ϕ</sup>* when operator Δ is given by (32) and B*<sup>ϕ</sup>* = {(*p*, *q*) ∈ P×Q : ||(*p*, *q*)|| ≤ *ϕ*}. For (*p*, *q*) ∈ B*ϕ*, *τ* ∈ E

$$\begin{aligned} |f(\pi, p(\tau), q(\tau))| &\leq \phi\_1 |p(\tau)| + \phi\_2 |q(\tau)| + \mathcal{S}\_1 \\ &\leq \phi\_1 ||p|| + \phi\_2 ||q|| + \mathcal{S}\_1 \end{aligned}$$

and

$$\begin{split} |g(\tau, p(\tau), q(\tau))| &\leq \hat{\phi}\_1 |p(\tau)| + \hat{\phi}\_2 |q(\tau)| + \mathcal{S}\_2 \\ &\leq \hat{\phi}\_1 ||p|| + \hat{\phi}\_2 ||q|| + \mathcal{S}\_2. \end{split} \tag{46}$$

This guides to

$$\begin{split} |\Delta(p,q)(\tau)| \leq & ^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}} \Big[ |f(\tau,p(\tau),q(\tau)) - f(\tau,0,0)| + |f(\tau,0,0)| \Big] \\ & + |\delta(\tau)| \Big( |\hat{\mathcal{E}}| \Big( |\varepsilon|^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}+\epsilon} \mathcal{G}[(\sigma,p(\sigma),q(\sigma)) - \mathcal{g}(\sigma,0,0)| + |\mathcal{g}(\alpha,0,0)| \Big) \Big] \\ & + ^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}} f[(\mathcal{T},p(\mathcal{T}),q(\mathcal{T})) - f(\mathcal{T},0,0)| + |f(\mathcal{T},0,0)| \Big) \Big] \\ & + |\mathcal{E}\_1| \Big( |\pi|^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}+\epsilon} f[f(\sigma,p(\sigma),q(\sigma)) - f(\sigma,0,0)| + |f(\sigma,0,0)| \Big) \Big] \\ & + ^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}} \Big[ |g(\mathcal{T},p(\mathcal{T}),q(\mathcal{T})) - g(\mathcal{T},0,0)| + |g(\mathcal{T},0,0)| \Big) \Big) \Big) \end{split}$$

$$\begin{split} &\leq (\phi\_{1}||p||+\phi\_{2}||q||+\mathcal{S}\_{1}) \left\{ \frac{\left(\mathscr{T}^{\rho\tilde{\varsigma}}(1+|\delta|\big|\hat{\mathcal{E}}|)\right)}{\rho^{\mathfrak{z}}\Gamma(\check{\varsigma}+1)} + \frac{|\delta|\big|\pi|\big|\mathcal{E}\_{1}|\sigma^{\rho(\check{\varsigma}+\emptyset)}}{\rho^{\mathfrak{z}+\emptyset}\Gamma(\check{\varsigma}+\emptyset+1)} \right\} \\ &+ (\phi\_{1}||p||+\phi\_{2}||q||+\mathcal{S}\_{2}) \left\{ |\delta|\left(\frac{|\mathcal{E}\_{1}|\mathcal{T}^{\rho\check{\varsigma}}}{\rho^{\mathfrak{z}}\Gamma(\check{\varsigma}+1)} + \frac{|\hat{\mathcal{E}}||\boldsymbol{\varepsilon}|\big|\boldsymbol{\varrho}^{\rho(\check{\varsigma}+\emptyset)}}{\rho^{\mathfrak{z}+\emptyset}\Gamma(\check{\varsigma}+\emptyset+1)}\right) \right\} \end{split}$$

$$||\Delta\_1(p,q)|| \le (\phi\_1||p|| + \phi\_2||q|| + \mathcal{S}\_1)\mathcal{J}\_1 + (\hat{\phi}\_1||p|| + \hat{\phi}\_2||q|| + \mathcal{S}\_2)\mathcal{K}\_1.\tag{47}$$

Similarly, we obtain

$$|\Delta\_{2}(p,q)(\tau)| \leq (\hat{\rho}\_{1}||p|| + \hat{\rho}\_{2}||q|| + \mathcal{S}\_{2}) \left\{ \frac{\left(\mathcal{T}^{p\tilde{\varsigma}}(1+|\delta|\|\hat{\mathcal{E}})\right)}{\rho^{\xi}\Gamma(\zeta+1)} + \frac{|\delta|\|\varepsilon\|\mathcal{S}\_{2}|\sigma^{p(\zeta+\varrho)}}{\rho^{\xi+\varrho}\Gamma(\zeta+\varrho+1)} \right\}$$

$$+ (\phi\_{1}||p|| + \phi\_{2}||q|| + \mathcal{S}\_{1}) \left\{ |\delta| \left(\frac{\mathcal{T}^{p\tilde{\varsigma}}|\mathcal{S}\_{2}|}{\rho^{\xi}\Gamma(\zeta+1)} + \frac{|\pi||\hat{\mathcal{E}}|\sigma^{q(\zeta+\varrho)}}{\rho^{\xi+\varrho}\Gamma(\zeta+\varrho+1)}\right) \right\}$$

$$||\Delta\_{2}(p,q)|| \leq (\phi\_{1}||p|| + \phi\_{2}||q|| + \mathcal{S}\_{2})\mathcal{S}\_{2} + (\phi\_{1}||p|| + \phi\_{2}||q|| + \mathcal{S}\_{1})\mathcal{J}\_{2}.\tag{48}$$

As a result, (47) and (48) follow ||Δ(*p*, *q*)|| ≤ *ϕ*, and thus ΔB*<sup>ϕ</sup>* ⊂ B*ϕ*. Now, for (*p*1, *q*1),(*p*2, *q*2) ∈P×Q and any *τ* ∈ E, we get


$$\begin{split} & \leq (\phi\_{1} || p\_{1} - p\_{2} || + \phi\_{2} || q\_{1} - q\_{2} ||) \left\{ \frac{\left( \mathcal{T}^{\rho\_{5}^{\xi}} (1 + |\delta| |\hat{\mathcal{E}} ) \right)}{\rho^{\xi} \Gamma(\zeta + 1)} + \frac{|\delta| |\pi| |\mathcal{E}\_{1} | \sigma^{\rho(\xi + \varrho)}}{\rho^{\xi + \varrho} \Gamma(\zeta + \varrho + 1)} \right\} \\ & + (\hat{\phi}\_{1} || p\_{1} - p\_{2} || + \hat{\phi}\_{2} || q\_{1} - q\_{2} ||) \left\{ |\delta| \left( \frac{|\mathcal{E}\_{1} | \mathcal{T}^{\rho \zeta}}{\rho^{\xi} \Gamma(\zeta + 1)} + \frac{|\hat{\mathcal{E}} | |\varepsilon| \sigma^{\rho(\zeta + \zeta)}}{\rho^{\xi + \varrho} \Gamma(\zeta + \varrho + 1)} \right) \right\} \end{split}$$

$$0 \le (\mathcal{J}\_1(\phi\_1 + \phi\_2) + \mathcal{K}\_1(\phi\_1 + \phi\_2))(||p\_1 - p\_2|| + ||q\_1 - q\_2||).$$

Similarly, we obtain

$$\begin{split} & |\Delta\_2(p\_1, q\_1)(\tau) - \Delta\_2(p\_2, q\_2)(\tau)| \\ \leq & (\not{q\_1} ||p\_1 - p\_2|| + \not{q\_2} ||q\_1 - q\_2||) \left\{ \frac{\left(\mathcal{T}^{p\tilde{\varsigma}}(1 + |\delta| |\hat{\mathcal{E}}|)\right)}{\rho^{\mathfrak{r}} \Gamma(\zeta + 1)} + \frac{|\delta| |\varepsilon| |\mathcal{E}\_2| \alpha^{\rho(\zeta + \varsigma)}}{\rho^{\mathfrak{f} + \mathfrak{s}} \Gamma(\zeta + \varsigma + 1)} \right\} \\ & + (\phi\_1 ||p\_1 - p\_2|| + \phi\_2 ||q\_1 - q\_2||) \left\{ |\delta| \left(\frac{\mathcal{T}^{p\tilde{\varsigma}}|\mathcal{E}\_2|}{\rho^{\mathfrak{s}} \Gamma(\zeta + 1)} + \frac{|\pi| |\hat{\mathcal{E}}| \sigma^{\rho(\zeta + \varsigma)}}{\rho^{\mathfrak{s} + \varrho} \Gamma(\zeta + \varrho + 1)} \right) \right\} \\ \leq & (\mathcal{J}\_2(\phi\_1 + \phi\_2) + \mathcal{K}\_2(\phi\_1 + \phi\_2)) (||p\_1 - p\_2|| + ||q\_1 - q\_2||). \end{split}$$

Thus we obtain

$$||\Delta\_1(p\_1, q\_1)(\tau) - \Delta\_1(p\_2, q\_2)(\tau)|| \le (\mathcal{J}\_1(\phi\_1 + \phi\_2) + \mathcal{K}\_1(\phi\_1 + \phi\_2))(||p\_1 - p\_2|| + ||q\_1 - q\_2||). \tag{49}$$
 
$$\text{In a similar manner,}$$

$$||\Delta\_2(p\_1, q\_1)(\tau) - \Delta\_2(p\_2, q\_2)(\tau)|| \le (\mathcal{J}\_2(\phi\_1 + \phi\_2) + \mathcal{K}\_2(\phi\_1 + \phi\_2))(||p\_1 - p\_2|| + ||q\_1 - q\_2||). \tag{50}$$

Hence, using (49) and (50) we can get

$$\begin{aligned} \left| \left| \Delta(p\_1, q\_1)(\tau) - \Delta(p\_2, q\_2)(\tau) \right| \right| &\leq \left( (\mathcal{J}\_1 + \mathcal{J}\_2)(\phi\_1 + \phi\_2) + (\mathcal{K}\_1 + \mathcal{K}\_2)(\hat{\phi}\_1 + \hat{\phi}\_2) \right) \\ &\qquad \qquad \qquad \qquad (||p\_1 - p\_2|| + ||q\_1 - q\_2||). \end{aligned}$$

As a consequence of condition ((J<sup>1</sup> + J2)(*φ*<sup>1</sup> + *<sup>φ</sup>*2)+(K<sup>1</sup> + K2)(*φ*<sup>ˆ</sup> <sup>1</sup> + *φ*ˆ <sup>2</sup>)) < 1, Δ is a contraction operator. As an outcome of the Banach fixed point theorem, we can conclude that operator has a unique fixed point, which is the unique solution of the problem (1), and (2).

For brevity's sake, we'll use the following notations:

$$
\hat{\Omega}\_1 = \mathcal{J}\_1 - \frac{\mathcal{T}^{\rho\xi}}{\rho^\sharp \Gamma(\xi + 1)} + \mathcal{K}\_{1\prime} \tag{51}
$$

$$
\hat{\mathcal{Q}}\_2 = \mathcal{J}\_2 - \frac{\mathcal{T}^{\rho\zeta}}{\rho^\zeta \Gamma(\zeta + 1)} + \mathcal{K}\_2. \tag{52}
$$

**Theorem 3.** *Assume that <sup>f</sup>* , *<sup>g</sup>* : E × <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> *are continuous functions satisfying the assumption* (A2) *in Theorem 2. Furthermore, there exist positive constants* U1, U<sup>2</sup> *such that* ∀*τ* ∈ E *and ri* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 1, 2*.*

$$|f(\mathbf{r}, r\_1, r\_2)| \le \mathcal{U}\_{1\prime} \qquad |g(\mathbf{r}, r\_1, r\_2)| \le \mathcal{U}\_2. \tag{53}$$

*If*

$$\frac{\mathcal{T}^{\rho^\sharp\_\sharp}(\phi\_1 + \phi\_2)}{\rho^\sharp \Gamma(\zeta + 1)} + \frac{\mathcal{T}^{\rho^\sharp\_\sharp}(\phi\_1 + \phi\_2)}{\rho^\sharp \Gamma(\zeta + 1)} < 1,\tag{54}$$

*then the BVP (1), and (2) has at least one solution on* E*.*

**Proof.** Let us define a closed ball B*<sup>ϕ</sup>* = {(*p*, *q*) ∈P×Q : ||(*p*, *q*)|| ≤ *ϕ*} as follows and split Δ1, Δ<sup>2</sup> as:

$$\Delta\_{1,1}(p,q)(\tau) = \delta(\tau) \left( \widehat{\mathcal{E}}\left(\varepsilon^{\rho} \mathcal{Z}\_{0+}^{\xi+\varepsilon} \mathcal{g}(\sigma, p(\sigma), q(\sigma)) - ^{\rho} \mathcal{Z}\_{0+}^{\xi} f(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T})) \right) \right.$$

$$+ \mathcal{E}\_{1} \Big(\pi^{\rho} \mathcal{Z}\_{0+}^{\xi+\varrho} f(\sigma, p(\sigma), q(\sigma)) - ^{\rho} \mathcal{Z}\_{0+}^{\xi} g(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T})) \Big) \Big). \tag{55}$$

$$
\Delta\_{1,1}(p,q)(\tau) = ^\rho \mathcal{T}\_{0+}^\sharp f(\tau, p(\tau), q(\tau)),
\tag{56}
$$

$$\Delta\_{2,1}(p,q)(\tau) = \delta(\tau) \left( \hat{\mathcal{E}}\left(\pi^{\rho} \mathcal{Z}\_{0+}^{\mathbb{T}+\varrho} f(\sigma, p(\sigma), q(\sigma)) - ^{\rho} \mathcal{Z}\_{0+}^{\mathbb{T}} \mathcal{g}\left(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T})\right) \right) \right.$$

$$+ \mathcal{E}\_2 \Big(\varepsilon^{\rho} \mathcal{Z}\_{0+}^{\mathbb{T}+\xi} \mathcal{g}\left(\sigma, p(\sigma), q(\sigma)\right) - ^{\rho} \mathcal{Z}\_{0+}^{\mathbb{T}} f(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T})) \Big) \Big), \tag{57}$$

$$
\Delta\_{2,2}(p,q)(\tau) = {^\rho}T\_{0+}^{\tilde{\tau}}g(\tau,p(\tau),q(\tau)).\tag{58}
$$

In the Banach space P×Q, Δ1(*p*, *q*)(*τ*) = Δ1,1(*p*, *q*)(*τ*) + Δ1,2(*p*, *q*)(*τ*), and Δ2(*p*, *q*) (*τ*) = Δ2,1(*p*, *q*)(*τ*) + Δ2,2(*p*, *q*)(*τ*) on B*<sup>ϕ</sup>* are closed, bounded and convex subsets of P×Q. Let us fix *ϕ* ≤ max{J1U<sup>1</sup> + K1U2, J2U<sup>1</sup> + K2U2} and show that ΔB*<sup>ϕ</sup>* ⊂ B*<sup>ϕ</sup>* to verify Krasnoselskii's theorem [27] condition (i), If we choose *p* = (*p*1, *p*2), *q* = (*q*1, *q*2) ∈ B*ϕ*, and utilizing condition (53), we obtain


In a similar manner, we can find that

$$|\Delta\_{2,1}(p,q)(\tau) + \Delta\_{2,2}(p,q)(\tau)| \le \mathcal{U}\_1 \mathcal{J}\_2 + \mathcal{U}\_2 \mathcal{K}\_2 \le \varphi.$$

Clearly the above two inequalities lead to the fact that Δ1(*p*, *q*) + Δ2(*p*, *q*) ∈ B*ϕ*. Thus, we define operator (Δ1,2, Δ2,2) as a contraction-satisfying condition (iii) of Krasnoselskii's theorem [27]. For (*p*1, *q*1),(*p*2, *q*2) ∈ B*ϕ*, we have

$$\begin{split} |\Delta|\_{1,2}(p\_1, q\_1)(\tau) - \Delta\_{1,2}(p\_2, q\_2)(\tau)| &\leq \frac{\rho^{1-\frac{\tau}{\xi}}}{\Gamma(\frac{\xi}{\xi})} \int\_0^\tau \frac{\theta^{\rho-1}}{(\tau^{\rho} - \theta^{\rho})^{1-\frac{\xi}{\xi}}} \\ &\quad \times |f(\theta, p\_1(\theta), q\_1(\theta)) - f(\theta, p\_2(\theta), q\_2(\theta))| d\theta \\ &\leq \frac{\mathcal{T}^{\rho\frac{\tau}{\xi}}}{\rho^{\xi}\Gamma(\xi+1)} (\phi\_1 ||p\_1 - p\_2|| + \phi\_2 ||q\_1 - q\_2||) \end{split} \tag{59}$$

and

$$\begin{split} |\Delta\_{2,1}(p\_1, q\_1)(\tau) - \Delta\_{2,1}(p\_2, q\_2)(\tau)| &\leq \frac{\rho^{1-\frac{\tau}{\zeta}}}{\Gamma(\zeta)} \int\_0^\tau \frac{\theta^{\rho-1}}{(\tau^\rho - \theta^\rho)^{1-\overline{\zeta}}} \\ &\quad \times |g(\theta, p\_1(\theta), q\_1(\theta)) - g(\theta, p\_2(\theta), q\_2(\theta))| d\theta \\ &\leq \frac{\mathcal{T}^{\rho\overline{\zeta}}}{\rho^{\zeta}\Gamma(\zeta+1)} (\hat{\phi}\_1^{\gamma}||p\_1 - p\_2|| + \hat{\phi}\_2^{\gamma}||q\_1 - q\_2||). \end{split} \tag{60}$$

As a result (59) and (60),

$$\begin{aligned} &| (\Delta\_{1,2}, \Delta\_{2,2}) (p\_1, q\_1) (\tau) - (\Delta\_{1,2}, \Delta\_{2,2}) (p\_2, q\_2) (\tau) | \\ \leq & \frac{\mathcal{T}^{\rho\_\zeta^\sharp} (\phi\_1 + \phi\_2)}{\rho^\sharp \Gamma(\zeta + 1)} + \frac{\mathcal{T}^{\rho\_\zeta^\sharp} (\phi\_1 + \phi\_2)}{\rho^\zeta \Gamma(\zeta + 1)} (||p\_1 - p\_2|| + ||q\_1 - q\_2||) .\end{aligned}$$

is a contraction by (54). Therefore, condition (iii) of the Theorem is satisfied. Following that, we can establish that the operator (Δ1,1, Δ2,1) satisfies the Krasnoselskii theorem's [27] condition (ii). We can infer the continuous existence of the (Δ1,1, Δ2,1) operator by examining the continuity of the *f* , *g* functions. For each (*p*, *q*) ∈ B*<sup>ϕ</sup>* we have

$$\begin{aligned} &|\Delta\_{1,1}(p,q)(\tau)| \\ \leq &|\delta(\tau)| \left( |\hat{\mathcal{E}}| \big( |\varepsilon| \,^{\rho} \mathcal{Z}\_{0+}^{\tilde{\zeta}+\zeta} \big| g(\sigma,p(\sigma),q(\sigma)) \big| \big( +^{\rho} \mathcal{Z}\_{0+}^{\tilde{\zeta}} \big| f(\mathcal{T},p(\mathcal{T}),q(\mathcal{T})) \big| \big) \right) \\ &+|\mathcal{E}\_1| \Big( |\pi| \,^{\rho} \mathcal{Z}\_{0+}^{\tilde{\zeta}+\varrho} |f(\sigma,p(\sigma),q(\sigma))| \,^{+\rho} \mathcal{Z}\_{0+}^{\tilde{\zeta}} \big| g(\mathcal{T},p(\mathcal{T}),q(\mathcal{T})) \big| \Big) \Big) \end{aligned}$$

$$\begin{split} &\leq \mathcal{U}\_{1} \left\{ \frac{\left(\mathcal{T}^{\rho\xi}(|\boldsymbol{\delta}| | \widehat{\mathcal{E}}|) \right)}{\rho^{\xi} \Gamma(\zeta + 1)} + \frac{|\boldsymbol{\delta}| |\boldsymbol{\pi}| |\boldsymbol{\mathcal{E}}\_{1}| \sigma^{\rho(\zeta + \varrho)}}{\rho^{\zeta + \varrho} \Gamma(\zeta + \varrho + 1)} \right\} \\ &\quad + \mathcal{U}\_{2} \left\{ |\boldsymbol{\delta}| \left(\frac{|\mathcal{E}\_{1}| \mathcal{T}^{\rho\zeta}}{\rho^{\zeta} \Gamma(\zeta + 1)} + \frac{|\widehat{\mathcal{E}}| |\boldsymbol{\varepsilon}| \boldsymbol{\alpha}^{\rho(\zeta + \zeta)}}{\rho^{\zeta + \varrho} \Gamma(\zeta + \varrho + 1)} \right) \right\} \\ &= \widehat{\Omega}\_{1\prime} \end{split}$$

$$\begin{split} |\Delta\_{2,1}(p,q)(\tau)| &\leq \mathcal{U}\_{2} \left\{ \frac{\left(\mathcal{T}^{p\tilde{\varsigma}}(|\boldsymbol{\delta}|\boldsymbol{|\mathcal{E}}|)\right)}{\rho^{\mathfrak{s}}\Gamma(\zeta+1)} + \frac{|\boldsymbol{\delta}| |\boldsymbol{\varepsilon}| |\mathcal{E}\_{2}| \boldsymbol{\varepsilon} \rho^{\rho(\zeta+\varrho)}}{\rho^{\mathfrak{s}+\mathfrak{s}}\Gamma(\zeta+\varrho+1)} \right\} \\ &+ \mathcal{U}\_{1} \left\{ |\boldsymbol{\delta}| \left(\frac{\mathcal{T}^{p\tilde{\varsigma}}|\mathcal{E}\_{2}|}{\rho^{\mathfrak{s}}\Gamma(\zeta+1)} + \frac{|\boldsymbol{\pi}| |\boldsymbol{\mathcal{E}}| \boldsymbol{\varepsilon}^{\rho(\zeta+\varrho)}}{\rho^{\mathfrak{s}+\mathfrak{q}}\Gamma(\zeta+\varrho+1)} \right) \right\} \\ &= \hat{\Omega}\_{2\prime} \end{split}$$

which leads to

$$||(\Delta\_{1,1}, \Delta\_{2,1})(p, q)|| \le \hat{\Omega}\_1 + \hat{\Omega}\_2.$$

From the above inequalities, the set (Δ1,1, Δ2,1)B*<sup>ϕ</sup>* is uniformly bounded. The following step will demonstrate that the set (Δ1,1, Δ2,1)B*<sup>ϕ</sup>* is equicontinuous. For *τ*1, *τ*<sup>2</sup> ∈ E with *τ*<sup>1</sup> < *τ*<sup>2</sup> and for any (*p*, *q*) ∈ B*<sup>ϕ</sup>* we get

$$|\Delta\_{1,1}(p\_\prime q)(\tau\_2) - \Delta\_{1,1}(p\_\prime q)(\tau\_1)|$$

$$\begin{split} \leq & |\delta(\tau\_{2}) - \delta(\tau\_{1})| \left( |\hat{\mathcal{E}}| \big( |\epsilon|^{\rho} \mathcal{I}\_{0+}^{\mathbb{S}+\varsigma} |\mathcal{g}(\omega, p(\omega), q(\omega))| + {}^{\rho} \mathcal{I}\_{0+}^{\mathbb{S}} |f(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T}))| \big) \right) \\ & + |\mathcal{E}\_{1}| \big( |\pi|^{\rho} \mathcal{I}\_{0+}^{\mathbb{S}+\varsigma} |f(\sigma, p(\sigma), q(\sigma))| + {}^{\rho} \mathcal{I}\_{0+}^{\mathbb{S}} |g(\mathcal{T}, p(\mathcal{T}), q(\mathcal{T}))| \big) \Big) \\ \leq & |\delta(\tau\_{2}) - \delta(\tau\_{1})| \left( \mathcal{U}\_{1} \Big( \frac{\mathcal{T}^{\rho\mathbb{S}}(|\delta| |\hat{\mathcal{E}}|)}{\rho^{\mathbb{S}} \Gamma(\xi+1)} + \frac{|\delta| |\pi| |\mathcal{E}\_{1}| \sigma^{\rho(\zeta+\varrho)}}{\rho^{\mathbb{S}+\varsigma} \Gamma(\xi+\varrho+1)} \right) \\ & + \mathcal{U}\_{2}|\delta| \big( \frac{|\mathcal{E}\_{1}| \mathcal{T}^{\rho\mathbb{S}}}{\rho^{\mathbb{S}} \Gamma(\zeta+1)} + \frac{|\hat{\mathcal{E}}| |\epsilon| \omega^{\rho(\zeta+\varsigma)}}{\rho^{\mathbb{S}+\varsigma} \Gamma(\zeta+\varsigma+1)} \Big). \end{split}$$

Likewise, we obtain

$$\begin{split} & \left| \Delta\_{2,1}(p,q)(\tau\_{2}) - \Delta\_{2,1}(p,q)(\tau\_{1}) \right| \\ & \leq \left| \delta(\tau\_{2}) - \delta(\tau\_{1}) \right| \left( \mathcal{U}\_{2} \left( \frac{\left( \mathcal{T}^{\rho^{\sharp}\_{\zeta}}(|\delta| |\hat{\mathcal{E}}|) \right)}{\rho^{\zeta} \Gamma(\zeta+1)} + \frac{|\delta| |\varepsilon| |\mathcal{E}\_{2}| \mathcal{O}^{\rho(\zeta+\zeta)}}{\rho^{\zeta+\varrho} \Gamma(\zeta+\varrho+1)} \right), \\ & \quad + \mathcal{U}\_{1} \left( |\delta| \left( \frac{\mathcal{T}^{\rho^{\sharp}\_{\zeta}}|\mathcal{E}\_{2}|}{\rho^{\zeta} \Gamma(\zeta+1)} + \frac{|\pi| |\hat{\mathcal{E}}| \mathcal{O}^{\rho(\zeta+\varrho)}}{\rho^{\zeta+\varrho} \Gamma(\zeta+\varrho+1)} \right) \right). \end{split}$$

Therefore |(Δ1,1, Δ2,1(*τ*2)) − (Δ1,1, Δ2,1(*τ*1))| → 0 as *τ*<sup>2</sup> → *τ*<sup>1</sup> independent of (*p*, *q*) ∈ B*<sup>ϕ</sup>* Thus the set (Δ1,1, Δ2,1)B*<sup>ϕ</sup>* is equicontinuous. As an outcome, the Arzela– Ascoli theorem implies that the operator (Δ1,1, Δ2,1) is compact on B*ϕ*. Krasnoselskii's theorem [27] statement leads us to the conclusion that the problem (1) and (2) has at least one solution on E.

#### **4. Example**

Consider the following Liouville–Caputo type generalized FDEs coupled system:

$$\begin{cases} \stackrel{\circ}{\mathcal{C}} \mathcal{D}\_{0+}^{\frac{5}{4}} p(\tau) = f(\tau, p(\tau), q(\tau)), \tau \in \mathcal{E} := [0, 1],\\ \stackrel{\circ}{\mathcal{C}} \mathcal{D}\_{0+}^{\frac{3}{4}} q(\tau) = g(\tau, p(\tau), q(\tau)), \tau \in \mathcal{E} := [0, 1], \end{cases} \tag{61}$$

supplemented with boundary conditions:

$$\begin{cases} p(0) = 0, \; q(0) = 0, \; p(1) = \frac{1}{6}^{\frac{3}{4}} \mathcal{L}^{\frac{13}{20}} q(\frac{7}{10}), \; q(1) = \frac{1}{7}^{\frac{3}{4}} \mathcal{L}^{\frac{17}{20}} p(\frac{1}{2}), \end{cases} \tag{62}$$

where *ξ* = <sup>5</sup> <sup>4</sup> , *<sup>ζ</sup>* <sup>=</sup> <sup>31</sup> <sup>20</sup> , *<sup>ρ</sup>* <sup>=</sup> <sup>3</sup> <sup>4</sup> , <sup>T</sup> <sup>=</sup> 1, <sup>=</sup> <sup>1</sup> <sup>6</sup> , <sup>=</sup> <sup>7</sup> <sup>10</sup> , *<sup>π</sup>* <sup>=</sup> <sup>1</sup> <sup>7</sup> , *<sup>σ</sup>* <sup>=</sup> <sup>1</sup> <sup>2</sup> , *<sup>ς</sup>* <sup>=</sup> <sup>13</sup> <sup>20</sup> , <sup>=</sup> <sup>17</sup> <sup>20</sup> and

$$f(\tau, p(\tau), q(\tau)) = \frac{(1+\tau)}{30} \left( \frac{|p(\tau)|}{1+|p(\tau)|} + \frac{1}{3} \cos(q(\tau)) + 3\tau \right),\tag{63}$$

$$g(\tau, p(\tau), q(\tau)) = \frac{e^{-\tau}}{25} \left( \frac{\sqrt{\tau} + 1}{5} + \frac{1}{6} \cos(p(\tau)) + \frac{|q(\tau)|}{1 + |q(\tau)|} \right). \tag{64}$$

With *ψ*<sup>0</sup> = <sup>1</sup> <sup>10</sup> , *<sup>ψ</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>30</sup> , *<sup>ψ</sup>*<sup>2</sup> <sup>=</sup> <sup>1</sup> <sup>90</sup> , *<sup>ψ</sup>*<sup>ˆ</sup> <sup>0</sup> = <sup>1</sup> <sup>125</sup> , *<sup>ψ</sup>*<sup>ˆ</sup> <sup>1</sup> = <sup>1</sup> <sup>25</sup> , and *<sup>ψ</sup>*<sup>ˆ</sup> <sup>2</sup> = <sup>1</sup> <sup>150</sup> , the functions *f* and *g* clearly satisfy the (A1) condition. Next, we find that (J1) = 2.5370237266984113, (K1) = 0.17111607453629377, J<sup>2</sup> = 0.0906406939922634, K<sup>2</sup> = 2.274156747108814, J*i*, K*<sup>i</sup>* (*i* = 1, 2) are respectively given by (35),(36),(37) and (38), based on the data available. Thus *<sup>ψ</sup>*1(J<sup>1</sup> + J2) + *<sup>ψ</sup>*<sup>ˆ</sup> <sup>1</sup>(K<sup>1</sup> <sup>+</sup> <sup>K</sup>2) 0.18539972688882678 <sup>&</sup>lt; 1, *<sup>ψ</sup>*2(J<sup>1</sup> <sup>+</sup> <sup>J</sup>2) + *<sup>ψ</sup>*<sup>ˆ</sup> <sup>2</sup>(K<sup>1</sup> <sup>+</sup> <sup>K</sup>2) 0.04549809015197488 < 1, all the conditions of Theorem 1 are satisfied, and there is at least one solution for problem (61) and (62) on [0, 1] with *f* and *g* given by (63) and (64) respectively.

In addition, we'll use

$$f(\tau, p(\tau), q(\tau)) = \frac{\tau}{3} + \frac{3}{4(\tau + 16)} + \frac{|p(\tau)|}{1 + |p(\tau)|} + \frac{2}{75} \cos(q(\tau)),\tag{65}$$

$$g(\tau, p(\tau), q(\tau)) = \frac{(1 + e^{-\tau})}{4} + \frac{19}{400} \cos(p(\tau)) + \frac{1}{60} \frac{|q(\tau)|}{1 + |q(\tau)|},\tag{66}$$

to demonstrate Theorem 2. It is simple to demonstrate that *f* and *g* are continuous and satisfy the assumption (A2) with *<sup>φ</sup>*<sup>1</sup> <sup>=</sup> <sup>3</sup> <sup>64</sup> , *<sup>φ</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup> <sup>75</sup> , *<sup>φ</sup>*<sup>ˆ</sup> <sup>1</sup> = <sup>19</sup> <sup>400</sup> and *<sup>φ</sup>*<sup>ˆ</sup> <sup>2</sup> = <sup>1</sup> <sup>60</sup> . All the assumptions of Theorem <sup>2</sup> are also satisfied with (J<sup>1</sup> + J2)(*φ*<sup>1</sup> + *<sup>φ</sup>*2)+(K<sup>1</sup> + K2)(*φ*<sup>ˆ</sup> <sup>1</sup> + *φ*ˆ <sup>2</sup>) 0.35014782699385444 < 1. As a result, Theorem 2 holds true, and the problem (61) and (62) with *f* and *g* given by (65) and (66) respectively, has a unique solution on [0,1].

#### **5. Ulam–Hyers Stability Results for the Problem (1) and (2)**

The U–H stability of the solutions to the BVP (1) and (2) will be discussed in this section using the integral representation of their solutions defined by

$$p(\tau) = \Delta\_1(p, q)(\tau), \ q(\tau) = \Delta\_2(p, q)(\tau), \tag{67}$$

where Δ<sup>1</sup> and Δ<sup>2</sup> are given by (33) and (34). Consider the following definitions of nonlinear operators

$$\mathcal{H}\_{1}, \mathcal{H}\_{2} \in \mathcal{C}(\mathcal{E}, \mathbb{R}) \times \mathcal{C}(\mathcal{E}, \mathbb{R}) \to \mathcal{C}(\mathcal{E}, \mathbb{R}),$$

$$\begin{cases} \stackrel{\rho}{\mathcal{C}} \mathcal{D}\_{0+}^{\mathbb{E}} p(\tau) - f(\tau, p(\tau), q(\tau)) = \mathcal{H}\_{1}(p, q)(\tau), \tau \in \mathcal{E},\\ \stackrel{\rho}{\mathcal{C}} \mathcal{D}\_{0+}^{\mathbb{E}} q(\tau) - g(\tau, p(\tau), q(\tau)) = \mathcal{H}\_{1}(p, q)(\tau), \tau \in \mathcal{E}. \end{cases}$$

It considered the following inequalities for some *λ*ˆ 1, *λ*ˆ <sup>2</sup> > 0 :

$$||\mathcal{H}\_1(p,q)|| \le \hat{\lambda}\_1 ||\mathcal{H}\_2(p,q)|| \le \hat{\lambda}\_2. \tag{68}$$

**Definition 4.** *The coupled system (1) and (2) is said to be U–H stable if* V1, V<sup>2</sup> > 0 *and there exists a unique solution* (*p*, *<sup>q</sup>*) ∈ C(E, <sup>R</sup>) *of a problem (1) and (2) with*

$$||(p\_\prime q) - (p^\*, q^\*)|| \le \mathcal{V}\_1 \hat{\lambda}\_1 + \mathcal{V}\_2 \hat{\lambda}\_2\hat{\lambda}\_1$$

<sup>∀</sup>(*p*, *<sup>q</sup>*) ∈ C(E, <sup>R</sup>) of inequality (68).

**Theorem 4.** *Assume that* (A2) *holds. Then the problem (1) and (2) is U–H stable.*

**Proof.** Let (*p*, *<sup>q</sup>*) ∈ C(E, <sup>R</sup>) × C(E, <sup>R</sup>) be the (1)–(2) solution of the problem that satisfies (33) and (34). Let (*p*, *q*) be any solution that meets the condition (68):

$$\begin{cases} \prescript{\rho}{}{\mathcal{D}}\_{0+}^{\sharp}p(\tau) = f(\tau, p(\tau), q(\tau)) + \mathcal{H}\_1(p, q)(\tau), \tau \in \mathcal{E}\_{\prime},\\ \prescript{\rho}{}{\mathcal{D}}\_{0+}^{\sharp}q(\tau) = g(\tau, p(\tau), q(\tau)) + \mathcal{H}\_1(p, q)(\tau), \tau \in \mathcal{E}\_{\prime}. \end{cases}$$

so,

$$\begin{split} p^\*(\tau) &= \Delta\_1(p^\*, q^\*)(\tau) + ^\rho \mathcal{Z}\_{0+}^\sharp \mathcal{H}\_1(p, q)(\tau) \\ &+ \delta(\tau) \Big( \hat{\mathcal{E}} \Big[ \varepsilon^\rho \mathcal{Z}\_{0+}^{\sharp + \varsigma} \mathcal{H}\_2(p, q)(\sigma) - ^\rho \mathcal{Z}\_{0+}^{\sharp \tau} \mathcal{H}\_1(p, q)(\mathcal{T}) \Big] \\ &+ \mathcal{E}\_1 \Big[ \pi^\rho \mathcal{Z}\_{0+}^{\sharp + \varsigma} \mathcal{H}\_1(p, q)(\sigma) - ^\rho \mathcal{Z}\_{0+}^{\sharp \varsigma} \mathcal{H}\_2(p, q)(\mathcal{T}) \Big] \Big). \end{split}$$

It follows that

$$\begin{split} |\Delta\_{1}(p^{\*},q^{\*})(\tau)-p^{\*}(\tau)| &\leq \ell^{\rho}\mathcal{Z}\_{0+}^{\frac{\delta}{2}}|\mathcal{H}\_{1}(p,q)(\tau)| \\ &\quad + |\delta(\tau)| \left( |\hat{\mathcal{E}}| \left| |\epsilon|^{\rho}\mathcal{Z}\_{0+}^{\frac{\delta}{2}+\epsilon}|\mathcal{H}\_{2}(p,q)(\sigma)| + {}^{\rho}\mathcal{Z}\_{0+}^{\frac{\delta}{2}}|\mathcal{H}\_{1}(p,q)(\mathcal{T})| \right| \right) \\ &\quad + |\mathcal{E}\_{1}| \left[ |\pi|\,^{\rho}\mathcal{Z}\_{0+}^{\frac{\delta}{2}+\epsilon}|\mathcal{H}\_{1}(p,q)(\sigma)| + {}^{\rho}\mathcal{Z}\_{0+}^{\frac{\delta}{2}}|\mathcal{H}\_{2}(p,q)(\mathcal{T})| \right] \Bigg) \\ &\leq \hat{\lambda}\_{1} \left\{ \frac{\left( \mathcal{T}^{\delta\xi}(1+|\mathcal{S}|\left|\hat{\mathcal{E}}\right|) \right)}{\rho^{\xi}\Gamma(\xi+1)} + \frac{|\delta|\left|\pi|\left|\mathcal{E}\_{1}|\sigma^{\rho(\xi+\epsilon)}\right|}{\rho^{\xi+\epsilon}\Gamma(\xi+\epsilon+1)} \right\} \\ &\quad + \hat{\lambda}\_{2} \left\{ |\delta|\left( \frac{|\mathcal{E}\_{1}|\mathcal{T}^{\theta\xi}}{\rho^{\xi}\Gamma(\xi+1)} + \frac{|\hat{\mathcal{E}}|\|\epsilon|\phi^{\rho(\xi+\epsilon)}}{\rho^{\xi+\epsilon}\Gamma(\xi+\epsilon+1)} \right) \right\} \\ &\leq \mathcal{J}\_{1}\hat{\lambda}\_{1} + \mathcal{K}\_{1}\hat{\lambda}\_{2}. \end{split}$$

Similarly, we obtain

$$\begin{split} |\Delta\_{2}(p^{\*},q^{\*})(\tau)-q^{\*}(\tau)| \leq &\hat{\lambda}\_{2}\left\{ \frac{\left(\mathcal{T}^{\rho^{\mathbb{F}}}(1+|\delta||\hat{\mathcal{E}})\right)}{\rho^{\mathbb{F}}\Gamma(\zeta+1)}+\frac{|\delta||\varepsilon||\mathcal{E}\_{2}|\alpha^{\rho(\zeta+\varrho)}}{\rho^{\mathbb{F}+\varrho}\Gamma(\zeta+\varrho+1)} \right\} \\ &+\hat{\lambda}\_{1}\left\{ |\delta|\left(\frac{\mathcal{T}^{\rho\mathbb{F}}|\mathcal{E}\_{2}|}{\rho^{\mathbb{F}}\Gamma(\zeta+1)}+\frac{|\pi||\hat{\mathcal{E}}|\sigma^{\rho(\zeta+\varrho)}}{\rho^{\mathbb{F}+\varrho}\Gamma(\zeta+\varrho+1)}\right) \right\} \\ \leq &\mathcal{I}\_{2}\hat{\lambda}\_{1}+\mathcal{K}\_{2}\hat{\lambda}\_{2\prime} \end{split}$$

where J1, K1, J2, and K<sup>2</sup> are defined in (35)–(38), respectively. As an outcome, we deduce from operator Δ's fixed-point property, which is defined by (33) and (34),

$$\begin{split} |p(\tau) - p^\*(\tau)| &= |p(\tau) - \Delta\_1(p^\*, q^\*)(\tau) + \Delta\_1(p^\*, q^\*)(\tau) - p^\*(\tau)| \\ &\le |\Delta\_1(p, q)(\tau) - \Delta\_1(p^\*, q^\*)(\tau)| + |\Delta\_1(p^\*, q^\*)(\tau) - p^\*(\tau)| \\ &\le ((\mathcal{J}\_1 \phi\_1 + \mathcal{K}\_1 \hat{\phi}\_1) + (\mathcal{J}\_1 \phi\_2 + \mathcal{K}\_1 \hat{\phi}\_2)) ||(p, q) - (p^\*, q^\*)|| \\ &+ \mathcal{J}\_1 \hat{\lambda}\_1 + \mathcal{K}\_1 \hat{\lambda}\_2. \end{split} \tag{69}$$

$$\begin{split} |q(\tau) - q^\*(\tau)| &= |q(\tau) - \Delta\_2(p^\*, q^\*)(\tau) + \Delta\_2(p^\*, q^\*)(\tau) - q^\*(\tau)| \\ &\le |\Delta\_2(p, q)(\tau) - \Delta\_2(p^\*, q^\*)(\tau)| + |\Delta\_2(p^\*, q^\*)(\tau) - q^\*(\tau)| \\ &\le ((\mathcal{J}\_2 \phi\_1 + \mathcal{K}\_2 \phi\_1) + (\mathcal{J}\_2 \phi\_2 + \mathcal{K}\_2 \phi\_2)) ||(p, q) - (p^\*, q^\*)|| \\ &+ \mathcal{J}\_2 \hat{\lambda}\_1 + \mathcal{K}\_2 \hat{\lambda}\_2. \end{split} \tag{70}$$

From the above Equations (69) and (70) it follows that

$$\begin{aligned} ||(\boldsymbol{p}, \boldsymbol{q}) - (\boldsymbol{p}^\*, \boldsymbol{q}^\*)|| &\leq (\mathcal{J}\_1 + \mathcal{J}\_2)\hat{\lambda}\_1 + (\mathcal{K}\_1 + \mathcal{K}\_2)\hat{\lambda}\_2 \\ &+ ((\mathcal{J}\_1 + \mathcal{J}\_2)(\boldsymbol{\phi}\_1 + \boldsymbol{\phi}\_2) + (\mathcal{K}\_1 + \mathcal{K}\_2)(\boldsymbol{\phi}\_1 + \boldsymbol{\phi}\_2))||(\boldsymbol{p}, \boldsymbol{q}) - (\boldsymbol{p}^\*, \boldsymbol{q}^\*)||. \end{aligned}$$

$$\begin{aligned} ||(p\_\prime q) - (p^\* \lrcorner q^\*)|| &\leq \frac{(\mathcal{J}\_1 + \mathcal{J}\_2)\hat{\lambda}\_1 + (\mathcal{K}\_1 + \mathcal{K}\_2)\hat{\lambda}\_2}{1 - ((\mathcal{J}\_1 + \mathcal{J}\_2)(\phi\_1 + \phi\_2) + (\mathcal{K}\_1 + \mathcal{K}\_2)(\hat{\phi}\_1 + \hat{\phi}\_2))} \\ &\leq \mathcal{V}\_1 \hat{\lambda}\_1 + \mathcal{V}\_2 \hat{\lambda}\_2. \end{aligned}$$

with

$$
\mathcal{V}\_1 = \frac{\mathcal{J}\_1 + \mathcal{J}\_2}{1 - ((\mathcal{J}\_1 + |\mathcal{J}\_2)(\phi\_1 + \phi\_2) + (\mathcal{K}\_1 + |\mathcal{K}\_2)(\hat{\phi}\_1 + \hat{\phi}\_2))},
$$

$$
\mathcal{V}\_2 = \frac{\mathcal{K}\_1 + \mathcal{K}\_2}{1 - ((\mathcal{J}\_1 + \mathcal{J}\_2)(\phi\_1 + \phi\_2) + (\mathcal{K}\_1 + \mathcal{K}\_2)(\hat{\phi}\_1 + \hat{\phi}\_2))}.
$$

Hence, the problem (1)–(2) is U–H stable.

#### **6. Example**

Consider the following Liouville–Caputo type generalized FDEs coupled system:

$$\begin{cases} \frac{13}{\zeta} \mathcal{D}\_{0+}^{\frac{5}{2}} p(\tau) = \frac{\sqrt{\tau}}{2} + \frac{1}{5(\tau + 25)} \frac{|p(\tau)|}{1 + |p(\tau)|} + \frac{3}{80} \cos(q(\tau)), \tau \in [0, 1],\\ \frac{13}{\zeta} \mathcal{D}\_{0+}^{\frac{31}{20}} q(\tau) = \frac{\tau}{5} + \frac{17}{300} \cos(p(\tau)) + \frac{1}{70} \frac{|q(\tau)|}{1 + |q(\tau)|}, \tau \in [0, 1], \end{cases} \tag{71}$$

supplemented with boundary conditions:

$$\begin{cases} p(0) = 0, \; q(0) = 0, \; p(1) = \frac{5}{6} \prescript{\frac{19}{20}}{}{\mathcal{Z}}^{\frac{13}{20}} q(\frac{9}{20}), \; q(1) = \prescript{6}{7}{}{\mathcal{Z}}^{\frac{17}{20}} p(\frac{13}{20}), \end{cases} \tag{72}$$

where *ξ* = <sup>5</sup> <sup>4</sup> , *<sup>ζ</sup>* <sup>=</sup> <sup>31</sup> <sup>20</sup> , *<sup>ρ</sup>* <sup>=</sup> <sup>19</sup> <sup>20</sup> , <sup>T</sup> <sup>=</sup> 1, <sup>=</sup> <sup>5</sup> <sup>6</sup> , <sup>=</sup> <sup>9</sup> <sup>20</sup> , *<sup>π</sup>* <sup>=</sup> <sup>6</sup> <sup>7</sup> , *<sup>σ</sup>* <sup>=</sup> <sup>13</sup> <sup>20</sup> , *<sup>ς</sup>* <sup>=</sup> <sup>13</sup> <sup>20</sup> , <sup>=</sup> <sup>17</sup> <sup>20</sup> and

$$|f(\tau, p\_1(\tau), q\_1(\tau)) - f(\tau, p\_2(\tau), q\_2(\tau))| = \frac{1}{125}|p\_1(\tau) - p\_2(\tau)| + \frac{3}{80}|q\_1(\tau) - q\_2(\tau)|\tag{73}$$

$$|g(\tau, p\_1(\tau), q\_1(\tau)) - g(\tau, p\_2(\tau), q\_2(\tau))| = \frac{17}{300}|p\_1(\tau) - p\_2(\tau)| + \frac{1}{70}|q\_1(\tau) - q\_2(\tau)|.\tag{74}$$

With *φ*<sup>1</sup> = <sup>1</sup> <sup>125</sup> , *<sup>φ</sup>*<sup>2</sup> <sup>=</sup> <sup>3</sup> <sup>80</sup> , *<sup>φ</sup>*<sup>ˆ</sup> <sup>1</sup> = <sup>17</sup> <sup>300</sup> , and *<sup>φ</sup>*<sup>ˆ</sup> <sup>2</sup> = <sup>1</sup> <sup>70</sup> , the functions *f* and *g* clearly satisfy the (A2) condition. Next, we find that (J1) = 1.9529307397739033,(K1) = 0.21135021378560123, J<sup>2</sup> = 0.42682560046779994, K<sup>2</sup> = 1.6225052940838325, J*i*, K*i*(*i* = 1, 2) are respectively given by (35),(36),(37) and (38), based on the data available. Thus ((J<sup>1</sup> + J2)(*φ*<sup>1</sup> + *<sup>φ</sup>*2)+(K<sup>1</sup> + K2)(*φ*<sup>ˆ</sup> <sup>1</sup> + *φ*ˆ <sup>2</sup>)) 0.2383953280869716 < 1, all the conditions of Theorem 5.2 are satisfied, and there is a unique solution for problem (71) and (72) on [0, 1], which is stable for Ulam–Hyers, with *f* and *g* given by (73) and (74) respectively.

#### **7. Existence Results for the Problem (1) and (75)**

Furthermore, we are investigating the system (1) under the following conditions:

$$\begin{cases} p(0) = 0, \quad q(0) = 0, \\ p(T) = \varepsilon^{\rho} \mathcal{T}\_{0^{+}}^{\mathbb{C}} q(\mathcal{o}) = \frac{\varepsilon \rho^{1-\mathfrak{c}}}{\Gamma(\mathfrak{c})} \int\_{0}^{\mathcal{O}} \frac{\theta^{\rho-1}}{(\alpha^{\rho} - \theta^{\rho})^{1-\mathfrak{c}}} q(\theta) d\theta, \\ q(T) = \pi^{\rho} \mathcal{T}\_{0+}^{\mathbb{C}} p(\mathcal{o}) = \frac{\pi \rho^{1-\mathfrak{c}}}{\Gamma(\mathfrak{o})} \int\_{0}^{\mathcal{O}} \frac{\theta^{\rho-1}}{(\alpha^{\rho} - \theta^{\rho})^{1-\mathfrak{c}}} p(\theta) d\theta, \\ 0 < \mathcal{O} < \mathcal{T}. \end{cases} \tag{75}$$

Bear in mind that the conditions (2) contain strips of varying lengths, whereas the one in (75) contains only one strip of the same length (0, ). We introduce the following notations for computational ease:

$$\mathcal{E}\_1 = \varepsilon \frac{\mathcal{O}^{\rho(\emptyset + 1)}}{\rho^{\emptyset + 1} \Gamma(\emptyset + 2)}, \; \mathcal{E}\_2 = \pi \frac{\mathcal{O}^{\rho(\emptyset + 1)}}{\rho^{\emptyset + 1} \Gamma(\emptyset + 2)}, \; \hat{\mathcal{E}} = \frac{\mathcal{T}^{\rho}}{\rho}, \tag{76}$$

$$
\mathcal{G} = \hat{\mathcal{E}}^2 - \mathcal{E}\_1 \mathcal{E}\_2 \neq 0,\tag{77}
$$

$$
\delta(\tau) = \left(\frac{\tau^{\rho}}{\rho \mathcal{G}}\right). \tag{78}
$$

**Lemma 4.** *Given the functions* ˆ *<sup>f</sup>* , *<sup>g</sup>*<sup>ˆ</sup> <sup>∈</sup> *<sup>C</sup>*(0, <sup>T</sup> ) ∩ L(0, <sup>T</sup> ), *<sup>p</sup>*, *<sup>q</sup>* ∈ AC<sup>2</sup> *<sup>γ</sup>*(E) *and* Λ = 0. *Then the solution of the coupled BVP:*

$$\begin{cases} \,^{\rho}\_{\mathcal{C}} \mathcal{D}^{\tilde{z}}\_{0+} p(\tau) = \,^{\rho}\_{\mathcal{I}}(\tau), \tau \in \mathcal{E} := [0, \mathcal{T}],\\ \,^{\rho}\_{\mathcal{C}} \mathcal{D}^{\tilde{z}}\_{0+} q(\tau) = \hat{\mathcal{g}}(\tau), \tau \in \mathcal{E} := [0, \mathcal{T}],\\ p(0) = 0, \; q(0) = 0, \; p(\mathcal{T}) = \epsilon^{\rho} \mathcal{D}^{\tilde{\xi}}\_{0+} q(\mathcal{o}), \; q(\mathcal{T}) = \pi^{\rho} \mathcal{D}^{\tilde{\xi}}\_{0+} p(\mathcal{o}), \; 0 < \mathcal{o} < \mathcal{T}, \end{cases} \tag{79}$$

*is given by*

$$p(\mathbf{r}) = ^\rho \mathcal{Z}\_{0+}^{\tilde{\mathbf{r}}} f(\mathbf{r}) + \delta(\mathbf{r}) \left( \left[ \varepsilon \, ^\rho \mathcal{Z}\_{0+}^{\tilde{\mathbf{r}}+\zeta} \mathcal{G}(\mathfrak{w}) - ^\rho \mathcal{Z}\_{0+}^{\tilde{\zeta}} f(\mathcal{T}) \right] + \left[ \pi \, ^\rho \mathcal{Z}\_{0+}^{\tilde{\mathbf{r}}+\zeta} f(\mathfrak{w}) - ^\rho \mathcal{Z}\_{0+}^{\tilde{\zeta}} \mathcal{G}(\mathcal{T}) \right] \right) \tag{80}$$

*and*

$$q(\mathbf{r}) = ^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}} \mathbf{\hat{g}}(\mathbf{r}) + \delta(\mathbf{r}) \left( \left[ \pi^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}+\zeta} \hat{f}(\mathcal{o}) - {^\rho \mathcal{Z}\_{0+}^{\zeta}} \mathcal{g}(\mathcal{T}) \right] + \left[ \varepsilon^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}+\zeta} \hat{g}(\mathcal{o}) - {^\rho \mathcal{Z}\_{0+}^{\tilde{\xi}}} \hat{f}(\mathcal{T}) \right] \right). \tag{81}$$

**Proof.** When *<sup>ρ</sup>*I*<sup>ξ</sup>* <sup>0</sup>+, *ρ* I*ζ* <sup>0</sup><sup>+</sup> are applied to the FDEs in (79) and Lemma 4 is used the solution of the FDEs in (79) for *τ* ∈ E is

$$p(\tau) = ^\rho \mathcal{Z}\_{0+}^\sharp f(\tau) + a\_1 + a\_2 \frac{\tau^\rho}{\rho} = \frac{\rho^{1-\mathfrak{f}}}{\Gamma(\mathfrak{f})} \int\_0^\tau \theta^{\rho-1} (\tau^\rho - \theta^\rho)^{\mathfrak{f}-1} f(\theta) d\theta + a\_1 + a\_2 \frac{\tau^\rho}{\rho}, \tag{82}$$

$$q(\tau) = ^\rho T\_{0+}^{\tilde{\zeta}} \xi(\tau) + b\_1 + b\_2 \frac{\tau^\rho}{\rho} = \frac{\rho^{1-\tilde{\zeta}}}{\Gamma(\tilde{\zeta})} \int\_0^\tau \theta^{\rho-1} (\tau^\rho - \theta^\rho)^{\tilde{\zeta}-1} \xi(\theta) d\theta + b\_1 + b\_2 \frac{\tau^\rho}{\rho},\tag{83}$$

respectively, for some *a*1, *a*2, *b*1, *b*<sup>2</sup> ∈ R. Making use of the boundary conditions *p*(0) = *q*(0) = 0 in (82) and (83) respectively, we get *a*<sup>1</sup> = *b*<sup>1</sup> = 0. We obtain by using the generalized integral operators *<sup>ρ</sup>*I <sup>0</sup>+, *ρ* I*ζ* <sup>0</sup><sup>+</sup> (82) and (83) respectively,

$$\prescript{\rho}{}{\mathcal{I}}\_{0+}^{\varrho} p(\tau) = \prescript{\rho}{}{\mathcal{I}}\_{0+}^{\tilde{\varsigma}+\varrho} \prescript{\rho}{}{f}(\tau) + a\_1 \frac{\tau^{\rho\varrho}}{\rho^{\varrho} \Gamma(\varrho+1)} + a\_2 \frac{\tau^{\rho(\varrho+1)}}{\rho^{\varrho+1} \Gamma(\varrho+2)},\tag{84}$$

$${}^{\rho}T\_{0+}^{\xi}q(\mathbf{r}) = {}^{\rho}T\_{0+}^{\tilde{\varsigma}+\xi}{}^{\xi}\hat{g}(\mathbf{r}) + b\_{1}\frac{\mathbf{r}^{\rho\xi}}{\rho^{\varsigma}\Gamma(\emptyset+1)} + b\_{2}\frac{\mathbf{r}^{\rho(\zeta+1)}}{\rho^{\varsigma+1}\Gamma(\emptyset+2)} {}^{\prime} \tag{85}$$

which, when combined with the boundary conditions *<sup>p</sup>*(<sup>T</sup> )=*ρ*I*<sup>ς</sup>* <sup>0</sup>+*q*(), *<sup>q</sup>*(<sup>T</sup> ) = *<sup>π</sup>ρ*I <sup>0</sup><sup>+</sup> *p*(), gives the following results:

$${}^{\rho}T\_{0+}^{\mathbb{E}}f(\mathcal{T}) + a\_1 + a\_2 \frac{\mathcal{T}^{\rho}}{\rho} = \varepsilon^{\rho} \mathcal{T}\_{0+}^{\mathbb{E}+\varsigma} \mathfrak{F}(\mathcal{a}) + b\_1 \frac{\varepsilon \mathcal{O}^{\rho\varsigma}}{\rho^{\varepsilon} \Gamma(\varsigma+1)} + b\_2 \frac{\varepsilon \mathcal{O}^{\rho(\varsigma+1)}}{\rho^{\varsigma+1} \Gamma(\varsigma+2)},\tag{86}$$

$${}^{\rho}\mathcal{Z}\_{0+}^{\tilde{\mathbb{S}}}\xi(\mathcal{T}) + b\_1 + b\_2 \frac{\mathcal{T}^{\rho}}{\rho} = \pi^{\rho}\mathcal{Z}\_{0+}^{\tilde{\mathbb{S}}+\varrho}f(\mathcal{o}) + a\_1 \frac{\pi\alpha\mathcal{o}^{\rho\varrho}}{\rho^{\varrho}\Gamma(\varrho+1)} + a\_2 \frac{\pi\alpha\mathcal{o}^{\rho(\varrho+1)}}{\rho^{\varrho+1}\Gamma(\varrho+2)}.\tag{87}$$

Next, we obtain

$$a\_2 \hat{\mathcal{E}} - b\_2 \mathcal{E}\_1 = \epsilon^\rho \mathcal{T}\_{0+}^{\tilde{\varsigma}+\zeta} \mathcal{G}(\mathcal{O}) - ^\rho \mathcal{T}\_{0+}^{\tilde{\varsigma}} f(\mathcal{T}),\tag{88}$$

$$b\_2 \hat{\mathcal{E}} - a\_2 \mathcal{E}\_2 = \pi^\rho \mathcal{Z}\_{0+}^{\sharp + \varrho} f(\mathcal{a}) - ^\rho \mathcal{Z}\_{0+}^{\sharp} \xi(\mathcal{T}),\tag{89}$$

by employing the notations (76)–(78) in (86) and (87) respectively. We find that when we solve the system of Equations (88) and (89) for *a*<sup>2</sup> and *b*2,

$$a\_2 = \frac{1}{\mathcal{G}} \Big[ \hat{\mathcal{E}} \Big( \epsilon^{\rho} \mathcal{Z}\_{0+}^{\mathsf{f}, + \epsilon} \circ (\mathcal{a} \circ) - \,^{\rho} \mathcal{Z}\_{0+}^{\mathsf{f}} f(\mathcal{T}) \Big) + \mathcal{E}\_1 \Big( \pi^{\rho} \mathcal{Z}\_{0+}^{\mathsf{f}, + \epsilon} f(\mathcal{a} \,) - \,^{\rho} \mathcal{Z}\_{0+}^{\mathsf{f}} \xi(\mathcal{T}) \Big) \Big],\tag{90}$$

$$b\_2 = \frac{1}{\mathcal{G}} \left[ \mathcal{E}\_2 \left( \epsilon^{\rho} \mathcal{Z}\_{0+}^{\mathbb{S}+\text{c}} \mathfrak{F}(\mathcal{O}) - ^{\rho} \mathcal{Z}\_{0+}^{\mathbb{S}} f(\mathcal{T}) \right) + \hat{\mathcal{E}} \left( \pi^{\rho} \mathcal{Z}\_{0+}^{\mathbb{S}+\text{c}} f(\mathcal{O}) - ^{\rho} \mathcal{Z}\_{0+}^{\mathbb{S}} \mathfrak{F}(\mathcal{T}) \right) \right]. \tag{91}$$

Substituting the values of *a*1, *a*2, *b*1, *b*<sup>2</sup> in (82) and (83) respectively, we get the solution for (79).

For brevity's sake, we'll use the following notations:

$$\mathcal{J}\_1 = \frac{\left(\mathcal{T}^{\rho \tilde{\varsigma}} (1 + |\delta| |\hat{\mathcal{E}}|) \right)}{\rho^{\mathfrak{z}} \Gamma(\tilde{\varsigma} + 1)} + \frac{|\delta| |\pi| |\mathcal{E}\_1| \mathcal{O}^{\rho(\tilde{\varsigma} + \varrho)}}{\rho^{\mathfrak{z} + \varrho} \Gamma(\tilde{\varsigma} + \varrho + 1)} \tag{92}$$

$$\mathcal{K}\_1 = |\delta| \left( \frac{|\mathcal{E}\_1| \mathcal{T}^{\rho \zeta}}{\rho^{\zeta} \Gamma(\zeta + 1)} + \frac{|\widehat{\mathcal{E}}| |\mathfrak{e}| \mathcal{a}^{\rho(\zeta + \zeta)}}{\rho^{\zeta + \zeta} \Gamma(\zeta + \zeta + 1)} \right), \tag{93}$$

$$\mathcal{J}\_2 = |\delta| \left( \frac{\mathcal{T}^{\rho \tilde{\varsigma}} |\mathcal{E}\_2|}{\rho^{\xi} \Gamma(\tilde{\varsigma} + 1)} + \frac{|\pi| |\hat{\mathcal{E}}| \mathcal{a}^{\rho(\zeta + \varrho)}}{\rho^{\xi + \varrho} \Gamma(\tilde{\varsigma} + \varrho + 1)} \right), \tag{94}$$

$$\mathcal{K}\_2 = \frac{\left(\mathcal{T}^{\rho\mathbb{\tilde{\zeta}}} (1 + |\delta| |\hat{\mathcal{E}}|) \right)}{\rho^{\mathbb{\tilde{\zeta}}} \Gamma(\mathbb{\zeta} + 1)} + \frac{|\delta| |\varepsilon| |\mathcal{E}\_2| \mathcal{a}^{\rho(\zeta + \emptyset)}}{\rho^{\mathbb{\tilde{\zeta}} + \emptyset} \Gamma(\mathbb{\zeta} + \emptyset + 1)}. \tag{95}$$

To finish up, we will go over the results of existence, uniqueness, and Ulam–Hyers stability for problems (1) and (75), respectively. For reasons that are similar to those in Sections 3–6, we are not providing the proof.

**Corollary 1.** *Assume that <sup>f</sup>* , *<sup>g</sup>* : E × <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> *are continuous functions satisfying the condition:* (A1) *there exists constants <sup>ψ</sup>m*, *<sup>ψ</sup>*<sup>ˆ</sup> *<sup>m</sup>* ≤ <sup>0</sup>(*<sup>m</sup>* = 1, 2) *and <sup>ψ</sup>*0, *<sup>ψ</sup>*<sup>ˆ</sup> <sup>0</sup> > 0 *such that*

$$\begin{aligned} |f(\pi, o\_1, o\_2)| &\le \psi\_0 + \psi\_1 |o\_1| + \psi\_2 |o\_2|, \\ |g(\pi, o\_1, o\_2)| &\le \hat{\psi}\_0 + \hat{\psi}\_1 |o\_1| + \hat{\psi}\_2 |o\_2|, \forall o\_m \in \mathbb{R}, m = 1, 2. \end{aligned}$$

*If <sup>ψ</sup>*1(J<sup>ˆ</sup> <sup>1</sup> <sup>+</sup> <sup>J</sup><sup>ˆ</sup> <sup>2</sup>) + *ψ*ˆ <sup>1</sup>(K<sup>ˆ</sup> <sup>1</sup> <sup>+</sup> <sup>K</sup><sup>ˆ</sup> <sup>2</sup>) <sup>&</sup>lt; 1, *<sup>ψ</sup>*2(J<sup>ˆ</sup> <sup>1</sup> <sup>+</sup> <sup>J</sup><sup>ˆ</sup> <sup>2</sup>) + *ψ*ˆ <sup>2</sup>(K<sup>ˆ</sup> <sup>1</sup> <sup>+</sup> <sup>K</sup><sup>ˆ</sup> <sup>2</sup>) < 1*. Then at least one solution for the BVP (1) and (75) on* <sup>E</sup>*, where* <sup>J</sup><sup>ˆ</sup> 1, <sup>K</sup><sup>ˆ</sup> 1, <sup>J</sup><sup>ˆ</sup> 2, <sup>K</sup><sup>ˆ</sup> <sup>2</sup> *are given by (92)–(95) respectively.*

**Corollary 2.** *Assume that <sup>f</sup>* , *<sup>g</sup>* : E × <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> *are continuous functions satisfying the condition: (*A2*) there exists constants <sup>φ</sup>m*, *<sup>φ</sup>*<sup>ˆ</sup> *<sup>m</sup>* ≤ 0(*m* = 1, 2) *such that*

$$\begin{aligned} |f(\boldsymbol{\tau},o\_1,o\_2) - f(\boldsymbol{\tau},\boldsymbol{\delta}\_1,\boldsymbol{\delta}\_2)| &\leq \boldsymbol{\phi}\_1|o\_1 - \boldsymbol{\delta}\_1| + \boldsymbol{\phi}\_2|o\_2 - \boldsymbol{\delta}\_2|,\\ |g(\boldsymbol{\tau},o\_1,o\_2) - g(\boldsymbol{\tau},\boldsymbol{\delta}\_1,\boldsymbol{\delta}\_2)| &\leq \boldsymbol{\phi}\_1|o\_1 - \boldsymbol{\delta}\_1| + \boldsymbol{\phi}\_2|o\_2 - \boldsymbol{\delta}\_2|, \forall o\_m,\boldsymbol{\delta}\_m \in \mathbb{R}, m = 1,2. \end{aligned}$$

*Moreover, there exist* S1, S<sup>2</sup> > 0 *such that* | *f*(*τ*, 0, 0)|≤S1, | *f*(*τ*, 0, 0)|≤S2, *Then, given that*

$$(\mathcal{J}\_1 + \mathcal{J}\_2)(\phi\_1 + \phi\_2) + (\mathcal{K}\_1 + \mathcal{K}\_2)(\phi\_1 + \phi\_2) < 1,\tag{96}$$

*the BVP (1) and (75) has a unique solution on* <sup>E</sup>*, where* <sup>J</sup><sup>ˆ</sup> 1, <sup>K</sup><sup>ˆ</sup> 1, <sup>J</sup><sup>ˆ</sup> 2, <sup>K</sup><sup>ˆ</sup> <sup>2</sup> *are given by (92)–(95) respectively.*

**Corollary 3.** *Assume that <sup>f</sup>* , *<sup>g</sup>* : E × <sup>R</sup> <sup>×</sup> <sup>R</sup> <sup>→</sup> <sup>R</sup> *are continuous functions satisfying the assumption* (A2) *in Theorem 2. Further more, there exist positive constants* U1, U<sup>2</sup> *such that* ∀*τ* ∈ E *and ri* <sup>∈</sup> <sup>R</sup>, *<sup>i</sup>* <sup>=</sup> 1, 2*.*

$$|f(\mathfrak{r}, r\_1, r\_2)| \le \mathcal{U}\_{1\prime} \qquad |\mathcal{g}(\mathfrak{r}, r\_1, r\_2)| \le \mathcal{U}\_{2\prime} \tag{97}$$

*If*

$$\frac{\mathcal{T}^{\rho\xi}(\phi\_1 + \phi\_2)}{\rho^{\xi}\Gamma(\xi+1)} + \frac{\mathcal{T}^{\rho\overline{\zeta}}(\hat{\phi}\_1 + \hat{\phi}\_2)}{\rho^{\overline{\zeta}}\Gamma(\zeta+1)} < 1,\tag{98}$$

*then the BVP (1), and (75) has at least one solution on* E*.*

**Corollary 4.** *Assume that (A2) holds. Then the problem (1) and (75) is Ulam–Hyers stable.*

#### **8. Asymmetric Cases**

**Remark 1.** *If ρ* = 1*, the problem (1) generalized Liouville–Caputo type reduces to the classical Caputo form.*

$$\begin{cases} \,^{\mathbb{C}}\mathcal{D}\_{0^{+}}^{\mathbb{E}}p(\tau) = f(\tau, p(\tau), q(\tau)), \tau \in \mathcal{G} := [0, \mathcal{T}],\\ \,^{\mathbb{C}}\mathcal{D}\_{0^{+}}^{\mathbb{E}}q(\tau) = g(\tau, p(\tau), q(\tau)), \tau \in \mathcal{G} := [0, \mathcal{T}].\end{cases} \tag{99}$$

**Remark 2.** *If ρ* = 1 *in the boundary conditions (2) and (75) generalized Riemann–Liouville integral boundary conditions reduces to the Riemann–Liouville integral conditions respectively.*

$$\begin{cases} p(0) = 0, \quad q(0) = 0, \\ p(\mathcal{T}) = \epsilon \mathcal{T}\_{0^+}^{\xi} q(\mathcal{o}) = \frac{\varepsilon}{\Gamma(\xi)} \int\_0^{\mathcal{O}} (\mathcal{o} - \theta)^{\varepsilon - 1} q(\theta) d\theta, \\ q(\mathcal{T}) = \pi \mathcal{L}\_{0+}^{\mathcal{O}} p(\sigma) = \frac{\pi}{\Gamma(\varrho)} \int\_0^{\mathcal{O}} (\sigma - \theta)^{\varrho - 1} p(\theta) d\theta, \\ 0 < \sigma < \mathcal{O} < \mathcal{T}, \end{cases} \tag{100}$$

*and*

$$\begin{cases} p(0) = 0, \quad q(0) = 0, \\ p(T) = \varepsilon \mathcal{T}\_{0^+}^\xi q(\mathcal{o}) = \frac{\varepsilon}{\Gamma(\emptyset)} \int\_0^\mathcal{O} (\mathcal{o} - \theta)^{\varepsilon - 1} q(\theta) d\theta, \\ q(T) = \pi \mathcal{L}\_{0+}^\xi p(\mathcal{o}) = \frac{\pi}{\Gamma(\emptyset)} \int\_0^\mathcal{O} (\mathcal{o} - \theta)^{\mathcal{O} - 1} p(\theta) d\theta, \\ 0 < \mathcal{o} < \mathcal{T}. \end{cases} \tag{101}$$

**Remark 3.** *If ρ* = 1 *and ς* = = 1 *in the boundary conditions (2) and (75) generalized Riemann– Liouville integral boundary conditions reduces to the classical integral conditions respectively.*

$$\begin{cases} p(0) = 0, \; q(0) = 0, \; p(T) = \epsilon \int\_0^\mathcal{\mathcal{O}} q(\theta) d\theta, \; q(T) = \pi \int\_0^\mathcal{\mathcal{O}} p(\theta) d\theta \; 0 < \sigma < \mathcal{O} < \mathcal{T} \end{cases} \tag{102}$$

*and*

$$\int p(0) = 0, \; q(0) = 0, \; p(\mathcal{T}) = \epsilon \int\_0^\mathcal{\mathcal{Q}} q(\theta) d\theta, \; q(\mathcal{T}) = \pi \int\_0^\mathcal{\mathcal{Q}} p(\theta) d\theta \, 0 < \mathcal{\mathcal{Q}} < \mathcal{T}. \tag{103}$$

#### **9. Conclusions**

This paper employs coupled nonlinear generalized Liouville–Caputo fractional differential equations and Katugampola fractional integral operators to solve a novel class of boundary value problems. Applying the techniques of fixed-point theory to discover the existence criterion for solutions is efficient. While the second outcome provides a sufficient criterion to establish the problem's unique solution, the first and third results define various criteria for the presence of solutions to the given problem. In the fourth section, the Hyers–Ulam stability of the solution was determined. In the remarks, we have shown the asymmetric cases of the assigned problem. Moreover, the form of the solution in these kinds of remarks can be used to study the positive solution and its asymmetry in more depth. We conclude that our results are novel and can be viewed as an expansion of the qualitative analysis of fractional differential equations. Our results are novel in this configuration and add to the literature on nonlinear coupled generalized Liouville–Caputo fractional differential equations with nonlocal boundary conditions utilizing Katugampolatype integral operators. Future research could focus on various conceptions of stability and existence in relation to a Lotka–Volterra prey-predator system/coupled logistic system.

**Author Contributions:** Conceptualization, M.S.; formal analysis, M.A. and M.S.; methodology, M.A, M.S., K.A. and M.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [Grant No.793].

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


### *Article* **The Fractional Hilbert Transform of Generalized Functions**

**Naheed Abdullah 1,2,\*,† and Saleem Iqbal 2,†**


**Abstract:** The fractional Hilbert transform, a generalization of the Hilbert transform, has been extensively studied in the literature because of its widespread application in optics, engineering, and signal processing. In the present work, we expand the fractional Hilbert transform that displays an odd symmetry to a space of generalized functions known as Boehmians. Moreover, we introduce a new fractional convolutional operator for the fractional Hilbert transform to prove a convolutional theorem similar to the classical Hilbert transform, and also to extend the fractional Hilbert transform to Boehmians. We also produce a suitable Boehmian space on which the fractional Hilbert transform exists. Further, we investigate the convergence of the fractional Hilbert transform for the class of Boehmians and discuss the continuity of the extended fractional Hilbert transform.

**Keywords:** convolution; Boehmian; fractional Hilbert transform; Hilbert transform; equivalence class; delta sequences; compact support

#### **1. Introduction**

The space of Boehmians is a class of generalized functions that include all regular operators and generalized functions or distributions, and other objects. The theory of Boehmians with two convergences, introduced by Mikusinski [1], broadens the concept of Boehme's regular operators [2]. In contrast to the theory of distributions in which generalized functions are treated as members of the dual space of any space of testing function, the space of Boehmians treats distributions more as algebraic objects. Several integral transforms for various spaces of Boehmians were studied and their properties were investigated in [3–13]. Currently, a large number of studies are available on the extension of classical integral transforms to Boehmians. Karunakaran and Roopkumar introduced the Hilbert transform as continuous linear mapping defined on some space of Boehmians into another space of Boehmians [7]. They also studied the Hilbert transform for the space of ultradistributions [8]. The pioneering work of Zayed [13], Al-Omari, and Agarwal [6] introduced an extension of fractional integral transform to Boehmians by extending the fractional Fourier and Sumudu transforms to the space of integrable Boehmians. The properties and generalizations of various quaternion integral transform [14] and fractional integral transforms were also studied from the perspective of q-calculus analysis [15,16] and rapidly decaying functions [17]. In recent years, the extension of fractional integral transforms to the space of Boehmians has been an active area of research. Many wellknown fractional integral transforms have been extended to the space of Boehmians, but an extension of the fractional Hilbert transform (FHT) has not yet been reported. So, the goal of this paper is to extend the FHT to some space of Boehmians. Different definitions of FHT exist in the literature [18–20], but in the generalization of the classical Hilbert transform, it might rightly be said that the fractionalization of Hilbert transform is given by Zayed and

**Citation:** Abdullah, N.; Iqbal, S. The Fractional Hilbert Transform of Generalized Functions. *Symmetry* **2022**, *14*, 2096. https://doi.org/ 10.3390/sym14102096

Academic Editors: Palle E.T. Jorgensen and Alexander Zaslavski

Received: 23 August 2022 Accepted: 21 September 2022 Published: 8 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

is mathematically elaborated in [21]. The fractional Hilbert transform of a function *f*(*x*), denoted by *Hα*[ *f*(*x*)], is defined as [20]

$$H\_{\mathfrak{a}}[f(\mathfrak{x})] = \frac{1}{\pi} \int\_{-\infty}^{\infty} \frac{e^{-i\frac{\mathfrak{x}^2 - t^2}{2}\cot a}}{\mathfrak{x} - t} f(t) dt \text{ for } \mathfrak{a} \neq 0, \ \pi/2, \ \pi,\tag{1}$$

where the integral is taken in the sense of the Cauchy principal value. The special case *α* = *π*/2 reduces FHT into the standard Hilbert transform. Indeed, the FHT allows for converting a real signal into a complex signal by suppressing the negative frequency. Such a signal has a wide variety of applications in optics, signal processing, and image processing [22–25]. It also does not flip the domain of the signal—the signal remains in the same domain. However, it lacks detailed mathematical analysis, so we require a thorough mathematical theory of FHT to understand its strengths and limitations. Consequently, we need to extend the existing theory on such a significant transformation in terms of generalized functions. An extension of FHT to some space of Boehmians may have applications in engineering and other sciences, as it may apply in converting functions with discontinuities into smooth functions that consequently lead to the description of various physical occurrences such as point charges [26].

The present paper is organized as follows: Section 1 covers the introduction. Section 2 covers the important definitions and theorems, and we also discuss the abstract construction of Boehmians to render the paper self-contained. Section 3 covers results that comprise a new convolutional operator and a new convolutional theorem for FHT, and proves auxiliary results required for the construction of two Boehmian spaces. Lastly, we extend the FHT to some spaces of Boehmians. Section 4 presents our conclusions.

#### **2. Preliminaries**

Let <sup>R</sup> be the set of all real numbers, <sup>L</sup>1(R) = <sup>L</sup><sup>1</sup> be the collection of complex-valued measurable functions *f* defined on R for which

$$\|f\|\_{1} = \int\_{-\infty}^{\infty} |f(\mathbf{x})|d\mathbf{x} < \infty$$

and <sup>C</sup><sup>∞</sup> <sup>=</sup> <sup>C</sup>∞(R) be the set of all infinitely differentiable functions defined on <sup>R</sup>, such that functions and their derivatives converge uniformly on compact sets in R.

**Theorem 1** ([27] Theorem 9.5)**.** *For any function f on* <sup>R</sup> *and for all t* <sup>∈</sup> <sup>R</sup>*, let ft be defined by*

$$f\_t(\mathfrak{x}) = f(\mathfrak{x} - t).$$

*If p* <sup>≥</sup> <sup>1</sup> *and f* ∈ L*p, then mapping t* <sup>→</sup> *ft is uniformly continuous from* <sup>R</sup> *into* <sup>L</sup>*p*(R)*.*

**Definition 1.** *Let <sup>f</sup> and <sup>g</sup> be any two functions on* <sup>R</sup>*; their convolution, denoted by <sup>f</sup>* <sup>∗</sup> *g, is defined as*

$$f \* g = \int\_{-\infty}^{\infty} f(t)g(\mathbf{x} - t)dt. \tag{2}$$

The Hilbert transform of convolutional operation ∗ is given as follows:

**Theorem 2.** *If <sup>f</sup>* , *<sup>g</sup>* ∈ L1(R) *with Hilbert transforms H f* , *Hg respectively, so that H f* , *Hg* <sup>∈</sup> <sup>L</sup>1(R)*, then*

$$H[f \ast g] = Hf \ast g = f \ast Hg.$$

The FHT may not act as agreeably with the classical convolutional operator as the classical Hilbert transform (Theorem 2).

#### *Boehmian Space*

The members of Boehmian spaces are called Boehmians, which are equivalence classes of "quotients of sequences". These equivalence classes are formulated from an integral domain of continuous functions. The integral domain operations for Boehmians are addition and convolution. This convolutional operation may differ from the standard convolutional operation given in Definition 2.

We now present a brief introduction to Boehmians.

Let *G* be a complex linear space, (*H*, .) is a commutative semigroup, and let ⊗ : *G* × *H* → *G*, so that the conditions given below hold:


A pair of sequences { *fn*, *<sup>φ</sup>n*} with *fn* <sup>∈</sup> *<sup>G</sup>* for all *<sup>n</sup>* <sup>∈</sup> <sup>N</sup> and {*φn*} ∈ <sup>Δ</sup> are a quotient of sequences, denoted by *fn <sup>φ</sup><sup>n</sup>* , if

$$f\_n \otimes \phi\_m = f\_m \otimes \phi\_n \,\,\forall m \, n \in \mathbb{N}.$$

Two quotients of sequences *fn <sup>φ</sup><sup>n</sup>* and *gn <sup>ψ</sup><sup>n</sup>* are equivalent (∼) if, for every *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>

$$f\_n \otimes \psi\_n = \emptyset \rtimes \phi\_n.$$

The equivalence class of *fn <sup>φ</sup><sup>n</sup>* induced by "∼" is denoted by *fn φn* . Every equivalence class is called a Boehmian. The space of all Boehmians is denoted by B = B(*G*, *H*, ⊗, Δ). B is a vector space under the operations of addition and scalar multiplication defined as follows:


If we define an isomorphism *f* → *<sup>f</sup>*⊗*φ<sup>n</sup> φn* , then *G* is a subspace of B. Therefore, every element of *G* can be expressed uniquely as a Boehmian.

#### **3. Results**

In this section, we define a new convolutional operation for FHT that yields a generalized result for Theorem 2. Moreover, to extend the FHT to the class of Boehmians, we define two classes of Boehmians. Two convergences of FHT are proved on <sup>C</sup>∞. Lastly, an extension of FHT on Boehmians is introduced.

#### *3.1. Convolutional Structure for Fractional Hilbert Transform*

The idea of convolutional operation makes it evident that, given any integral transform, we can associate a convolutional operation to it [28]. So, we introduce a new fractional convolutional operator that helps us in extending FHT to the space of Boehmians.

**Definition 2.** *Let f* , *<sup>g</sup>* ∈ L1(R)*. We define a fractional convolution* (*<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>g</sup>*) *as*

$$(f \ast\_{\mathfrak{a}} g)(\mathfrak{x}) = \int\_{-\infty}^{\infty} f(\mathfrak{x} - t) g(t) e^{-i t (\mathfrak{x} - t) \cot \mathfrak{a}} dt. \tag{3}$$

**Lemma 1.** *Let f* , *<sup>g</sup>* ∈ L1*. Then,* (*<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>g</sup>*) *is also in* <sup>L</sup>1*.*

**Proof.** To prove that *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>g</sup>* ∈ L1, we consider its <sup>L</sup><sup>1</sup> norm.

$$\begin{aligned} \|f \ast\_{\mathfrak{a}} \mathcal{g}\|\_{1} &= \int\_{-\infty}^{\infty} |f \ast\_{\mathfrak{a}} \mathcal{g}| d\mathfrak{x} \\ &\leq \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} |f(\mathfrak{x} - t)| |\mathfrak{g}(t)| dt d\mathfrak{x}. \end{aligned}$$

By using Fubini's theorem, we have

$$\|f \ast\_{\mathfrak{a}} g\|\_{1} \leq \int\_{-\infty}^{\infty} |f(\mathfrak{x} - t)| d\mathfrak{x} \int\_{-\infty}^{\infty} |g(t)| dt.$$

Since the <sup>L</sup><sup>1</sup> norm is translation invariance, so <sup>∞</sup> <sup>−</sup><sup>∞</sup> <sup>|</sup> *<sup>f</sup>*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*)|*dx* <sup>=</sup> *ft*<sup>1</sup> <sup>=</sup> *<sup>f</sup>* 1. Therefore,

$$\|f \ast\_{\mathfrak{a}} \mathcal{g}\|\_{1} \le \|f\|\_{1} \|\mathcal{g}\|\_{1} \cdot$$

Since *<sup>f</sup>* , *<sup>g</sup>* ∈ L1,

$$||f \ast\_{\mathfrak{a}} g||\_1 \le ||f||\_1 ||g||\_1 < \infty,$$

which proves that *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>g</sup>* ∈ L1.

To extend the FHT to the case of Boehmians, the essential step is to prove the convolutional theorem, and suitable Boehmian spaces can then be constructed by proving the supplementary results. Now, we state and prove the convolutional theorem for FHT.

**Theorem 3.** *(convolutional Theorem) Assume that f* , *<sup>g</sup>* ∈ L1*. Then,*

$$H\_{\mathfrak{a}}[f \ast\_{\mathfrak{a}} \mathfrak{g}] = H\_{\mathfrak{a}}[f] \ast\_{\mathfrak{a}} \mathfrak{g} = f \ast\_{\mathfrak{a}} H\_{\mathfrak{a}}[\mathfrak{g}].\tag{4}$$

*In addition,* (*f* ∗*<sup>α</sup> g*) = −(*Hα*[ *f* ] ∗*<sup>α</sup> Hα*[*g*]).

**Proof.**

$$\begin{split} H\_{\mathfrak{a}}[(f\*\_{\mathfrak{a}}\,\mathfrak{g})(\mathfrak{x})] &= \frac{1}{\pi} \int\_{-\infty}^{\infty} \frac{e^{-i\frac{\mathfrak{x}^{2}-\mathfrak{r}^{2}}{2}\cot a}}{\mathfrak{x}-t} (f\*\_{\mathfrak{a}}\,\mathfrak{g})(t) \,dt \\ &= \frac{1}{\pi} \int\_{-\infty}^{\infty} \frac{e^{-i\frac{\mathfrak{x}^{2}-\mathfrak{r}^{2}}{2}\cot a}}{\mathfrak{x}-t} \int\_{-\infty}^{\infty} f(t-y)g(y)e^{-iy(t-y)\cot a} dy dt. \end{split}$$

By changing variables *t* − *y* = *ν*, the above equation can be simplified to

$$\begin{split} H\_{\mathfrak{a}}[(f\*\_{\mathfrak{a}}g)(\mathfrak{x})] &= \frac{1}{\pi} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \frac{e^{-i\frac{\mathbf{x}^{2}-2\mathbf{y}+\mathbf{y}^{2}-\mathbf{v}^{2}}{2}\cot a}}{(\mathbf{x}-\mathbf{y})-\boldsymbol{\nu}} f(\boldsymbol{\nu})g(\boldsymbol{y})e^{-i(\boldsymbol{y}\cdot\mathbf{x}-\mathbf{y}^{2})\cot a}d\boldsymbol{\nu}d\boldsymbol{y} \\ &= \int\_{-\infty}^{\infty} H\_{\mathfrak{a}}[f(\mathbf{x}-\mathbf{y})]g(\boldsymbol{y})e^{-i\boldsymbol{y}(\mathbf{x}-\mathbf{y})\cot a}d\boldsymbol{y} \\ &= (H\_{\mathfrak{a}}[f]\*\_{\mathfrak{a}}g)(\boldsymbol{x}). \end{split}$$

Similarly,

$$H\_{\mathfrak{a}}[(f\*\_{\mathfrak{a}}\mathfrak{g})(\mathfrak{x})] = H\_{\mathfrak{a}}[(\mathfrak{g}\*\_{\mathfrak{a}}f)(\mathfrak{x})] = (H\_{\mathfrak{a}}[\mathfrak{g}]\*\_{\mathfrak{a}}f)(\mathfrak{x}) = (f\*\_{\mathfrak{a}}H\_{\mathfrak{a}}[\mathfrak{g}])(\mathfrak{x}).\tag{5}$$

If we substitute *g* by *Hα*[*g*] in (4), we can write

$$\begin{aligned} H\_{\mathfrak{a}}[(f \ast\_{\mathfrak{a}} H\_{\mathfrak{a}}[\mathfrak{g}])(\mathfrak{x})] &= (H\_{\mathfrak{a}}[f] \ast\_{\mathfrak{a}} H\_{\mathfrak{a}}[\mathfrak{g}])(\mathfrak{x}), \\ (f \ast\_{\mathfrak{a}} H\_{\mathfrak{a}}[H\_{\mathfrak{a}}[\mathfrak{g}]])(\mathfrak{x}) &= (H\_{\mathfrak{a}}[f] \ast\_{\mathfrak{a}} H\_{\mathfrak{a}}[\mathfrak{g}])(\mathfrak{x}), \quad (\text{by (5)}), \\ f \ast\_{\mathfrak{a}} \mathfrak{g} &= -(H\_{\mathfrak{a}}[f] \ast\_{\mathfrak{a}} H\_{\mathfrak{a}}[\mathfrak{g}]), \end{aligned}$$

where *H*<sup>2</sup> *<sup>α</sup>* = −*I*, and this proves the theorem.

#### *3.2. Abstract Construction of Boehmians*

Now, we construct the Boehmian space required for extending the theory of the fractional Hilbert transform to some space of Boehmians. Here, we refer to only two spaces of Boehmians needed to develop the theory of FHT. Now to define the space of Boehmians, we introduce a class of identities as follows: Let space D constitute all infinitely differentiable functions with compact support in R. Let

$$\mathcal{S} = \{ \phi \in \mathcal{D} : \phi \ge 0 \text{ and } \int\_{\mathbb{R}} \phi = 1 \}.$$

Then, the space of Boehmians is given by

$$\mathcal{B}\_1 = \mathcal{B}\_1(\mathcal{L}^1(\mathbb{R}), \mathcal{S}\_{\prime} \*\_{\alpha} \Delta)\_{\prime}$$

where Δ is the collection of all sequences of real-valued functions {*φn*(*x*)} ⊂ *S*, such that


These sequences are *delta sequences*. We now state and prove the results that are needed to build the desired space for Boehmians.

#### **Lemma 2.** *The operation* ∗*<sup>α</sup> is both commutative and associative.*

**Proof.** To prove that ∗*<sup>α</sup>* is commutative, consider

$$(f \ast\_{\alpha} g)(x) = \int\_{-\infty}^{\infty} f(x - t)g(t)e^{-i(x - t)\cot\alpha}dt.$$

By changing variable *x* − *t* = *τ*, we can simplify the above equation to

$$(f \ast\_{\mathfrak{a}} g)(\mathfrak{x}) = \int\_{-\infty}^{\infty} f(\tau) g(\mathfrak{x} - \tau) e^{-i(\mathfrak{x} - \tau)\tau \cot a} d\tau = (g \ast\_{\mathfrak{a}} f)(\mathfrak{x}).$$

To prove the associativity, let us consider

$$\begin{aligned} ((f \ast\_{\mathfrak{a}} g) \ast\_{\mathfrak{a}} h)(\mathfrak{x}) &= \int\_{-\infty}^{\infty} (f \ast\_{\mathfrak{a}} g)(\mathfrak{x} - t) h(t) e^{-i(\mathfrak{x} - t) \cot \mathfrak{a}} dt \\ &= \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} f(\mathfrak{x} - t - u) g(u) h(t) e^{-i u(\mathfrak{x} - t - u) \cot \mathfrak{a}} e^{-i t(\mathfrak{x} - t) \cot \mathfrak{a}} dt du. \end{aligned}$$

By changing variables *t* + *u* = *y*, we can write the above equation as

$$((f \ast\_{\mathfrak{a}} \mathfrak{g}) \ast\_{\mathfrak{a}} h)(\mathfrak{x}) = \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} f(\mathfrak{x} - y) g(y - t) h(t) e^{-i(y - t)(\mathfrak{x} - y) \cot a} e^{-it(\mathfrak{x} - t) \cot a} dt dy.$$

As an application of Fubini's theorem, we have

$$\begin{split} (\left(f\*\_{\mathfrak{a}}\mathfrak{g}\right)\*\_{\mathfrak{a}}h)(\mathfrak{x}) &= \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \mathfrak{g}(y-t)h(t)e^{-i(-tx+yt+tx-t^{2})\cot\mathfrak{a}}f(\mathfrak{x}-y)e^{-iy(\mathfrak{x}-y)\cot\mathfrak{a}}dtdydx \\ &= \int\_{-\infty}^{\infty} f(\mathfrak{x}-y)(\mathfrak{g}\*\_{\mathfrak{a}}h)(y)e^{-iy(\mathfrak{x}-y)\cot\mathfrak{a}}dy \\ &= (f\*\_{\mathfrak{a}}\mathfrak{g}\*\_{\mathfrak{a}}h))(\mathfrak{x}). \end{split}$$

Thus, ((*f* ∗*<sup>α</sup> g*) ∗*<sup>α</sup> h*)(*x*) = (*f* ∗*<sup>α</sup>* (*g* ∗*<sup>α</sup> h*))(*x*).

**Lemma 3.** *Assume that* {*φn*} *and* {*ψn*} *are in* Δ*. Then, their convolution* {*φ<sup>n</sup>* ∗*<sup>α</sup> ψn*} *is also in* Δ*.*

**Proof.** To prove that {*φ<sup>n</sup>* ∗*<sup>α</sup> ψn*} ∈ Δ, we must show that the three conditions for delta sequences are fulfilled.

1. <sup>R</sup> *<sup>e</sup>it*(*x*−*t*) cot *<sup>α</sup>*(*φ<sup>n</sup>* <sup>∗</sup>*<sup>α</sup> <sup>ψ</sup>n*)(*x*)*dx* <sup>=</sup> <sup>R</sup> *<sup>e</sup>it*(*x*−*t*) cot *<sup>α</sup>* <sup>∞</sup> −∞ *<sup>φ</sup>n*(*<sup>x</sup>* <sup>−</sup> *<sup>t</sup>*)*ψn*(*t*)*e*−*it*(*x*−*t*) cot *<sup>α</sup> dtdx*. By using Fubini's theorem, we can write

$$\int\_{\mathbb{R}} e^{it(\mathbf{x}-\mathbf{t})\cot a} (\phi\_n \*\_{\mathbf{t}} \psi\_n)(\mathbf{x}) d\mathbf{x} = \int\_{\mathbb{R}} e^{it(\mathbf{x}-\mathbf{t})\cot a} e^{-it(\mathbf{x}-\mathbf{t})\cot a} \phi\_n(\mathbf{x}-\mathbf{t}) d\mathbf{x} \int\_{-\infty}^{\infty} \psi\_n(\mathbf{t}) dt.$$

Since {*φn*}, {*ψn*} ∈ Δ, then

$$\int\_{\mathbb{R}} e^{it(\chi-t)\cot\alpha} (\phi\_n \*\_{\alpha} \psi\_n)(\mathfrak{x}) d\mathfrak{x} = 1.$$

2.

$$\begin{split} \|\!\!\|\phi\_{n}\ast\_{\alpha}\psi\_{n}\|\|\_{1} &= \int\_{-\infty}^{\infty} |(\phi\_{n}\ast\_{\alpha}\psi\_{n})(\mathbf{x})|d\mathbf{x} \\ &= \int\_{-\infty}^{\infty} \left| \int\_{-\infty}^{\infty} \phi\_{n}(\mathbf{x}-t)\psi\_{n}(t)e^{-it(\mathbf{x}-t)\cot\alpha}dt \right| d\mathbf{x} \\ &\leq \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \left| \phi\_{n}(\mathbf{x}-t)\psi\_{n}(t)e^{-it(\mathbf{x}-t)\cot\alpha}dt \right| d\mathbf{x} \\ &= \|\!\!\|\phi\_{n}\|\_{1} \|\psi\_{n}\|\_{1} \\ &\leq M^{2}, \quad \forall n \in \mathbb{N}. \end{split}$$

Thus, *φ<sup>n</sup>* <sup>∗</sup>*<sup>α</sup> <sup>ψ</sup>n*<sup>1</sup> <sup>≤</sup> *<sup>M</sup>*2.

3.

$$\begin{aligned} \lim\_{n \to \infty} \int\_{|t| > \varepsilon} |(\phi\_n \*\_n \psi\_n)(x)| dx &\le \lim\_{n \to \infty} \int\_{|t| > \varepsilon} \int\_{-\infty}^{\infty} |\phi\_n(x - t)\psi\_n(t)| dt dx \\ &= ||\phi\_n||\_1 \lim\_{n \to \infty} \int\_{|t| > \varepsilon} |\psi\_n(t)| dt. \end{aligned}$$

Since {*ψn*} ∈ Δ, then

$$\lim\_{n \to \infty} \int\_{|t| > \epsilon} |\psi\_n(t)| dt = 0, \qquad \text{for } \epsilon > 0.$$

Hence,

$$\int\_{|t|>\varepsilon} |(\phi\_n \*\_{\alpha} \psi\_n)(\mathfrak{x})| d\mathfrak{x} \to 0 \quad \text{as } n \to \infty, \text{ for } \varepsilon > 0.$$

This completes the proof.

**Lemma 4.** *If f* ∈ L<sup>1</sup> *and <sup>φ</sup><sup>n</sup>* <sup>∈</sup> <sup>Δ</sup> *then the convolution f* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup><sup>n</sup>* ∈ L1*.*

**Proof.** Let *<sup>f</sup>* ∈ L<sup>1</sup> and *<sup>φ</sup><sup>n</sup>* <sup>∈</sup> <sup>Δ</sup>. To show that *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup><sup>n</sup>* ∈ L1, we consider the <sup>L</sup>1-norm.

$$\begin{split} ||f \ast\_{\alpha} \phi\_{n}||\_{1} &= \int\_{\mathbb{R}} |(f \ast\_{\alpha} \phi\_{n})(\mathbf{x})| d\mathbf{x}, \\ &= \int\_{\mathbb{R}} \left| \int\_{-\infty}^{\infty} f(\mathbf{x} - t) \phi\_{n}(t) e^{-i t (\mathbf{x} - t) \cot \alpha} dt \right| d\mathbf{x}, \\ &\leq \int\_{\mathbb{R}} \int\_{-\infty}^{\infty} \left| f(\mathbf{x} - t) \phi\_{n}(t) e^{-i t (\mathbf{x} - t) \cot \alpha} \right| dt d\mathbf{x}, \\ &= \int\_{-\infty}^{\infty} |f(\mathbf{x} - t)| d\mathbf{x} \int\_{-\infty}^{\infty} |\phi\_{n}(t)| dt, \\ &= ||f||\_{1} ||\phi\_{n}||\_{1}. \end{split}$$

Since *<sup>f</sup>* ∈ L<sup>1</sup> and {*φn*} ∈ <sup>Δ</sup>, *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>n*<sup>1</sup> ≤ *<sup>f</sup>* 1*φn*<sup>1</sup> <sup>&</sup>lt; <sup>∞</sup>, which proves that *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup><sup>n</sup>* ∈ L1.

**Lemma 5.** *If f* , *<sup>g</sup>* ∈ L1*, <sup>φ</sup>* <sup>∈</sup> *S, then* (*<sup>f</sup>* <sup>+</sup> *<sup>g</sup>*) <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* <sup>=</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* <sup>+</sup> *<sup>g</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>*.

The proof of this lemma is straightforward. Therefore, we omitted the details.

**Lemma 6.** *Let fn* <sup>→</sup> *f in* <sup>L</sup><sup>1</sup> *as n* <sup>→</sup> <sup>∞</sup> *and <sup>φ</sup>* <sup>∈</sup> *S. Then fn* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* <sup>→</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup> in* <sup>L</sup>1*.*

**Proof.** From Lemma 4, we can write

$$\begin{aligned} \|(f\_n \*\_{\mathfrak{a}} \phi) - (f \*\_{\mathfrak{a}} \phi)\|\_1 &= \|(f\_n - f) \*\_{\mathfrak{a}} \phi\|\_1 \\ &\le \|f\_n - f\|\_1 \|\phi\|\_1 \\ &\le M \|f\_n - f\|\_1 \to 0 \text{ as } n \to \infty \text{ for } M > 0. \end{aligned}$$

Hence, *fn* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* <sup>→</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* in <sup>L</sup><sup>1</sup> whenever *fn* <sup>→</sup> *<sup>f</sup>* in <sup>L</sup>1.

**Lemma 7.** *Let fn* <sup>→</sup> *f in* <sup>L</sup><sup>1</sup> *and* {*φn*} ∈ <sup>Δ</sup>*. Then fn* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup><sup>n</sup>* <sup>→</sup> *f in* <sup>L</sup>1*.*

**Proof.** Let {*φn*} ∈ <sup>Δ</sup> then <sup>∞</sup> <sup>−</sup><sup>∞</sup> *<sup>φ</sup>n*(*t*)*eit*(*x*−*t*)*dt* <sup>=</sup> 1; therefore, we can write

$$\begin{split} (f\_{\boldsymbol{n}} \*\_{\boldsymbol{n}} \phi\_{\boldsymbol{n}})(\mathbf{x}) - f(\mathbf{x}) &= \int\_{-\infty}^{\infty} f\_{\boldsymbol{n}}(\mathbf{x} - t) \phi\_{\boldsymbol{n}}(t) e^{-i\mathbf{i}(\mathbf{x} - t)\cot\mathfrak{a}} dt - f(\mathbf{x}) \int\_{-\infty}^{\infty} \phi\_{\boldsymbol{n}}(t) e^{i\mathbf{i}(\mathbf{x} - t)\cot\mathfrak{a}} dt \\ &= \int\_{-\infty}^{\infty} \left( f\_{\boldsymbol{n}}(\mathbf{x} - t) e^{-2i\mathbf{i}(\mathbf{x} - t)\cot\mathfrak{a}} - f(\mathbf{x}) \right) e^{i\mathbf{i}(\mathbf{x} - t)\cot\mathfrak{a}} \phi\_{\boldsymbol{n}}(t) dt. \end{split}$$

Now, we consider the *L*1-norm of the above equation:

$$\begin{split} \|f\_{\boldsymbol{n}} \ast\_{\boldsymbol{n}} \phi\_{\boldsymbol{n}} - f\|\_{1} &= \int\_{-\infty}^{\infty} \left| \int\_{-\infty}^{\infty} \left( f\_{\boldsymbol{n}}(\boldsymbol{x} - t) e^{-2i\boldsymbol{t}(\boldsymbol{x} - t)\cot a} - f(\boldsymbol{x}) \right) e^{i\boldsymbol{t}(\boldsymbol{x} - t)\cot a} \phi\_{\boldsymbol{n}}(t) dt \right| d\boldsymbol{x} \\ &\leq \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} |f\_{\boldsymbol{n}}(\boldsymbol{x} - t) e^{-2i\boldsymbol{t}(\boldsymbol{x} - t)\cot a} - f(\boldsymbol{x})| |\phi\_{\boldsymbol{n}}(t)| dt d\boldsymbol{x}. \end{split}$$

As an application of Fubini's theorem and via Property 2 of delta sequences, we have

$$\begin{aligned} ||f\_n \*\_{\mathfrak{a}} \phi\_{\mathfrak{n}} - f||\_1 &\leq \int\_{-\infty}^{\infty} |\phi\_{\mathfrak{n}}(t)| dt \int\_{-\infty}^{\infty} |f\_{\mathfrak{n}}(\mathbf{x} - t) e^{-2it(\mathbf{x} - t)\cot\mathfrak{a}} - f(\mathbf{x})| d\mathbf{x} \\ &\leq M ||(f\_n)\_t e^{-2it(\mathbf{x} - t)\cot\mathfrak{a}} - f||\_{1\prime} \quad (M > 0). \end{aligned}$$

Using the triangular inequality of normed spaces,

$$\begin{aligned} \|f\_{\boldsymbol{n}} \*\_{\boldsymbol{n}} \phi\_{\boldsymbol{n}} - f\|\_{1} &\leq M \|(f\_{\boldsymbol{n}})\_{\boldsymbol{t}} e^{-2i\boldsymbol{t}(\mathbf{x}-\boldsymbol{t})\cot\boldsymbol{a}} - f\_{\boldsymbol{t}} e^{-2i\boldsymbol{t}(\mathbf{x}-\boldsymbol{t})\cot\boldsymbol{a}}\|\_{1} + \|f\_{\boldsymbol{f}} e^{-2i\boldsymbol{t}(\mathbf{x}-\boldsymbol{t})\cot\boldsymbol{a}} - f\|\_{1} \\ &\leq M \|(f\_{\boldsymbol{n}})\_{\boldsymbol{t}} e^{-2i\boldsymbol{t}(\mathbf{x}-\boldsymbol{t})\cot\boldsymbol{a}} - f\_{\boldsymbol{t}} e^{-2i\boldsymbol{t}(\mathbf{x}-\boldsymbol{t})\cot\boldsymbol{a}}\|\_{1} + M \|f\_{\boldsymbol{f}} e^{-2i\boldsymbol{t}(\mathbf{x}-\boldsymbol{t})\cot\boldsymbol{a}} - f\|\_{1}. \end{aligned}$$

By using the convergence of *fn* ∈ L<sup>1</sup> and Theorem 1, we have

$$||(f\_n)\_t e^{-2it(\mathbf{x}-t)\cot\mathfrak{a}} - f\_t e^{-2it(\mathbf{x}-t)\cot\mathfrak{a}}||\_1 \to 0 \text{ as } n \to \infty$$

and

$$\|f\_t e^{-2\text{if}(\mathbf{x}-t)\cot a} - f\|\_1 \to 0 \text{ as } t \to 0.$$

$$\text{Therefore, } \|f\_n \*\_{\sf a} \phi\_{\sf n} - f\|\_1 \to 0 \text{ as } n \to \infty \text{, hence, } f\_{\sf n} \*\_{\sf a} \phi\_{\sf n} \to f \text{ in } \mathcal{L}^1.$$

In order to extend the FHT to the class of Boehmians, we define another class of Boehmians (as the codomain of the extended fractional Hilbert transform) <sup>B</sup><sup>2</sup> <sup>=</sup> <sup>B</sup>2(C∞, *<sup>S</sup>*, <sup>∗</sup>*α*, <sup>Δ</sup>) [7]. The notion of delta sequences, quotients, and their equivalence classes remains the same as

that in the prior case. We also retain the definitions of addition and scalar multiplication. Now, we define

$$D^m \left[ \frac{f\_n}{\phi\_n} \right] = \left[ \frac{D^m f\_n}{\phi\_n} \right] \text{ for any } \left[ \frac{f\_n}{\phi\_n} \right] \in \mathcal{B}\_2.$$
 
$$\text{for all } n, \text{ and } \phi\_n \text{ for } \phi\_n, \phi\_n \text{ is }$$

In addition,

$$
\left[\frac{f\_n}{\phi\_n}\right] \ast\_\alpha \left[\frac{\mathcal{G}\_n}{\psi\_n}\right] = \left[\frac{f\_n \ast\_\alpha \mathcal{g}\_n}{\phi\_n \ast\_\alpha \psi\_n}\right].
$$

Since a concept of convergence is required to construct a Boehmian space, we prove two convergences on <sup>C</sup>∞.

**Lemma 8.** *Let fn* <sup>→</sup> *<sup>f</sup> as <sup>n</sup>* <sup>→</sup> <sup>∞</sup> *in* <sup>C</sup><sup>∞</sup> *then fn* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* <sup>→</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup> in* <sup>C</sup><sup>∞</sup> *for all <sup>φ</sup>* <sup>∈</sup> *D; further, for each delta sequence* {*δn*}*, fn* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup><sup>n</sup>* <sup>→</sup> *f as n* <sup>→</sup> <sup>∞</sup> *in* <sup>C</sup>∞*.*

**Proof.** Let *<sup>K</sup>* <sup>⊂</sup> <sup>R</sup> be any compact set, such that *<sup>x</sup>* <sup>∈</sup> *<sup>K</sup>*. To prove the convergence of a sequence of functions in <sup>C</sup>∞, we must show that the functions and their derivatives converge uniformly on compact sets.

First, we prove that *fn* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* <sup>→</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* in <sup>C</sup>∞. For this, consider

$$|(f\_{\mathfrak{n}} \*\_{\mathfrak{a}} \phi - f \*\_{\mathfrak{a}} \phi)(\mathbf{x})| = |((f\_{\mathfrak{n}} - f) \*\_{\mathfrak{a}} \phi)(\mathbf{x})| \le \int\_{-\infty}^{\infty} |(f\_{\mathfrak{n}} - f)(\mathbf{x} - t)| \phi(t) dt.$$

Since *t* varies over the compact support of *φ*; therefore, *x* − *t* also varies over a compact set in <sup>R</sup>. So, <sup>|</sup>((*fn* <sup>−</sup> *<sup>f</sup>*) <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>*)(*x*)| → 0 as *<sup>n</sup>* <sup>→</sup> <sup>∞</sup> uniformly on compact sets. Then,

$$|(f\_n \*\_{\alpha} \phi - f \*\_{\alpha} \phi)(x)| \to 0 \text{ as } n \to \infty,$$

or we can write

$$f\_n \*\_{\alpha} \phi \to f \*\_{\alpha} \phi \text{ as } n \to \infty,\tag{6}$$

uniformly on compact sets.

In addition,

$$D^{\mathfrak{m}}((f\_n \ast\_{\mathfrak{a}} \phi) - (f \ast\_{\mathfrak{a}} \phi)) = (D^{\mathfrak{m}}f\_n \ast\_{\mathfrak{a}} \phi) - (D^{\mathfrak{m}}f \ast\_{\mathfrak{a}} \phi). \tag{7}$$

Replacing *D<sup>m</sup> fn* by *fn* and *D<sup>m</sup> f* by *f* in (7), we have

$$D^m((f\_n \*\_a \phi) - (f \*\_a \phi)) = (f\_n \*\_a \phi) - (f \*\_a \phi),\tag{8}$$

the right-hand side of (8) approaches zero by (6). Thus,

$$D^m(f\_n \ast\_\alpha \phi) \to D^m(f \ast\_\alpha \phi)$$

uniformly on compact sets. Hence, *fn* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* <sup>→</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>φ</sup>* as *<sup>n</sup>* <sup>→</sup> <sup>∞</sup> in <sup>C</sup>∞.

Next, without any loss of generality, let us suppose that {*δn*} ∈ Δ is such that it has a compact support. Then,

$$\begin{split} |(f\_{\boldsymbol{n}} \*\_{\mathcal{U}} \delta\_{\boldsymbol{n}} - f)(\boldsymbol{x})| &= \left| \int\_{-\infty}^{\infty} f\_{\boldsymbol{n}}(\boldsymbol{x} - t) \delta\_{\boldsymbol{n}}(t) e^{-2i(\boldsymbol{x} - t)\cot\boldsymbol{x}} dt - f(\boldsymbol{x}) \int\_{-\infty}^{\infty} e^{2i(\boldsymbol{x} - t)\cot\boldsymbol{x}} \delta\_{\boldsymbol{n}}(t) dt \right| \\ &\leq \int\_{-\infty}^{\infty} |f\_{\boldsymbol{n}}(\boldsymbol{x} - t) e^{-2i(\boldsymbol{x} - t)\cot\boldsymbol{x}} - f(\boldsymbol{x})| \delta\_{\boldsymbol{n}}(t) dt, \\ &\leq \int\_{-\infty}^{\infty} \left( |f\_{\boldsymbol{n}}(\boldsymbol{x} - t) e^{-2i(\boldsymbol{x} - t)\cot\boldsymbol{x}} - f(\boldsymbol{x} - t) e^{-2i(\boldsymbol{x} - t)\cot\boldsymbol{x}}| + |f(\boldsymbol{x} - t) e^{-2i(\boldsymbol{x} - t)\cot\boldsymbol{x}} - f(\boldsymbol{x})| \right) \delta\_{\boldsymbol{n}}(t) dt. \end{split}$$

Now, both *x* and *t* vary over compact sets; therefore, *x* − *t* also varies over a compact set. Thus,

$$\int\_{-\infty}^{\infty} \left( |f\_{\mathbf{n}}(\mathbf{x}-t)e^{-2it(\mathbf{x}-t)\cot a} - f(\mathbf{x}-t)e^{-2it(\mathbf{x}-t)\cot a}| + |f(\mathbf{x}-t)e^{-2it(\mathbf{x}-t)\cot a} - f(\mathbf{x})| \right) \delta\_{\mathbf{n}}(t)dt \to 0$$

as *n* → ∞ and *t* → 0.

We have *fn* ∗*<sup>α</sup> δ<sup>n</sup>* → *f* uniformly on compact sets. Similarly, *<sup>D</sup>m*(*fn* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>n*) <sup>→</sup> *<sup>D</sup>m*(*f*) uniformly on compact sets. Hence, *fn* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup><sup>n</sup>* <sup>→</sup> *<sup>f</sup>* as *<sup>n</sup>* <sup>→</sup> <sup>∞</sup> in <sup>C</sup>∞.

**Lemma 9.** *If fn* <sup>→</sup> *f as n* <sup>→</sup> <sup>∞</sup> *in* <sup>L</sup>1*, then fn* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>* <sup>→</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup> as n* <sup>→</sup> <sup>∞</sup> *in* <sup>C</sup><sup>∞</sup> *for every <sup>δ</sup>* <sup>∈</sup> *S.*

**Proof.** To show the convergence in <sup>C</sup>∞, we assume that *<sup>x</sup>* varies over a compact set *<sup>K</sup>*.

$$\begin{aligned} |(f\_n \*\_{\alpha} \delta - f \*\_{\alpha} \delta)(\mathbf{x})| &= |((f\_n \* - f) \*\_{\alpha} \delta)(\mathbf{x})| \\ &= \left| \int\_{-\infty}^{\infty} (f\_n - f)(\mathbf{x} - t) \delta(t) e^{-it(\mathbf{x} - t) \cot \alpha} dt \right| \\ &\leq \int\_{-\infty}^{\infty} |(f\_n - f)(\mathbf{x} - t)| |\delta(t)| dt \\ &\leq ||f\_n - f||\_1 ||\delta||\_{\infty} .\end{aligned}$$

Since *fn* <sup>→</sup> *<sup>f</sup>* in <sup>L</sup><sup>1</sup> and *<sup>δ</sup>* <sup>∈</sup> *<sup>S</sup>* has a compact support, *<sup>x</sup>* <sup>−</sup> *<sup>t</sup>* varies over a compact set, and |(*fn* ∗*<sup>α</sup> δ* − *f* ∗*<sup>α</sup> δ*)(*x*)| → 0 as *n* → ∞ on compact sets. Similarly, we have

$$|D^m[(f\_n \*\_{\mathfrak{a}} \delta - f \*\_{\mathfrak{a}} \delta)](\mathfrak{x})| \le ||f\_n - f||\_1 ||D^m \delta||\_{\infty}.$$

Thus, *<sup>D</sup>m*(*fn* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>*) <sup>→</sup> *<sup>D</sup>m*(*<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>*) on compact sets. Hence, *fn* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>* <sup>→</sup> *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>* as *<sup>n</sup>* <sup>→</sup> <sup>∞</sup> in <sup>C</sup>∞.

#### *3.3. Fractional Hilbert Transform on Boehmians*

The following result is very important in the aftermath. The proof of the following theorem is similar to the proof of convolution theorem for FHT as in Theorem 2; we omitted the details.

**Theorem 4.** *If f* ∈ L<sup>1</sup> *and <sup>δ</sup>* <sup>∈</sup> <sup>Δ</sup>*, then Hα*[ *<sup>f</sup>* <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>*] = *<sup>H</sup>α*[ *<sup>f</sup>* ] <sup>∗</sup>*<sup>α</sup> <sup>δ</sup>.*

**Definition 3.** *The fractional Hilbert transform* H*<sup>α</sup>* : B<sup>1</sup> → B<sup>2</sup> *on Boehmians is defined by*

$$\mathcal{H}\_{\alpha} \left[ \frac{f\_n}{\phi\_n} \right] = \left[ \frac{\mathcal{H}\_{\alpha} f\_n}{\phi\_n} \right] \rho$$

where *fn <sup>φ</sup><sup>n</sup>* is an arbitrary representative of any given Boehmian *B* ∈ B1. Since

$$f\_n \*\_{\alpha} \phi\_m = f\_m \*\_{\alpha} \phi\_n \quad \forall m \, \, n \in \mathbb{N}.$$

By Theorem 4, we can write <sup>H</sup>*α*[ *fn*] <sup>∗</sup>*<sup>α</sup> <sup>φ</sup><sup>m</sup>* <sup>=</sup> <sup>H</sup>*α*[ *fm*] <sup>∗</sup>*<sup>α</sup> <sup>φ</sup><sup>n</sup>* <sup>∀</sup>*m*, *<sup>n</sup>* <sup>∈</sup> <sup>N</sup>.

Therefore, <sup>H</sup>*α*[ *fn*] *<sup>φ</sup><sup>n</sup>* represents a Boehmian in <sup>B</sup>2. In a similar manner, let *gn <sup>ψ</sup><sup>n</sup>* be another representative of *B*; then, again, with an application of Theorem 4,

$$\frac{\mathcal{H}\_{\alpha}[f\_n]}{\phi\_n} \sim \frac{\mathcal{H}\_{\alpha}[g\_n]}{\psi\_n}.$$

thus the extended FHT on Boehmians H*<sup>α</sup>* : B<sup>1</sup> → B<sup>2</sup> is well-defined.

**Theorem 5.** *Let* H*<sup>α</sup>* : B<sup>1</sup> → B<sup>2</sup> *be the extended FHT; then, 1. If fn <sup>φ</sup><sup>n</sup>* ∈ B<sup>1</sup> *then* <sup>H</sup>*<sup>α</sup> fn <sup>φ</sup><sup>n</sup>* ∈ B2*.*


**Proof.** The proof of the above theorem is similar to those of Hilbert transform on Boehmians; we omitted the details. For details, the reader is referred to [7].

#### **4. Conclusions**

This paper gave an extension of the fractional Hilbert transform to a class of generalized functions known as Boehmians. It introduces a new convolutional operator, and the consequent convolutional theorem was also presented. In addition, the extended fractional Hilbert transform is a well-defined map between the spaces of Boehmians having properties, such as continuity and linearity, identical to the classical properties of their corresponding classical versions. Lastly, convergence concerning *δ* and Δ was also examined.

The methods of this paper can also be utilized to extend FHT to the space of ultradistributions. We suggest that readers consider the expansion of the fractional Hilbert transform to q-calculus and develop the theory of the quaternion fractional Hilbert transform.

**Author Contributions:** Both authors contributed equally to this work. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank the reviewers for taking the time to read and improve the present article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviation is used in this manuscript:

FHT Fractional Hilbert transform

#### **References**

