Analysis of Implicit Neutral-Tempered Caputo Fractional Volterra–Fredholm Integro-Differential Equations Involving Retarded and Advanced Arguments

Abstract

This paper investigates a class of implicit neutral fractional integro-differential equations of Volterra–Fredholm type. The equations incorporate a tempered fractional derivative in the Caputo sense, along with both retarded (delay) and advanced arguments. The problem is formulated on a time domain segmented into past, present, and future intervals and includes nonlinear mixed integral operators. Using Banach’s contraction mapping principle and Schauder’s fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions within the space of continuous functions. The study is then extended to general Banach spaces by employing Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Ulam–Hyers–Rassias stability is also analyzed under appropriate conditions. To demonstrate the practical applicability of the theoretical framework, explicit examples with specific nonlinear functions and integral kernels are provided. Furthermore, detailed numerical simulations are conducted using MATLAB-based specialized algorithms, illustrating solution convergence and behavior in both finite-dimensional and Banach space contexts.

1. Introduction

Fractional differential equations now constitute a vital modeling approach for describing intricate systems characterized by memory effects and hereditary behavior within various fields such as physics, engineering, biology, and finance [1,2,3]. In contrast to traditional integer-order differential models, fractional-order operators naturally account for non-local interactions and extended temporal dependencies. This makes them particularly suitable for representing processes like anomalous diffusion, viscoelastic responses, and other phenomena where past states significantly influence current dynamics. In recent years, considerable progress has been made in the theoretical analysis of fractional differential equations, particularly regarding solution existence, uniqueness, and stability across various classes [4,5,6]. Extensive research has been conducted on equations incorporating diverse fractional operators—including Riemann–Liouville, Caputo, and Hadamard derivatives—subject to a wide range of boundary and initial conditions [7,8,9].

A notable development in this domain is tempered fractional calculus, which incorporates an exponential tempering factor into traditional fractional operators [10,11]. This adjustment retains the non-local nature of fractional derivatives while reducing the slow algebraic decay typical of conventional models, thereby offering more accurate representations of systems exhibiting tempered power-law memory [12,13].

The tempered fractional framework has proven valuable in applications where the influence of past states decays exponentially at large times, such as in finance, geophysics, and turbulent flows [14]. Recent studies have begun to investigate the qualitative properties of tempered fractional differential equations, including well-posedness and numerical approximations.

Within the broad context of fractional dynamics, implicit differential equations present distinctive analytical challenges, as the highest-order derivative is defined implicitly by the unknown function itself. This characteristic often leads to nonlinear and non-local formulations that resist conventional solution methods [15]. The analysis of such implicit fractional equations has attracted significant interest, leading to the application of diverse fixed point theorems and measure theoretic methodologies to prove existence and stability under various assumptions [16,17]. A substantial body of recent literature has expanded the scope of these models by integrating retarded and advanced arguments, nonlocal conditions, and integral boundary terms [18,19]. Specific studies in this evolving field include the examination of implicit neutral-tempered fractional equations with mixed delays [20,21], the analysis of systems featuring nonlocal conditions in weighted function spaces [22], and the exploration of implicit fractional difference equations [23]. Further contributions involve models based on the AB-Caputo derivative [24] and other advanced configurations [25]. Building upon this foundation, recent advancements have also addressed more generalized fractional frameworks, such as the investigation of existence and Ulam stability for k-generalized $\mathfrak{F}$-Hilfer fractional problems [26]. Additionally, the study of continuity properties for the fractional derivative in time-fractional semilinear pseudo-parabolic systems [27] highlights the ongoing effort to understand the analytical behavior of complex fractional models. Collectively, this work underscores the increasing sophistication and practical importance of implicit fractional models.

From an applied standpoint, the class of implicit neutral-tempered fractional Volterra–Fredholm integro-differential equations considered in this work is motivated by models arising in viscoelasticity, control theory, and neural dynamics. In viscoelastic materials, tempered Caputo fractional derivatives provide an accurate description of hereditary stress–strain behavior, where long-range memory effects persist but decay exponentially over time. The presence of Volterra-type integral terms reflects history-dependent material responses, while Fredholm-type integrals account for spatially distributed or global interactions. Moreover, neutral structures with retarded and advanced arguments naturally occur in systems involving delayed feedback and anticipation effects, such as transmission lines and neural networks with predictive mechanisms. These modeling considerations demonstrate that the proposed mathematical framework is not only theoretically significant but also well suited for the analysis of complex real-world systems characterized by memory, delay, and nonlocal effects.

However, a significant research gap remains regarding implicit neutral fractional differential equations that simultaneously incorporate tempered derivatives and mixed integral operators of Volterra and Fredholm types in a unified model. While individual aspects—such as implicit structures [5], neutral terms [20], and integral operators [9,28]—have been studied separately, their combined analysis remains largely unexplored.For instance, recent work on weak solutions for fractional Langevin equations involving two fractional orders in Banach spaces [29] demonstrates the relevance of functional-analytic techniques for handling fractional models in abstract spaces, yet the incorporation of mixed integral operators and advanced-delayed structures introduces additional layers of complexity. Neutral equations, where the derivative appears with a delayed argument, are essential for modeling systems with propagation delays, such as in neural networks and transmission lines. The inclusion of both Volterra (history-dependent) and Fredholm (global) integral operators further allows the modeling of distributed past effects and global constraints simultaneously. However, the coupling of these features with the non-local and tempered fractional derivative introduces significant mathematical complexities, particularly in establishing the existence of solutions under nonlocal and advanced-retarded boundary conditions. To address these challenges, methodologies from the theory of nonlinear integral equations in abstract spaces [30] provide a robust foundation for analyzing the operator structures that arise in such composite problems.

Inspired by this unresolved issue and drawing on techniques from [5,12,20], this research aims to examine a new category of implicit neutral Caputo tempered fractional differential equations that include combined Volterra–Fredholm integral operators and involve both advanced and delayed arguments. Specifically, we intend to

1.

Frame the problem in a suitable Banach space setting, considering both historical and advanced segments.

2.

Derive sufficient criteria for solution existence using the Kuratowski measure of noncompactness [31], Darbo’s fixed point theorem [32], and key properties of tempered fractional calculus [10,33].

3.

Additionally, the solution map’s continuity guarantees that trajectories vary only slightly under small perturbations in initial conditions or system parameters—a fundamental property for assessing stability.

In this paper, we study existence, uniqueness, and stability properties for the following implicit neutral fractional integro-differential equation of Volterra–Fredholm type:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill {}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}[\mathbb{k}\left(\rho \right)-& \aleph (\rho ,{\mathbb{k}}^{\rho })]=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^{\xi }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right),\rho \in \phi =[0,\xi ],\hfill \\ \mathbb{k}\left(\rho \right)=\varphi \left(\rho \right),\hfill & \rho \in [-\mathfrak{a},0],\hfill \\ \mathbb{k}\left(\rho \right)=\widehat{\varphi }\left(\rho \right),\hfill & \rho \in [\xi ,\xi +\mathfrak{g}].\hfill \end{array}\right.\end{array}$$

(1)

where ${}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}$ is the Caputo-tempered fractional derivative of order $\mathfrak{w}\in (0,1)$, ${\mathfrak{w}}_1\ge 0$, $\mathfrak{a},\mathfrak{g}>0$, $\mathfrak{F}:\phi \times C\left(\right[-\mathfrak{a},\mathfrak{g}],\mathbb{R})\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is a given function, $\aleph :\phi \times C\left(\right[-\mathfrak{a},\mathfrak{g}],\mathbb{R})\to \mathbb{R}$ is a given function, ${\hslash }_1,{\hslash }_2:\phi \times \phi \times \mathbb{R}\to \mathbb{R}$ are integral kernels, $\varphi \in C\left(\right[-\mathfrak{a},0],\mathbb{R})$, and $\widehat{\varphi }\in C([\xi ,\xi +\mathfrak{g}],\mathbb{R})$. We denote by ${\mathbb{k}}^{\rho }$ the element of $C\left(\right[-\mathfrak{a},\mathfrak{g}],\mathbb{R})$ defined by

$${\mathbb{k}}^{\rho }=\mathbb{k}(\rho +\ell ):\ell \in [-\mathfrak{a},\mathfrak{g}].$$

The paper is arranged as follows: In Section 2, we provide some notations, definitions, and auxiliary results used throughout the work. Section 3 presents existence and uniqueness results for problem (1) based on Banach’s contraction principle and Schauder’s fixed point theorem, along with a demonstrative example. A similar problem in Banach spaces is also treated in Section 3 using Darbo’s fixed point theorem and the Kuratowski measure of noncompactness, followed by an analysis of Ulam–Hyers–Rassias stability and corresponding application results for problem (19). Section 4 is devoted to numerical simulations of the proposed problems. Finally, Section 5 provides conclusions and suggests directions for future research.

2. Preliminaries

This section introduces the foundational mathematical framework, notations, and preliminary concepts that will be utilized throughout this investigation. Our analysis is conducted within a Banach space, denoted by $(E,\parallel ·\parallel )$. Consider a fixed interval $\phi :=[0,\xi ]$. The space of all continuous functions defined on the compact set $\phi$ and taking values in a Banach space E, equipped with the supremum norm, forms a Banach space denoted by $\mathbb{C}(\phi ,E)$.  

$${\parallel \mathbb{k}\parallel }_{\infty }=\underset{\rho \in \phi }{max}\parallel \mathbb{k}\left(\rho \right)\parallel .$$

Additionally, we need specific Banach spaces for functions defined on delayed initial intervals and advanced terminal intervals. For a given non-negative constant $\mathfrak{a}$, the space is defined as

$${C([-\mathfrak{a},0],E),\mathrm{with}\mathrm{its}\mathrm{norm}\mathrm{given}\mathrm{by}\parallel \mathbb{k}\parallel }_{[-\mathfrak{a},0]}=\underset{\rho \in [-\mathfrak{a},0]}{sup}\parallel \mathbb{k}\left(\rho \right)\parallel .$$

Correspondingly, for a positive constant $\mathfrak{g}$, we define

$${C([T,\xi +\mathfrak{g}],E),\mathrm{with}\mathrm{its}\mathrm{norm}\mathrm{given}\mathrm{by}\parallel \mathbb{k}\parallel }_{[T,\xi +\mathfrak{g}]}=\underset{\rho \in [T,\xi +\mathfrak{g}]}{sup}\parallel \mathbb{k}\left(\rho \right)\parallel .$$

For the comprehensive interval, let $C\left(\right[-\mathfrak{a},\xi +\mathfrak{g}],E)$ be the Banach space with the norm

$${\parallel \mathbb{k}\parallel }_{[-\mathfrak{a},\xi +\mathfrak{g}]}=sup\{\parallel \mathbb{k}\left(\rho \right)\parallel :\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]\}.$$

We also require the space of Bochner integrable functions, denoted by $L^1(\phi ,E)$, with the norm

$${\parallel \mathbb{k}\parallel }_{L^1}={\int }_0^{\xi }\parallel \mathbb{k}\left(\rho \right)\parallel d\rho .$$

For the integral kernels, we consider the space $C(\phi \times \phi \times E,E)$ of continuous functions under appropriate boundedness conditions.

Now, define the function space $\mathrm{{\rm Y}}$ as

$$\mathrm{{\rm Y}}=\{\mathbb{k}:[-\mathfrak{a},\xi +\mathfrak{g}]\to E:\mathbb{k}{|}_{[0,\xi ]}\in C(\phi ,E),\mathbb{k}{|}_{[-\mathfrak{a},0]}\in C([-\mathfrak{a},0],E),\mathbb{k}{|}_{[\xi ,\xi +\mathfrak{g}]}\in C([T,\xi +\mathfrak{g}],E)\}.$$

This space $\mathrm{{\rm Y}}$ is a Banach space when endowed with the norm

$${\parallel \mathbb{k}\parallel }_{\mathrm{{\rm Y}}}=\underset{\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]}{sup}\parallel \mathbb{k}\left(\rho \right)\parallel .$$

Definition 1

([10,11,14]). For a prescribed function $\mathfrak{F}$ belonging to the space $C(\phi ,E)$ and a fixed tempering coefficient ${\mathfrak{w}}_1\ge 0$, the Riemann–Liouville-tempered fractional integral of order $\mathfrak{w}$ is formally defined by the following representation:

$${}_{0}I_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathfrak{F}\left(\rho \right)=e^{-{\mathfrak{w}}_1\rho }{}_{0}I_{\rho }^{\mathfrak{w}}\left(e^{{\mathfrak{w}}_1\rho }\mathfrak{F}\left(\rho \right)\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{\displaystyle \frac{e^{-{\mathfrak{w}}_1(\rho -\varpi )}\mathfrak{F}\left(\varpi \right)}{{(\rho -\varpi )}^{1-\mathfrak{w}}}}d\varpi ,$$

where ${}_{0}I_{\rho }^{\mathfrak{w}}$ refers to the classical Riemann–Liouville fractional integral, defined by

$${}_{0}I_{\rho }^{\mathfrak{w}}\mathfrak{F}\left(\rho \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{\displaystyle \frac{\mathfrak{F}\left(\varpi \right)}{{(\rho -\varpi )}^{1-\mathfrak{w}}}}d\varpi .$$

We observe that when ${\mathfrak{w}}_1=0$, the tempered form simplifies to its classical Riemann–Liouville counterpart.

Definition 2.

Let $\mathfrak{w}$ be a real number satisfying $\beta -1<\mathfrak{w}<\beta$, where β is a positive integer, and let ${\mathfrak{w}}_1$ be a non-negative constant. The Riemann–Liouville-tempered fractional derivative of a function $\mathfrak{F}\left(\rho \right)$ with respect to the variable ρ is defined as

$${}_{0}D_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathfrak{F}\left(\rho \right)=e^{-{\mathfrak{w}}_1\rho }{}_{0}D_{\rho }^{\mathfrak{w}}\left(e^{{\mathfrak{w}}_1\rho }\mathfrak{F}\left(\rho \right)\right)={\displaystyle \frac{e^{-{\mathfrak{w}}_1\rho }}{\mathrm{\Gamma }(\beta -\mathfrak{w})}}{\displaystyle \frac{d^{\beta }}{d{\rho }^{\beta }}}{\int }_0^{\rho }{\displaystyle \frac{e^{{\mathfrak{w}}_1\varpi }\mathfrak{F}\left(\varpi \right)}{{(\rho -\varpi )}^{\mathfrak{w}-\beta +1}}}d\varpi ,$$

where ${}_{0}D_{\rho }^{\mathfrak{w}}$ denotes the classical Riemann–Liouville fractional derivative operator.

Definition 3.

Given parameters $\mathfrak{w}$ and ${\mathfrak{w}}_1$ satisfying $\beta -1<\mathfrak{w}<\beta$ for some $\beta \in \mathbb{N}$ and ${\mathfrak{w}}_1\ge 0$, the Caputo-tempered fractional derivative of a function $\mathfrak{F}\left(\rho \right)$ is expressed as

$${}_0{}^{C}D_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathfrak{F}\left(\rho \right)=e^{-{\mathfrak{w}}_1\rho }{}_0{}^{C}D_{\rho }^{\mathfrak{w}}\left(e^{{\mathfrak{w}}_1\rho }\mathfrak{F}\left(\rho \right)\right).$$

Equivalently, it can be written in integral form as

$${}_0{}^{C}D_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathfrak{F}\left(\rho \right)={\displaystyle \frac{e^{-{\mathfrak{w}}_1\rho }}{\mathrm{\Gamma }(\beta -\mathfrak{w})}}{\int }_0^{\rho }{(\rho -\varpi )}^{\beta -\mathfrak{w}-1}·{\displaystyle \frac{d^{\beta }}{d{\varpi }^{\beta }}}\left(e^{{\mathfrak{w}}_1\varpi }\mathfrak{F}\left(\varpi \right)\right)d\varpi ,$$

where ${}_0{}^{C}D_{\rho }^{\mathfrak{w}}$ represents the classical Caputo fractional derivative.

Lemma 1

([10]). For a constant C, we have

$${}_{0}D_{\theta }^{\sigma ,\varrho }C=C{e}^{-\theta \varrho }{}_{0}D_{\theta }^{\sigma }{e}^{\theta \varrho },{}_0{}^{C}D_{\theta }^{\sigma ,\varrho }C=C{e}^{-\theta \varrho }{}_0{}^{C}D_{\theta }^{\sigma }{e}^{\theta \varrho }.$$

Obviously,

$${}_{0}D_{\theta }^{\sigma ,\varrho }\left(C\right)\ne {}_0{}^{C}D_{\theta }^{\sigma ,\varrho }\left(C\right).$$

Moreover, ${}_0{}^{C}D_{\theta }^{\sigma ,\varrho }\left(C\right)$ is no longer equal to zero, which is different from ${}_0{}^{C}D_{\theta }^{\sigma }\left(C\right)\equiv 0$.

Lemma 2

([10,14]). Assume $\mathfrak{F}\in C^{\beta }(\phi ,E)$ where $\beta -1<\omega <\beta$ with $\beta \in \mathbb{N}$. The sequential application of the tempered Caputo derivative and the tempered Riemann–Liouville integral is governed by specific, well-defined compositional rules:

$${}_{0}I_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\left[{}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathfrak{F}\left(\rho \right)\right]=\mathfrak{F}\left(\rho \right)-\sum _{j=0}^{\beta -1}{e}^{-{\mathfrak{w}}_1\rho }{\displaystyle \frac{{\rho }^j}{j!}}{\left.{\displaystyle \frac{d^j\left(e^{{\mathfrak{w}}_1\rho }\mathfrak{F}\left(\rho \right)\right)}{d{\rho }^j}}\right|}_{\rho =0},$$

and for $\mathfrak{w}\in (0,1)$,

$${}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\left[{}_{0}I_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathfrak{F}\left(\rho \right)\right]=\mathfrak{F}\left(\rho \right).$$

Definition 4

([31]). Consider a Banach space X and denote by ${\Delta }_X$ the collection of all bounded subsets of X. The Kuratowski measure of noncompactness is a function

$$\mathcal{E}:{\Delta }_X\to [0,\infty )$$

that assigns to each bounded subset a non-negative real number, defined as follows:

$$\mathcal{E}\left(\mathcal{M}\right)=inf\left\{\epsilon >0:\mathcal{M}\subset \bigcup _{i=1}^n{\mathcal{M}}_i,\mathrm{diam}\left({\mathcal{M}}_i\right)\le \epsilon \right\},$$

for $\mathcal{M}\in {\Delta }_X$. This measure satisfies the following properties for $\mathcal{M},{\mathcal{M}}_1,{\mathcal{M}}_2\in {\Delta }_X$ and $c\in \mathbb{R}$:

• $\mathcal{E}\left(\mathcal{M}\right)=0$ if and only if $\overline{M}$ is compact.
• $\mathcal{E}\left(\mathcal{M}\right)=\mathcal{E}\left(\overline{M}\right)$.
• ${\mathcal{M}}_1\subset {\mathcal{M}}_2$ implies $\mathcal{E}\left({\mathfrak{z}}_1\right)\le \mathcal{E}\left({\mathfrak{z}}_2\right)$.
• $\mathcal{E}({\mathcal{M}}_1+{\mathcal{M}}_2)\le \mathcal{E}\left({\mathcal{M}}_1\right)+\mathcal{E}\left({\mathcal{M}}_2\right)$.
• $\mathcal{E}\left(c\mathcal{M}\right)=\left|c\right|\mathcal{E}\left(\mathcal{M}\right)$.
• $\mathcal{E}\left(\mathrm{conv}\right(\mathcal{M}\left)\right)=\mathcal{E}\left(\mathcal{M}\right)$.

Lemma 3.

The set $\mathfrak{B}$ is taken as a subset of the function space Υ. Whenever $\mathfrak{B}$ is both bounded and equicontinuous, the ensuing attributes are satisfied:

(a) 

The mapping $\rho \mapsto \mathcal{E}\left(\mathfrak{B}\left(\rho \right)\right)$ is continuous on the interval $[-\mathfrak{a},\xi +\mathfrak{g}]$. Moreover, the measure of noncompactness of $\mathfrak{B}$ in Υ satisfies

$${\mathcal{E}}_{\mathrm{{\rm Y}}}\left(\mathfrak{B}\right)=\underset{\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]}{sup}\mathcal{E}\left(\mathfrak{B}\left(\rho \right)\right).$$

(b) 

For any collection of functions $\mathbb{k}\in \mathfrak{B}$, the measure of noncompactness of the set of their integrals is bounded by the integral of the measures:

$$\mathcal{E}\left(\left\{{\int }_0^{\xi }\mathbb{k}\left(\varpi \right)d\varpi :\mathbb{k}\in \mathfrak{B}\right\}\right)\le {\int }_0^{\xi }\mathcal{E}\left(\mathfrak{B}\left(\varpi \right)\right)d\varpi ,$$

where $\mathfrak{B}\left(\rho \right)=\left\{\mathbb{k}\right(\rho )\mid \mathbb{k}\in \mathfrak{B}\}$ for each ρ in the domain φ.

We recall several fundamental fixed point theorems that will be employed in our analysis:

Theorem 1

(Banach’s Fixed Point Theorem [34]). Let X be a Banach space and $\mathbb{k}:X\to X$ a contraction mapping, i.e., there exists $j\in [0,1)$ such that

$$\parallel \mathbb{k}\left({\mathbb{k}}_1\right)-\mathbb{k}\left({\mathbb{k}}_2\right)\parallel \le j\parallel {\mathbb{k}}_1-{\mathbb{k}}_2\parallel \mathit{for}\mathit{all}{\mathbb{k}}_1,{\mathbb{k}}_2\in X.$$

Then $\mathbb{k}$ possesses a unique fixed point in X.

Theorem 2

(Schauder’s Fixed Point Theorem [34]). Let X be a Banach space and let $\mathcal{Q}$ be a subset of X that is nonempty, bounded, closed, and convex. If $\mathbb{k}:\mathcal{Q}\to \mathcal{Q}$ is a continuous operator such that the image $\mathbb{k}\left(\mathcal{Q}\right)$ is relatively compact in X, then there exists an element $x^{*}\in \mathcal{Q}$ satisfying $\mathbb{k}\left(x^{*}\right)=x^{*}$.

Theorem 3

(Darbo’s Fixed Point Theorem [32]). Consider a Banach space X and let $\mathcal{Q}\subset X$ be nonempty, closed, bounded, and convex. Assume $\phi :\mathcal{Q}\to \mathcal{Q}$ is a continuous operator and let $\mathcal{E}$ denote the Kuratowski measure of noncompactness on X. If there exists a constant $\mathfrak{a}\in [0,1)$ such that the inequality

$$\mathcal{E}\left(\phi \right(\mathfrak{B}\left)\right)\le \mathfrak{a}\mathcal{E}\left(\mathfrak{B}\right)$$

For any subset $\mathfrak{B}\subset \mathcal{Q}$, if the given condition is satisfied, then the operator $\mathbb{k}$ has a fixed point in $\mathcal{Q}$.

Lemma 4.

For the integral operators present in our main problem, we have the following estimates:

Consider the Volterra integral operator $V_1\left[\mathbb{k}\right]\left(\rho \right)={\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi$. If ${\hslash }_1$ satisfies a Lipschitz condition in its third argument, that is, if there exists a function ${\mathcal{A}}_1(\rho ,\varpi )$ such that

$$\parallel {\hslash }_1(\rho ,\varpi ,u)-{\hslash }_1(\rho ,\varpi ,v)\parallel \le {\mathcal{A}}_1(\rho ,\varpi )\parallel u-v\parallel ,$$

then

$$\parallel V_1\left[\mathbb{k}\right]\left(\rho \right)-V_1\left[z\right]\left(\rho \right)\parallel \le {\int }_0^{\rho }{\mathcal{A}}_1(\rho ,\varpi )\parallel \mathbb{k}\left(\varpi \right)-\mathfrak{u}\left(\varpi \right)\parallel d\varpi .$$

Similarly, for the Fredholm integral operator $V_2\left[\mathbb{k}\right]\left(\rho \right)={\int }_0^{\xi }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi$, if there exists ${\mathcal{A}}_2(\rho ,\varpi )$ such that

$$\parallel {\hslash }_2(\rho ,\varpi ,u)-{\hslash }_2(\rho ,\varpi ,v)\parallel \le {\mathcal{A}}_2(\rho ,\varpi )\parallel u-v\parallel ,$$

then

$$\parallel V_2\left[\mathbb{k}\right]\left(\rho \right)-V_2\left[z\right]\left(\rho \right)\parallel \le {\int }_0^{\xi }{\mathcal{A}}_2(\rho ,\varpi )\parallel \mathbb{k}\left(\varpi \right)-\mathfrak{u}\left(\varpi \right)\parallel d\varpi .$$

Lemma 5.

If $B\subset \mathrm{{\rm Y}}$ is a bounded and equicontinuous set, the noncompactness measure of the related Volterra integral operator can be determined as follows:

$$\mathcal{E}\left(\left\{{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi :\mathbb{k}\in \mathfrak{B}\right\}\right)\le {\int }_0^{\rho }{\mathcal{A}}_1(\rho ,\varpi )\mathcal{E}\left(\mathfrak{B}\left(\varpi \right)\right)d\varpi .$$

For the Fredholm integral operator,

$$\mathcal{E}\left(\left\{{\int }_0^{\xi }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi :\mathbb{k}\in \mathfrak{B}\right\}\right)\le {\int }_0^{\xi }{\mathcal{A}}_2(\rho ,\varpi )\mathcal{E}\left(\mathfrak{B}\left(\varpi \right)\right)d\varpi ,$$

where $\mathfrak{B}\left(\varpi \right)=\left\{\mathbb{k}\right(\varpi ):\mathbb{k}\in \mathfrak{B}\}$.

3. Main Results

This section presents the essential mathematical groundwork for our analysis, including definitions of the function spaces and tempered fractional operators to be used, along with key lemmas and fixed point theorems. With these preliminaries established, we now proceed to investigate the existence, uniqueness, and stability of solutions to the implicit neutral-tempered fractional integro-differential problem in the next section.

3.1. Unique and Existence Results

Consider the fractional integro-differential equation given by

$${}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}[\mathbb{k}\left(\rho \right)-\mathcal{G}\left(\rho \right)]=\mathrm{\Lambda }\left(\rho \right),\mathrm{if}\rho \in \phi ,0<\mathfrak{w}<1,$$

(2)

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\varphi \left(\rho \right),\mathrm{if}\rho \in [-\mathfrak{a},0],r>0,\end{array}$$

(3)

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\tilde{\varphi }\left(\rho \right),\mathrm{if}\rho \in [\xi ,\xi +\mathfrak{g}],\mu >0,\end{array}$$

(4)

The functional $\mathrm{\Lambda }:\phi \to \mathbb{R}$ is continuous, while $\mathcal{G}:\phi \to \mathbb{R}$ is an arbitrary mapping. The functions $\varphi$ and $\tilde{\varphi }$ are continuous over the intervals $[-\mathfrak{a},0]$ and $[\xi ,\xi +\mathfrak{g}]$, respectively, with both taking values in $\mathbb{R}$.

Lemma 6.

For $\mathfrak{w}\in (0,1)$ and a continuous function $\mathrm{\Lambda }:\phi \to \mathbb{R}$, the system given by (2)–(4) possesses a unique solution represented as

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\left\{\begin{array}{cc}\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\mathcal{G}\left(0\right)e^{-{\mathfrak{w}}_1\rho }+\mathcal{G}\left(\rho \right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi ,\hfill & \rho \in \phi ,\hfill \\ \varphi \left(\rho \right),\hfill & \rho \in [-\mathfrak{a},0],\hfill \\ \tilde{\varphi }\left(\rho \right),\hfill & \rho \in [\xi ,\xi +\mathfrak{g}].\hfill \end{array}\right.\end{array}$$

(5)

Proof. 

Suppose that $\mathbb{k}$ satisfies (2)–(4). From Lemma 2, we have

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)-\mathcal{G}\left(\rho \right)-e^{-{\mathfrak{w}}_1\rho }[\varphi \left(0\right)-\mathcal{G}\left(0\right)]={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi .\end{array}$$

Then,

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)-\mathcal{G}\left(\rho \right)=e^{-{\mathfrak{w}}_1\rho }[\varphi \left(0\right)-\mathcal{G}\left(0\right)]+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi .\end{array}$$

Finally, we have

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\mathcal{G}\left(0\right)e^{-{\mathfrak{w}}_1\rho }+\mathcal{G}\left(\rho \right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi .\end{array}$$

Conversely, we can easily show by Definition 3 and Lemmas 1 and 2 that if $\mathbb{k}$ verifies (5), then it satisfies problems (2)–(4).    □

Let

$$\begin{array}{c}\hfill {\mathrm{{\rm Y}}}_{\mathbb{R}}=\{\mathbb{k}:[-\mathfrak{a},\xi +\mathfrak{g}]\to \mathbb{R}:\mathbb{k}{{|}_{[0,\xi ]}\in C(\rho ,\mathbb{R}),\mathbb{k}|}_{[-\mathfrak{a},0]}\in C([-\mathfrak{a},0],\mathbb{R})\\ \hfill {\mathrm{and}\mathbb{k}|}_{[\xi ,\xi +\mathfrak{g}]}\in C([\xi ,\xi +\mathfrak{g}],\mathbb{R})\}\end{array}$$

be a Banach space with the norm

$$\begin{array}{c}\hfill {\parallel \mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}=\underset{\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]}{sup}\left|\mathbb{k}\left(\rho \right)\right|.\end{array}$$

(6)

Definition 5.

We define a solution to problem (1) as a function $\mathbb{k}\in {\mathrm{{\rm Y}}}_{\mathbb{R}}$ fulfilling Equation (1)

Lemma 7.

Let $\mathfrak{F}:\phi \times C\left(\right[-\mathfrak{a},\mathfrak{g}],\mathbb{R})\times \mathbb{R}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and ${\hslash }_1,{\hslash }_2:\phi \times \phi \times \mathbb{R}\to \mathbb{R}$ be continuous functions. Then, the problem (1) is equivalent to the following integral equation:

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\left\{\begin{array}{c}\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1\rho }+\aleph (\rho ,{\mathbb{k}}^{\rho })+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\phi }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)\mathit{if}\rho \in \phi ,\hfill \\ \varphi \left(\rho \right),\mathit{if}\rho \in [-\mathfrak{a},0],\hfill \\ \widehat{\varphi }\left(\rho \right),\mathit{if}\rho \in [T,\xi +\mathfrak{g}],\hfill \end{array}\right.\end{array}$$

(7)

where $\mathrm{\Lambda }\in C(\rho ,\mathbb{R})$ satisfies the following functional equation:

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right).\end{array}$$

(8)

The following conditions are assumed throughout our analysis:

($\mathcal{H}$1)

Continuity is a property held by the functions $\mathfrak{F}$, ℵ, ${\hslash }_1$, and ${\hslash }_2$

($\mathcal{H}$2)

Let $\mu >0$ and $0<\mathcal{A}<1$ be constants

$$\begin{array}{ccc}\hfill |\mathfrak{F}(\rho ,\beta ,\gamma ,{\nu }_1,{\nu }_2)-\mathfrak{F}(\rho ,\overline{\beta },\overline{\gamma },{\overline{\nu }}_1,{\overline{\nu }}_2)|& \le & \mu \parallel \beta -\overline{\beta }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\mathcal{A}|\gamma -\overline{\gamma }|+{\mathcal{A}}_1|{\nu }_1-{\overline{\nu }}_1|\hfill \\ & +& {\mathcal{A}}_2|{\nu }_2-{\overline{\nu }}_2|,\hfill \end{array}$$

for any $\beta ,\overline{\beta }\in C([-\mathfrak{a},\mathfrak{g}],\mathbb{R})$, $\gamma ,\overline{\gamma },{\nu }_1,{\overline{\nu }}_1,{\nu }_2,{\overline{\nu }}_2\in \mathbb{R}$ and $\rho \in \phi$, where ${\mathcal{A}}_1,{\mathcal{A}}_2>0$.

($\mathcal{H}$3)

There exists constant $\lambda >0$ such that

$$\begin{array}{c}\hfill |\aleph (\rho ,\beta )-\aleph (\rho ,\overline{\beta })|\le \lambda \parallel \beta -\overline{\beta }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]},\end{array}$$

for any $\beta ,\overline{\beta }\in C([-\mathfrak{a},\mathfrak{g}],\mathbb{R})$, $\rho \in \phi$.

($\mathcal{H}$4)

There exist constants ${\delta }_1,{\delta }_2>0$ such that for all $\rho ,\ell \in \phi$ and $u,v\in \mathbb{R}$,

$$\begin{array}{cc}\hfill |{\hslash }_1(\rho ,\varpi ,u)-{\hslash }_1(\rho ,\varpi ,v)|& \le {\delta }_1|u-v|,\end{array}$$

(9)

$$\begin{array}{cc}\hfill |{\hslash }_2(\rho ,\varpi ,u)-{\hslash }_2(\rho ,\varpi ,v)|& \le {\delta }_2|u-v|.\end{array}$$

(10)

($\mathcal{H}$5)

There exist functions ${\mathfrak{z}}_1,{\mathfrak{z}}_2\in L^1(\rho ,{\mathbb{R}}_{+})$ such that for all $\rho ,\ell \in \phi$ and $u\in \mathbb{R}$,

$$\begin{array}{cc}\hfill |{\hslash }_1(\rho ,\varpi ,u)|& \le {\mathfrak{z}}_1\left(\varpi \right)(1+|u\left|\right),\end{array}$$

(11)

$$\begin{array}{cc}\hfill |{\hslash }_2(\rho ,\varpi ,u)|& \le {\mathfrak{z}}_2\left(\varpi \right)(1+|u\left|\right).\end{array}$$

(12)

($\mathcal{H}$6)

There exist functions ${\eta }_1,{\eta }_2\in C(\rho ,{\mathbb{R}}_{+})$ such that

$$\begin{array}{cc}\hfill {\int }_0^{\rho }|{\hslash }_1(\rho ,\varpi ,u)-{\hslash }_1(\rho ,\varpi ,v)|d\varpi & \le {\eta }_1\left(\rho \right)|u-v|,\end{array}$$

(13)

$$\begin{array}{cc}\hfill {\int }_0^{\xi }|{\hslash }_2(\rho ,\varpi ,u)-{\hslash }_2(\rho ,\varpi ,v)|d\varpi & \le {\eta }_2\left(\rho \right)|u-v|.\end{array}$$

(14)

Theorem 4.

Provided that assumptions ($\mathcal{H}1$)–($\mathcal{H}6)$ are satisfied, and if

$$\begin{array}{c}\hfill \lambda +{\displaystyle \frac{(\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi ){\xi }^{\mathfrak{w}}}{(1-\mathcal{A})\mathrm{\Gamma }(\mathfrak{w}+1)}}<1,\end{array}$$

(15)

then the implicit fractional integro-differential problem (1) has a unique solution.

Proof. 

Consider the operator $\mathcal{S}:{\mathrm{{\rm Y}}}_{\mathbb{R}}⟶{\mathrm{{\rm Y}}}_{\mathbb{R}}$ defined by

$$\begin{array}{c}\hfill \mathcal{S}\mathbb{k}\left(\rho \right)=\left\{\begin{array}{cc}\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1\rho }+\aleph (\rho ,{\mathbb{k}}^{\rho })+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi ,\hfill & \mathrm{if}\rho \in \phi ,\hfill \\ \varphi \left(\rho \right),\mathrm{if}\rho \in [-\mathfrak{a},0],\hfill \\ \widehat{\varphi }\left(\rho \right),\mathrm{if}\rho \in [\xi ,\xi +\mathfrak{g}].\hfill \end{array}\right.\end{array}$$

where $\mathrm{\Lambda }\left(\rho \right)$ satisfies

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right).\end{array}$$

Obviously, the fixed points of the operator $\mathcal{S}$ are solutions of the problem (1). Let $\mathbb{k},\mathfrak{u}\in {\mathrm{{\rm Y}}}_{\mathbb{R}}$. If $\rho \in [-\mathfrak{a},0]$ or $\rho \in [\xi ,\xi +\mathfrak{g}]$, then

$$\begin{array}{c}\hfill \left|\mathcal{S}\mathbb{k}\right(\rho )-\mathcal{S}\mathfrak{u}(\rho \left)\right|=0.\end{array}$$

If $\rho \in \phi$, we have

$$\begin{array}{c}\hfill |\mathcal{S}\mathbb{k}\left(\rho \right)-\mathcal{S}\mathfrak{u}\left(\rho \right)|\le |\aleph (\rho ,{\mathbb{k}}^{\rho })-\aleph (\rho ,{\mathfrak{u}}^{\rho })|+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}|\mathrm{\Lambda }\left(\varpi \right)-\zeta \left(\varpi \right)|d\varpi ,\end{array}$$

where $\mathrm{\Lambda }$ and $\zeta$ are two functions verifying

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right),\end{array}$$

and

$$\begin{array}{c}\hfill \zeta \left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathfrak{u}}^{\rho },\zeta \left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right))d\mathfrak{s}\right).\end{array}$$

Through conditions ($\mathcal{H}2$) and ($\mathcal{H}4$), we find that

$$\begin{array}{ccc}\hfill \left|\mathrm{\Lambda }\right(\rho )-\zeta (\rho \left)\right|& =& \left|\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\right.\hfill \\ & -& \left.\mathfrak{F}\left(\rho ,{\mathfrak{u}}^{\rho },\zeta \left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\right|\hfill \\ & \le & \mu \parallel {\mathbb{k}}^{\rho }-{\mathfrak{u}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\mathcal{A}|\mathrm{\Lambda }\left(\rho \right)-\zeta \left(\rho \right)|\hfill \\ & +& {\mathcal{A}}_1{\int }_0^{\rho }|{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))-{\hslash }_1(\rho ,\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right))|d\mathfrak{s}\hfill \\ & +& {\mathcal{A}}_2{\int }_0^{\xi }|{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))-{\hslash }_2(\rho ,\mathfrak{s},\mathfrak{u}\left(\mathfrak{s}\right))|d\mathfrak{s}\hfill \\ & \le & \mu \parallel {\mathbb{k}}^{\rho }-{\mathfrak{u}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\mathcal{A}|\mathrm{\Lambda }\left(\rho \right)-\zeta \left(\rho \right)|\hfill \\ & +& {\mathcal{A}}_1{\delta }_1{\int }_0^{\rho }|\mathbb{k}\left(\mathfrak{s}\right)-\mathfrak{u}\left(\mathfrak{s}\right)|d\mathfrak{s}+{\mathcal{A}}_2{\delta }_2{\int }_0^{\xi }|\mathbb{k}\left(\mathfrak{s}\right)-\mathfrak{u}\left(\mathfrak{s}\right)|d\mathfrak{s}\hfill \\ & \le & \mu \parallel {\mathbb{k}}^{\rho }-{\mathfrak{u}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\mathcal{A}|\mathrm{\Lambda }\left(\rho \right)-\zeta \left(\rho \right)|+({\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_{2}T){\parallel \mathbb{k}-\mathfrak{u}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}},\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill |\mathrm{\Lambda }\left(\rho \right)-\zeta \left(\rho \right)|\le {\displaystyle \frac{\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi }{1-\mathcal{A}}}{\parallel \mathbb{k}-\mathfrak{u}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}.\end{array}$$

Then, for each $\rho \in \phi$, we have

$$\begin{array}{ccc}\hfill \left|\mathcal{S}\mathbb{k}\right(\rho )-\mathcal{S}\mathfrak{u}(\rho \left)\right|& \le & \lambda \parallel {\mathbb{k}}^{\rho }-{\mathfrak{u}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}\hfill \\ & +& {\displaystyle \frac{(\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi )}{(1-\mathcal{A})\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}{\parallel {\mathbb{k}}^{\varpi }-{\mathfrak{u}}^{\varpi }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}d\varpi \hfill \\ & \le & \left[\lambda +{\displaystyle \frac{(\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi ){\xi }^{\mathfrak{w}}}{(1-\mathcal{A})\mathrm{\Gamma }(\mathfrak{w}+1)}}\right]{\parallel \mathbb{k}-\mathfrak{u}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill {\parallel \mathcal{S}\mathbb{k}-\mathcal{S}\mathfrak{u}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}\le \left[\lambda +{\displaystyle \frac{(\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi ){\xi }^{\mathfrak{w}}}{(1-\mathcal{A})\mathrm{\Gamma }(\mathfrak{w}+1)}}\right]{\parallel \mathbb{k}-\mathfrak{u}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}.\end{array}$$

According to condition (15), the operator A is a contraction. Consequently, according to the Banach’s contraction principle, the operator A has a unique fixed point which is a solution of the fractional integro-differential problem (1).    □

Remark 1.

Let us put

$$\begin{array}{c}\hfill {\mathfrak{K}}_1\left(\rho \right)=\left|\mathfrak{F}(\rho ,0,0,0,0)\right|,\mu ={\mathfrak{K}}_2^{*},\mathcal{A}={\mathfrak{K}}_3^{*},{\mathcal{A}}_1={\mathfrak{K}}_4^{*},{\mathcal{A}}_2={\mathfrak{K}}_5^{*}.\end{array}$$

Then, condition ($\mathcal{H}2$) implies that

$$\begin{array}{c}\hfill |\mathfrak{F}(\rho ,\beta ,\gamma ,{\nu }_1,{\nu }_2)|\le {\mathfrak{K}}_1\left(\rho \right)+{\mathfrak{K}}_2^{*}{\parallel \beta \parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\mathfrak{K}}_3^{*}\left|\gamma \right|+{\mathfrak{K}}_4^{*}|{\nu }_1|+{\mathfrak{K}}_5^{*}\left|{\nu }_2\right|,\end{array}$$

for $\rho \in \phi$, $\beta \in C\left(\right[-\mathfrak{a},\mathfrak{g}],\mathbb{R})$, $\gamma ,{\nu }_1,{\nu }_2\in \mathbb{R}$ with ${\mathfrak{K}}_1\in C(\rho ,{\mathbb{R}}_{+})$, such that

$$\begin{array}{c}\hfill {\mathfrak{K}}_1^{*}=\underset{\rho \in \phi }{sup}{\mathfrak{K}}_1\left(\rho \right).\end{array}$$

Our second existence result for the problem (1) is based on Schauder’s fixed point theorem.

Theorem 5.

Assume that in addition to ($\mathcal{H}1$)–($\mathcal{H}6$), the following hypotheses hold:

($\mathcal{H}7$)

 For each $\rho \in \phi$ and bounded set $B\in C\left(\right[-\mathfrak{a},\mathfrak{g}],\mathbb{R})$, the set $\{\phi ⟼\aleph (\rho ,{\mathbb{k}}^{\rho }),\mathbb{k}\in \mathfrak{B}\}$ is equicontinuous.

($\mathcal{H}8$)

 There exist two functions ${\mathfrak{N}}_1,{\mathfrak{N}}_2\in C(\rho ,{\mathbb{R}}_{+})$, such that

$$\begin{array}{c}\hfill |\aleph (\rho ,\beta )|\le {\mathfrak{N}}_1\left(\rho \right){\parallel \beta \parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\mathfrak{N}}_2\left(\rho \right),\end{array}$$

for each $\beta \in C\left(\right[-\mathfrak{a},\mathfrak{g}],\mathbb{R})$, where ${\mathfrak{N}}_i^{*}={sup}_{\rho \in \phi }{\mathfrak{N}}_i\left(\rho \right);i=1,2$.

($\mathcal{H}9$)

 There exist functions ${\mathfrak{z}}_1,{\mathfrak{z}}_2\in L^1(\rho ,{\mathbb{R}}_{+})$ such that

$$\begin{array}{cc}\hfill {\int }_0^{\rho }\left|{\hslash }_1(\rho ,\varpi ,u)\right|d\varpi & \le {\mathfrak{z}}_1\left(\rho \right)(1+|u\left|\right),\\ \hfill {\int }_0^{\xi }\left|{\hslash }_2(\rho ,\varpi ,u)\right|d\varpi & \le {\mathfrak{z}}_2\left(\rho \right)(1+|u\left|\right),\end{array}$$

for all $\rho \in \phi$ and $u\in \mathbb{R}$.

If

$$\begin{array}{c}\hfill {\mathfrak{N}}_1^{*}+{\displaystyle \frac{({\mathfrak{K}}_2^{*}+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{L^1}+{\mathfrak{K}}_5^{*}\parallel {\mathfrak{z}}_2{\parallel }_{L^1}){\xi }^{\mathfrak{w}}}{(1-{\mathfrak{K}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}<1,\end{array}$$

then the implicit fractional integro-differential problem (1) has at least one solution.

Proof. 

The proof will be given in several steps.

Step 1: The operator $A:{\mathrm{{\rm Y}}}_{\mathbb{R}}⟶{\mathrm{{\rm Y}}}_{\mathbb{R}}$ is continuous. Let ${\left\{{\mathbb{k}}_{\beta }\right\}}_{\beta \in \mathbb{N}}$ be a sequence such that ${\mathbb{k}}_{\beta }⟶y$ in ${\mathrm{{\rm Y}}}_{\mathbb{R}}$. If $\rho \in [-\mathfrak{a},0]$ or $\rho \in [\xi ,\xi +\mathfrak{g}]$, then

$$\begin{array}{c}\hfill |\mathcal{S}{\mathbb{k}}_{\beta }\left(\rho \right)-\mathcal{S}\mathbb{k}\left(\rho \right)|=0.\end{array}$$

If $\rho \in \phi$, we have

$$\begin{array}{c}\hfill |\mathcal{S}{\mathbb{k}}_{\beta }\left(\rho \right)-\mathcal{S}\mathbb{k}\left(\rho \right)|\le |\aleph (\rho ,{\mathbb{k}}_{\beta }^{\rho })-\aleph (\rho ,{\mathbb{k}}^{\rho })|+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}|{\mathrm{\Lambda }}_{\beta }\left(\varpi \right)-\mathrm{\Lambda }\left(\varpi \right)|d\varpi ,\end{array}$$

where ${\mathrm{\Lambda }}_{\beta }$ and h are two functions satisfying the following functional equations:

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\beta }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}_{\beta }^{\rho },{\mathrm{\Lambda }}_{\beta }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,s,{\mathbb{k}}_{\beta }\left(s\right))ds,{\int }_0^{\xi }{\hslash }_2(\rho ,s,{\mathbb{k}}_{\beta }\left(s\right))ds\right),\end{array}$$

and

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)..\end{array}$$

By ($\mathcal{H}2$) and ($\mathcal{H}4$), we have

$$\begin{array}{ccc}\hfill |{\mathrm{\Lambda }}_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)|& =& |\mathfrak{F}\left(\rho ,{\mathbb{k}}_{\beta }^{\rho },{\mathrm{\Lambda }}_{\beta }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))d\mathfrak{s}\right)\hfill \\ & -& \mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)|\hfill \\ & \le \mu & \parallel {\mathbb{k}}_{\beta }^{\rho }-{\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\mathcal{A}|{\mathrm{\Lambda }}_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)|\hfill \\ & +& {\mathcal{A}}_1{\int }_0^{\rho }|{\hslash }_1(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))-{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))|d\mathfrak{s}\hfill \\ & +& {\mathcal{A}}_2{\int }_0^{\xi }|{\hslash }_2(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))-{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))|d\mathfrak{s}\hfill \\ & \le & \mu \parallel {\mathbb{k}}_{\beta }^{\rho }-{\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\mathcal{A}|{\mathrm{\Lambda }}_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)|\hfill \\ & +& {\mathcal{A}}_1{\delta }_1{\int }_0^{\rho }|{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right)-\mathbb{k}\left(\mathfrak{s}\right)|d\mathfrak{s}+{\mathcal{A}}_2{\delta }_2{\int }_0^{\xi }|{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right)-\mathbb{k}\left(\mathfrak{s}\right)|d\mathfrak{s}.\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill |{\mathrm{\Lambda }}_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)|\le {\displaystyle \frac{\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi }{1-\mathcal{A}}}{\parallel {\mathbb{k}}_{\beta }-\mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}.\end{array}$$

Thus,

$$\begin{array}{ccc}\hfill |\mathcal{S}{\mathbb{k}}_{\beta }\left(\rho \right)-\mathcal{S}\mathbb{k}\left(\rho \right)|& \le & |\aleph (\rho ,{\mathbb{k}}_{\beta }^{\rho })-\aleph (\rho ,{\mathbb{k}}^{\rho })|\hfill \\ & +& {\displaystyle \frac{(\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi )}{(1-\mathcal{A})\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}{\parallel {\mathbb{k}}_{\beta }^{\varpi }-{\mathbb{k}}^{\varpi }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}d\varpi .\hfill \end{array}$$

An application of the Lebesgue Dominated Convergence Theorem directly yields the result

$$\begin{array}{c}\hfill |\mathcal{S}{\mathbb{k}}_{\beta }\left(\rho \right)-\mathcal{S}\mathbb{k}\left(\rho \right)|⟶0\mathrm{as}\beta ⟶\infty ,\end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel \mathcal{S}{\mathbb{k}}_{\beta }{-A\mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}⟶0\mathrm{as}\beta ⟶\infty .\end{array}$$

Consequently, operator $\mathcal{S}$ satisfies the conditions for continuity. To proceed, we choose a positive constant R for which

$$\begin{array}{c}\hfill \mathcal{Z}\ge max\left\{{\displaystyle \frac{\left|\varphi \left(0\right)\right|+2{\mathfrak{N}}_2^{*}+\frac{({\mathfrak{K}}_1^{*}+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{L^1}+{\mathfrak{K}}_5^{*}\parallel {\mathfrak{z}}_2{\parallel }_{L^1}){\xi }^{\mathfrak{w}}}{(1-{\mathfrak{K}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}{1-{\mathfrak{N}}_1^{*}-\frac{({\mathfrak{K}}_2^{*}+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{L^1}+{\mathfrak{K}}_5^{*}\parallel {\mathfrak{z}}_2{\parallel }_{L^1}){\xi }^{\mathfrak{w}}}{(1-{\mathfrak{K}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}}{,\parallel \varphi \parallel }_{[-\mathfrak{a},0]},{\parallel \widehat{\varphi }\parallel }_{[\xi ,\xi +\mathfrak{g}]}\right\}.\end{array}$$

Define the ball

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\mathcal{Z}}=\{\mathbb{k}\in {\mathrm{{\rm Y}}}_{\mathbb{R}}{:\parallel \mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}\le \mathcal{Z}\}.\end{array}$$

It is clear that ${\mathcal{Q}}_{\mathcal{Z}}$ is a bounded, closed and convex subset of ${\mathrm{{\rm Y}}}_{\mathbb{R}}$.

Step 2: $\mathcal{S}\left({\mathcal{Q}}_{\mathcal{Z}}\right)\subseteq {\mathcal{Q}}_{\mathcal{Z}}$. Let $\mathbb{k}\in {\mathcal{Q}}_{\mathcal{Z}}$. If $\rho \in [-\mathfrak{a},0]$, then

$$\begin{array}{c}\hfill \left|\mathcal{S}\mathbb{k}\left(\rho \right)\right|\le {\parallel \varphi \parallel }_{[-\mathfrak{a},0]}\le \mathcal{Z},\end{array}$$

and if $\rho \in [T,\xi +\mathfrak{g}]$, then

$$\begin{array}{c}\hfill \left|\mathcal{S}\mathbb{k}\left(\rho \right)\right|\le \parallel \widehat{\varphi }{\parallel }_{[T,\xi +\mathfrak{g}]}\le \mathcal{Z}.\end{array}$$

For each $\rho \in \phi$, we have

$$\begin{array}{ccc}\hfill \left|\mathcal{S}\mathbb{k}\right(\rho \left)\right|& \le & \left|\varphi \left(0\right)\right|e^{-{\mathfrak{w}}_1\rho }+|\aleph (0,\varphi \left(0\right))|e^{-{\mathfrak{w}}_1\rho }+|\aleph (\rho ,{\mathbb{k}}^{\rho })|\hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\phi }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\left|\mathrm{\Lambda }\left(\varpi \right)\right|d\varpi .\hfill \end{array}$$

From hypothesis ($\mathcal{H}2$) and ($\mathcal{H}9$), we have

$$\begin{array}{ccc}\hfill \left|\mathrm{\Lambda }\right(\rho \left)\right|& =& \left|\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\right|\hfill \\ & \le & {\mathfrak{K}}_1\left(\rho \right)+{\mathfrak{K}}_2^{*}\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\mathfrak{K}}_3^{*}\left|\mathrm{\Lambda }\left(\rho \right)\right|+{\mathfrak{K}}_4^{*}{\int }_0^{\rho }|{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))|d\mathfrak{s}+{\mathfrak{K}}_5^{*}{\int }_0^{\xi }\left|{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))\right|d\mathfrak{s}\hfill \\ & \le & {\mathfrak{K}}_1^{*}+{\mathfrak{K}}_2^{*}{\parallel \mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}+{\mathfrak{K}}_3^{*}\left|\mathrm{\Lambda }\left(\rho \right)\right|+{\mathfrak{K}}_4^{*}{\mathfrak{z}}_1{\left(\rho \right)(1+\parallel \mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}})+{\mathfrak{K}}_5^{*}{\mathfrak{z}}_2{\left(\rho \right)(1+\parallel \mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}})\hfill \\ & \le & {\mathfrak{K}}_1^{*}+{\mathfrak{K}}_2^{*}\mathcal{Z}+{\mathfrak{K}}_3^{*}\left|\mathrm{\Lambda }\left(\rho \right)\right|+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{\infty }(1+\mathcal{Z})+{\mathfrak{K}}_5^{*}{\parallel {\mathfrak{z}}_2\parallel }_{\infty }(1+\mathcal{Z}),\hfill \end{array}$$

where $\parallel {\mathfrak{z}}_i{\parallel }_{\infty }={sup}_{\rho \in \phi }{\mathfrak{z}}_i\left(\rho \right)$ for $i=1,2$. Then,

$$\begin{array}{c}\hfill \left|\mathrm{\Lambda }\left(\rho \right)\right|\le {\displaystyle \frac{{\mathfrak{K}}_1^{*}+{\mathfrak{K}}_2^{*}\mathcal{Z}+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{\infty }(1+\mathcal{Z})+{\mathfrak{K}}_5^{*}{\parallel {\mathfrak{z}}_2\parallel }_{\infty }(1+\mathcal{Z})}{1-{\mathfrak{K}}_3^{*}}}.\end{array}$$

Finally, we get

$$\begin{array}{c}\hfill \left|\mathcal{S}\mathbb{k}\left(\rho \right)\right|\le \left|\varphi \left(0\right)\right|+2{\mathfrak{N}}_2^{*}+{\mathfrak{N}}_1^{*}\mathcal{Z}+{\displaystyle \frac{[{\mathfrak{K}}_1^{*}+{\mathfrak{K}}_2^{*}\mathcal{Z}+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{\infty }(1+\mathcal{Z})+{\mathfrak{K}}_5^{*}\parallel {\mathfrak{z}}_2{\parallel }_{\infty }(1+\mathcal{Z})]{\xi }^{\mathfrak{w}}}{(1-{\mathfrak{K}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}\le \mathcal{Z}.\end{array}$$

Thus, for each $\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]$,

$$\begin{array}{c}\hfill \left|\mathcal{S}\mathbb{k}\right(\rho \left)\right|\le \mathcal{Z},\end{array}$$

which implies that

$$\begin{array}{c}\hfill {\parallel \mathcal{S}\mathbb{k}\parallel }_{{\mathrm{{\rm Y}}}_{\mathbb{R}}}\le \mathcal{Z}.\end{array}$$

Consequently, $\mathcal{S}\left({\mathcal{Q}}_{\mathcal{Z}}\right)\subset {\mathcal{Q}}_{\mathcal{Z}}$.

Step 3: $\mathcal{S}\left({\mathcal{Q}}_{\mathcal{Z}}\right)$ is equicontinuous.

Let ${\rho }_1,{\rho }_2\in \phi$, where ${\rho }_1<{\rho }_2$ and $\mathbb{k}\in {\mathcal{Q}}_{\mathcal{Z}}$. Then,

$$\begin{array}{ccc}\hfill |\mathcal{S}\mathbb{k}\left({\rho }_2\right)-\mathcal{S}\mathbb{k}\left({\rho }_1\right)|& =& |\varphi \left(0\right)e^{-{\mathfrak{w}}_1{\rho }_2}-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1{\rho }_2}+\aleph ({\rho }_2,{\mathbb{k}}^{{\rho }_2})\hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{{\rho }_2}{e}^{-{\mathfrak{w}}_1({\rho }_2-\varpi )}{({\rho }_2-\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi \hfill \\ & -& \varphi \left(0\right)e^{-{\mathfrak{w}}_1{\rho }_1}+\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1{\rho }_1}-\aleph ({\rho }_1,{\mathbb{k}}^{{\rho }_1})\hfill \\ & -& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{{\rho }_1}{e}^{-{\mathfrak{w}}_1({\rho }_1-\varpi )}{({\rho }_1-\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi |\hfill \\ & \le & \left|\varphi \left(0\right)\right||e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}|+|\aleph (0,\varphi \left(0\right))\left|\right|e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}|\hfill \\ & +& |\aleph ({\rho }_2,{\mathbb{k}}^{{\rho }_2})-\aleph ({\rho }_1,{\mathbb{k}}^{{\rho }_1})|\hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{{\rho }_1}[e^{-{\mathfrak{w}}_1({\rho }_2-\varpi )}{({\rho }_2-\varpi )}^{\mathfrak{w}-1}-e^{-{\mathfrak{w}}_1({\rho }_1-\varpi )}{({\rho }_1-\varpi )}^{\mathfrak{w}-1}]\left|\mathrm{\Lambda }\left(\varpi \right)\right|d\varpi \hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_{{\rho }_1}^{{\rho }_2}{e}^{-{\mathfrak{w}}_1({\rho }_2-\varpi )}{({\rho }_2-\varpi )}^{\mathfrak{w}-1}\left|\mathrm{\Lambda }\left(\varpi \right)\right|d\varpi \hfill \\ & \le & \left|\varphi \left(0\right)\right||e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}|+|\aleph (0,\varphi \left(0\right))\left|\right|e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}|\hfill \\ & +& |\aleph ({\rho }_2,{\mathbb{k}}^{{\rho }_2})-\aleph ({\rho }_1,{\mathbb{k}}^{{\rho }_1})|\hfill \\ & +& {\displaystyle \frac{[{\mathfrak{K}}_1^{*}+{\mathfrak{K}}_2^{*}\mathcal{Z}+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{\infty }(1+\mathcal{Z})+{\mathfrak{K}}_5^{*}\parallel {\mathfrak{z}}_2{\parallel }_{\infty }(1+\mathcal{Z})]({\rho }_2^{\mathfrak{w}}-{\rho }_1^{\mathfrak{w}})}{(1-{\mathfrak{K}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}\hfill \\ & +& {\displaystyle \frac{[{\mathfrak{K}}_1^{*}+{\mathfrak{K}}_2^{*}\mathcal{Z}+{\mathfrak{K}}_4^{*}\parallel {\mathfrak{z}}_1{\parallel }_{\infty }(1+\mathcal{Z})+{\mathfrak{K}}_5^{*}\parallel {\mathfrak{z}}_2{\parallel }_{\infty }(1+\mathcal{Z})]{({\rho }_2-{\rho }_1)}^{\mathfrak{w}}}{(1-{\mathfrak{K}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}\hfill \\ & & ⟶0\mathrm{as}{\rho }_1⟶{\rho }_2\hfill \end{array}$$

Hence, the operator $\mathcal{S}$ is equicontinuous. Therefore, by applying Schauder’s fixed point theorem, we conclude that the implicit fractional integro-differential problem (1) admits at least one solution.    □

An Application

Consider the following implicit fractional integro-differential problem:

$$\begin{array}{ccc}\hfill & & {}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\left[\mathbb{k}\left(\rho \right)-\aleph (\rho ,{\mathbb{k}}^{\rho })\right]=\hfill \\ & & \mathfrak{F}(\rho ,{\mathbb{k}}^{\rho },{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^1{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ),\rho \in [0,1],\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\varphi \left(\rho \right),\rho \in [-1,0],\end{array}$$

(17)

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\widehat{\varphi }\left(\rho \right),\rho \in [1,2],\end{array}$$

(18)

where $\varphi \in C\left(\right[-1,0],\mathbb{R})$ and $\widehat{\varphi }\in C([1,2],\mathbb{R})$. Set

$$\begin{array}{c}\hfill \mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\mathbb{k}\left(\rho \right),{\nu }_1,{\nu }_2\right)={\displaystyle \frac{ln(\rho +1)+2\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\frac{2}{3}\left|{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\mathbb{k}\left(\rho \right)\right|+\frac{1}{5}{\nu }_1+\frac{1}{6}{\nu }_2}{\left(15+e^{2\phi }\right)\left(1+\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\left|{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\mathbb{k}\left(\rho \right)\right|+|{\nu }_1|+|{\nu }_2|\right)}},\end{array}$$

$$\begin{array}{c}\hfill \aleph (\rho ,{\mathbb{k}}^{\rho })={\displaystyle \frac{\phi +\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}}{100}},\end{array}$$

and define the integral kernels

$$\begin{array}{c}\hfill {\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))={\displaystyle \frac{sin\left(\rho \varpi \right)}{10(1+{\ell }^2)}}\mathbb{k}\left(\varpi \right),{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))={\displaystyle \frac{e^{-\phi \varpi }}{20(1+\ell )}}\mathbb{k}\left(\varpi \right).\end{array}$$

Clearly, the functions $\mathfrak{F}$, ℵ, ${\hslash }_1$, and ${\hslash }_2$ are continuous, and hypothesis ($\mathcal{H}1$) is satisfied.

For any $\beta ,\overline{\beta }\in C([-\mathfrak{a},\mathfrak{g}],\mathbb{R})$, $\gamma ,\overline{\gamma },{\nu }_1,{\overline{\nu }}_1,{\nu }_2,{\overline{\nu }}_2\in \mathbb{R}$ and $\rho \in [0,1]$, we have

$$\begin{array}{cc}& \left|\mathfrak{F}(\rho ,\beta ,\gamma ,{\nu }_1,{\nu }_2)-\mathfrak{F}(\rho ,\overline{\beta },\overline{\gamma },{\overline{\nu }}_1,{\overline{\nu }}_2)\right|\\ & \le {\displaystyle \frac{1}{15}}\left[2\parallel \beta -\overline{\beta }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\displaystyle \frac{2}{3}}|\gamma -\overline{\gamma }|+{\displaystyle \frac{1}{5}}|{\nu }_1-{\overline{\nu }}_1|+{\displaystyle \frac{1}{6}}|{\nu }_2-{\overline{\nu }}_2|\right].\end{array}$$

Then, hypothesis ($\mathcal{H}2$) is satisfied with $\mu ={\displaystyle \frac{2}{15}}$, $L={\displaystyle \frac{2}{45}}$, ${\mathcal{A}}_1={\displaystyle \frac{1}{75}}$, and ${\mathcal{A}}_2={\displaystyle \frac{1}{90}}$.

Also, we have

$$\begin{array}{c}\hfill \left|\aleph (\rho ,\beta )-\aleph (\rho ,\overline{\beta })\right|\le {\displaystyle \frac{1}{100}}{\parallel \beta -\overline{\beta }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}.\end{array}$$

So, condition ($\mathcal{H}3$) is satisfied with $C={\displaystyle \frac{1}{100}}$.

For the integral kernels, we have the Lipschitz conditions:

$$\begin{array}{cc}\hfill |{\hslash }_1(\rho ,\varpi ,u)-{\hslash }_1(\rho ,\varpi ,v)|& \le {\displaystyle \frac{1}{10(1+{\ell }^2)}}|u-v|\le {\displaystyle \frac{1}{10}}|u-v|,\\ \hfill |{\hslash }_2(\rho ,\varpi ,u)-{\hslash }_2(\rho ,\varpi ,v)|& \le {\displaystyle \frac{1}{20(1+\ell )}}|u-v|\le {\displaystyle \frac{1}{20}}|u-v|.\end{array}$$

Thus, ($\mathcal{H}4$) is satisfied with ${\delta }_1={\displaystyle \frac{1}{10}}$ and ${\delta }_2={\displaystyle \frac{1}{20}}$.

For the growth conditions, we have

$$\begin{array}{cc}\hfill |{\hslash }_1(\rho ,\varpi ,u)|& \le {\displaystyle \frac{1}{10(1+{\ell }^2)}}(1+|u\left|\right),\\ \hfill |{\hslash }_2(\rho ,\varpi ,u)|& \le {\displaystyle \frac{1}{20(1+\ell )}}(1+|u\left|\right),\end{array}$$

so ($\mathcal{H}5$) is satisfied with ${\mathfrak{z}}_1\left(\varpi \right)={\displaystyle \frac{1}{10(1+{\ell }^2)}}$ and ${\mathfrak{z}}_2\left(\varpi \right)={\displaystyle \frac{1}{20(1+\ell )}}$.

For the integral Lipschitz conditions,

$$\begin{array}{cc}\hfill {\int }_0^{\rho }|{\hslash }_1(\rho ,\varpi ,u)-{\hslash }_1(\rho ,\varpi ,v)|d\varpi & \le {\displaystyle \frac{\pi }{20}}|u-v|,\\ \hfill {\int }_0^1|{\hslash }_2(\rho ,\varpi ,u)-{\hslash }_2(\rho ,\varpi ,v)|d\varpi & \le {\displaystyle \frac{ln2}{20}}|u-v|,\end{array}$$

so ($\mathcal{H}6$) is satisfied with ${\eta }_1\left(\rho \right)={\displaystyle \frac{\pi }{20}}$ and ${\eta }_2\left(\rho \right)={\displaystyle \frac{ln2}{20}}$.

Now we compute the contraction condition from Theorem 4:

$$\begin{array}{cc}& \lambda +{\displaystyle \frac{(\mu +{\mathcal{A}}_1{\delta }_1\xi +{\mathcal{A}}_2{\delta }_2\xi ){\xi }^{\mathfrak{w}}}{(1-\mathcal{A})\mathrm{\Gamma }(\mathfrak{w}+1)}}\\ & ={\displaystyle \frac{1}{100}}+{\displaystyle \frac{\left(\frac{2}{15}+\frac{1}{75}·\frac{1}{10}·1+\frac{1}{90}·\frac{1}{20}·1\right)·1^{\frac{1}{3}}}{(1-\frac{2}{45})\mathrm{\Gamma }\left(\frac{4}{3}\right)}}\\ & ={\displaystyle \frac{1}{100}}+{\displaystyle \frac{\left(\frac{2}{15}+\frac{1}{750}+\frac{1}{1800}\right)}{(1-\frac{2}{45})\mathrm{\Gamma }\left(\frac{4}{3}\right)}}\\ & \approx {\displaystyle \frac{1}{100}}+{\displaystyle \frac{0.133333+0.001333+0.000556}{0.955556·0.892979}}\\ & \approx 0.01+{\displaystyle \frac{0.135222}{0.853181}}\\ & \approx 0.01+0.1585\\ & \approx 0.1685<1.\end{array}$$

Since the condition of Theorem 4 is verified, the implicit fractional integro-differential problem (16)–(18) has a unique solution.

3.2. Implicit Neutral Fractional Integro-Differential Problems with Retarded and Advanced Arguments in Banach Spaces

In this section, we focus on establishing the existence and stability results for a class of implicit neutral fractional integro-differential systems, formulated in a Banach space setting. The considered model extends the structure of problem (1) by incorporating both delayed and advanced arguments within the fractional framework. The mathematical formulation of this problem is described as follows:

$$\left\{\begin{array}{c}{}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\left[\mathbb{k}\left(\rho \right)-\aleph (\rho ,{\mathbb{k}}^{\rho })\right]=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^{\xi }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right),\hfill \\ \rho \in \phi :=[0,\xi ],\hfill \\ \mathbb{k}\left(\rho \right)=\varphi \left(\rho \right),\rho \in [-\mathfrak{a},0],\hfill \\ \mathbb{k}\left(\rho \right)=\widehat{\varphi }\left(\rho \right),\rho \in [\xi ,\xi +\mathfrak{g}].\hfill \end{array}\right.$$

(19)

where ${}_0{}^{C}D_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}$ is the Caputo-tempered fractional derivative of order $\mathfrak{w}\in (0,1)$, ${\mathfrak{w}}_1\ge 0$, $\mathfrak{a},\mathfrak{g}>0$, $\mathfrak{F}:\phi \times C\left(\right[-\mathfrak{a},\mathfrak{g}],E)\times E\times E\times E\to E$, $\aleph :\phi \times C\left(\right[-\mathfrak{a},\mathfrak{g}],E)\to E$, ${\hslash }_1,{\hslash }_2:\phi \times \phi \times E\to E$ are given functions, $\varphi \in C\left(\right[-\mathfrak{a},0],E)$, and $\widehat{\varphi }\in C([\xi ,\xi +\mathfrak{g}],E)$. We denote by ${\mathbb{k}}^{\rho }$ the element of $C\left(\right[-\mathfrak{a},\mathfrak{g}],E)$ defined by

$$\begin{array}{c}\hfill {\mathbb{k}}^{\rho }=\mathbb{k}(\rho +\ell ):\ell \in [-\mathfrak{a},\mathfrak{g}].\end{array}$$

Lemma 8.

Let $\mathfrak{F}:\phi \times C\left(\right[-\mathfrak{a},\mathfrak{g}],E)\times E\times E\times E\to E$ and ${\hslash }_1,{\hslash }_2:\phi \times \phi \times E\to E$ be continuous functions. Then, problem (19) is equivalent to the following integral equation:

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\left\{\begin{array}{c}\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1\rho }+\aleph (\rho ,{\mathbb{k}}^{\rho })+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right),if\rho \in \phi ,\hfill \\ \varphi \left(\rho \right),if\rho \in [-\mathfrak{a},0],\hfill \\ \widehat{\varphi }\left(\rho \right),if\rho \in [\xi ,\xi +\mathfrak{g}],\hfill \end{array}\right.\end{array}$$

where $h\in C(\rho ,E)$ satisfies the following functional equation

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right).\end{array}$$

Let us input the following conditions:

($\mathcal{H}7$)

 The functions $\mathfrak{F}$, ℵ, ${\hslash }_1$, and ${\hslash }_2$ are continuous.

($\mathcal{H}8$)

There exist constants $\widehat{\mu }>0$ and $0<\widehat{\mathcal{A}}<1$ such that

$$\begin{array}{ccc}\hfill \parallel \mathfrak{F}(\rho ,\beta ,\gamma ,{\nu }_1,{\nu }_2)-\mathfrak{F}(\rho ,\overline{\beta },\overline{\gamma },{\overline{\nu }}_1,{\overline{\nu }}_2)\parallel & \le & \widehat{\mu }\parallel \beta -\overline{\beta }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\widehat{\mathcal{A}}\parallel \gamma -\overline{\gamma }\parallel +{\widehat{\mathcal{A}}}_1\parallel {\nu }_1-{\overline{\nu }}_1\parallel \hfill \\ & +& {\widehat{\mathcal{A}}}_2\parallel {\nu }_2-{\overline{\nu }}_2\parallel ,\hfill \end{array}$$

for any $\beta ,\overline{\beta }\in C([-\mathfrak{a},\mathfrak{g}],E)$, $\gamma ,\overline{\gamma },{\nu }_1,{\overline{\nu }}_1,{\nu }_2,{\overline{\nu }}_2\in E$ and $\rho \in \phi$, where ${\widehat{\mathcal{A}}}_1,{\widehat{\mathcal{A}}}_2>0$.

($\mathcal{H}9$)

 There exists constant $\widehat{\lambda }>0$ such that

$$\begin{array}{c}\hfill \parallel \aleph (\rho ,\beta )-\aleph (\rho ,\overline{\beta })\parallel \le \widehat{\lambda }{\parallel \beta -\overline{\beta }\parallel }_{[-\mathfrak{a},\mathfrak{g}]},\end{array}$$

for any $\beta ,\overline{\beta }\in C([-\mathfrak{a},\mathfrak{g}],E)$, $\rho \in \phi$.

($\mathcal{H}10$)

There exist constants ${\widehat{\delta }}_1,{\widehat{\delta }}_2>0$ such that for all $\rho ,\ell \in \phi$ and $u,v\in E$,

$$\begin{array}{cc}\hfill \parallel {\hslash }_1(\rho ,\varpi ,u)-{\hslash }_1(\rho ,\varpi ,v)\parallel & \le {\widehat{\delta }}_1\parallel u-v\parallel ,\\ \hfill \parallel {\hslash }_2(\rho ,\varpi ,u)-{\hslash }_2(\rho ,\varpi ,v)\parallel & \le {\widehat{\delta }}_2\parallel u-v\parallel .\end{array}$$

($\mathcal{H}1$1)

  There exist functions ${\widehat{\mathfrak{z}}}_1,{\widehat{\mathfrak{z}}}_2\in L^1(\rho ,{\mathbb{R}}_{+})$ such that for all $\rho ,\ell \in \phi$ and $u\in E$,

$$\begin{array}{cc}\hfill \parallel {\hslash }_1(\rho ,\varpi ,u)\parallel & \le {\widehat{\mathfrak{z}}}_1\left(\varpi \right)(1+\parallel u\parallel ),\\ \hfill \parallel {\hslash }_2(\rho ,\varpi ,u)\parallel & \le {\widehat{\mathfrak{z}}}_2\left(\varpi \right)(1+\parallel u\parallel ).\end{array}$$

($\mathcal{H}12$)

  For each $\rho \in \phi$ and bounded set $B\in C\left(\right[-\mathfrak{a},\mathfrak{g}],E)$, the set $\{\phi ⟼\aleph (\rho ,{\mathbb{k}}^{\rho }),\mathbb{k}\in \mathfrak{B}\}$ is equicontinuous.

($\mathcal{H}1$3)

  For each $\rho \in \phi$ and bounded sets $B_1\subseteq C([-\mathfrak{a},\mathfrak{g}],E)$, $B_2\subseteq E$, $B_3,B_4\subseteq E$, we have

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathfrak{F}(\rho ,B_1,B_2,B_3,B_4)\right)\le \widehat{\mu }\underset{\ell \in [-\mathfrak{a},\mathfrak{g}]}{sup}\mathcal{E}\left({\mathfrak{B}}_1\left(\varpi \right)\right)+\widehat{\mathcal{A}}\mathcal{E}\left({\mathfrak{B}}_2\right)+{\widehat{\mathcal{A}}}_1\mathcal{E}\left({\mathfrak{B}}_3\right)+{\widehat{\mathcal{A}}}_2\mathcal{E}\left({\mathfrak{B}}_4\right).\end{array}$$

($\mathcal{H}1$4)

  For each $\rho \in \phi$ and bounded set $B_1\subseteq C([-\mathfrak{a},\mathfrak{g}],E)$, we have

$$\begin{array}{c}\hfill \mathcal{E}(\aleph (\rho ,B_1))\le \widehat{\lambda }\underset{\ell \in [-\mathfrak{a},\mathfrak{g}]}{sup}\mathcal{E}\left({\mathfrak{B}}_1\left(\varpi \right)\right).\end{array}$$

Remark 2

([22]). We remark that conditions $\left(\mathcal{H}8\right)$ and $\left(\mathcal{H}13\right)$ are mutually equivalent. Similarly, the equivalence also holds for $\left(\mathcal{H}9\right)$ and $\left(\mathcal{H}14\right)$.

Remark 3.

Let us put

$$\begin{array}{c}\hfill {\widehat{\mathfrak{K}}}_1\left(\rho \right)=\parallel \mathfrak{F}(\rho ,0,0,0,0)\parallel ,\widehat{\mu }={\widehat{\mathfrak{K}}}_2^{*},\widehat{\mathcal{A}}={\widehat{\mathfrak{K}}}_3^{*},{\widehat{\mathcal{A}}}_1={\widehat{\mathfrak{K}}}_4^{*},{\widehat{\mathcal{A}}}_2={\widehat{\mathfrak{K}}}_5^{*},\widehat{\lambda }={\widehat{\mathfrak{N}}}_1^{*},{\widehat{\mathfrak{N}}}_2\left(\rho \right)=\parallel \aleph (\rho ,0)\parallel .\end{array}$$

Then, condition ($\mathcal{H}7$) implies that

$$\begin{array}{c}\hfill \parallel \mathfrak{F}(\rho ,\beta ,\gamma ,{\nu }_1,{\nu }_2)\parallel \le {\widehat{\mathfrak{K}}}_1\left(\rho \right)+{\widehat{\mathfrak{K}}}_2^{*}{\parallel \beta \parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\widehat{\mathfrak{K}}}_3^{*}\parallel \gamma \parallel +{\widehat{\mathfrak{K}}}_4^{*}\parallel {\nu }_1\parallel +{\widehat{\mathfrak{K}}}_5^{*}\parallel {\nu }_2\parallel ,\end{array}$$

for $\rho \in \phi$, $\beta \in C\left(\right[-\mathfrak{a},\mathfrak{g}],E)$, $\gamma ,{\nu }_1,{\nu }_2\in E$ with ${\widehat{\mathfrak{K}}}_1\in C(\rho ,{\mathbb{R}}_{+})$, such that

$$\begin{array}{c}\hfill {\widehat{\mathfrak{K}}}_1^{*}=\underset{\rho \in \phi }{sup}{\widehat{\mathfrak{K}}}_1\left(\rho \right).\end{array}$$

And from hypothesis ($\mathcal{H}9$), we have

$$\begin{array}{c}\hfill \parallel \aleph (\rho ,\beta )\parallel \le {\widehat{\mathfrak{N}}}_1^{*}{\parallel \beta \parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\widehat{\mathfrak{N}}}_2\left(\rho \right),\end{array}$$

for each $\beta \in C\left(\right[-\mathfrak{a},\mathfrak{g}],E)$ with ${\widehat{\mathfrak{N}}}_2\in C(\rho ,{\mathbb{R}}_{+})$ such that

$$\begin{array}{c}\hfill {\widehat{\mathfrak{N}}}_2^{*}=\underset{\rho \in \phi }{sup}{\widehat{\mathfrak{N}}}_2\left(\rho \right).\end{array}$$

Similarly, from ($\mathcal{H}11$) we have

$$\begin{array}{cc}\hfill {\int }_0^{\rho }\parallel {\hslash }_1(\rho ,\varpi ,u)\parallel d\varpi & \le \parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}(1+\parallel u\parallel ),\\ \hfill {\int }_0^{\xi }\parallel {\hslash }_2(\rho ,\varpi ,u)\parallel d\varpi & \le \parallel {\widehat{\mathfrak{z}}}_2{\parallel }_{L^1}(1+\parallel u\parallel ),\end{array}$$

where $\parallel {\widehat{\mathfrak{z}}}_i{\parallel }_{L^1}={\int }_0^{\xi }{\widehat{\mathfrak{z}}}_i\left(\varpi \right)d\varpi$ for $i=1,2$.

Theorem 6.

Suppose that hypotheses $\left(\mathcal{H}7\right)$–$\left(\mathcal{H}12\right)$ are satisfied. Moreover, assume that

$$\begin{array}{c}\hfill {\widehat{\mathfrak{N}}}_1^{*}+{\displaystyle \frac{\left({\widehat{\mathfrak{K}}}_2^{*}+{\widehat{\mathfrak{K}}}_4^{*}{\widehat{\delta }}_1\xi +{\widehat{\mathfrak{K}}}_5^{*}{\widehat{\delta }}_2\xi \right){\xi }^{\mathfrak{w}}}{\left(1-{\widehat{\mathfrak{K}}}_3^{*}\right)\mathrm{\Gamma }(\mathfrak{w}+1)}}<1.\end{array}$$

Under these conditions, the implicit fractional Volterra–Fredholm integro-differential Equation (19) possesses at least one solution.

To establish the existence of a solution for the implicit fractional integro-differential equation in (19), our approach relies on the framework of measures of noncompactness together with the application of Darbo’s fixed point principle.

Proof. 

To reformulate problem (19) as a fixed point equation, we define the operator $S:\mathrm{{\rm Y}}\to \mathrm{{\rm Y}}$ by

$$\begin{array}{c}\hfill S\mathbb{k}\left(\rho \right)=\left\{\begin{array}{cc}\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1\rho }+\aleph (\rho ,{\mathbb{k}}^{\rho })+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi ,\hfill & \mathrm{if}\rho \in \phi ,\hfill \\ \varphi \left(\rho \right),\mathrm{if}\rho \in [-\mathfrak{a},0],\hfill \\ \widehat{\varphi }\left(\rho \right),\mathrm{if}\rho \in [\xi ,\xi +\mathfrak{g}].\hfill \end{array}\right.\end{array}$$

where $\mathrm{\Lambda }\left(\rho \right)$ satisfies the functional equation

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right).\end{array}$$

The proof will be given in several steps.

Step 1: The operator $S:\mathrm{{\rm Y}}⟶\mathrm{{\rm Y}}$ is continuous.

Let ${\left\{{\mathbb{k}}_{\beta }\right\}}_{\beta \in \mathbb{N}}$ be a sequence such that ${\mathbb{k}}_{\beta }⟶y$ in $\mathrm{{\rm Y}}$. If $\rho \in [-\mathfrak{a},0]$ or $\rho \in [\xi ,\xi +\mathfrak{g}]$, then

$$\begin{array}{c}\hfill \parallel S{\mathbb{k}}_{\beta }\left(\rho \right)-S\mathbb{k}\left(\rho \right)\parallel =0.\end{array}$$

If $\rho \in \phi$, we have

$$\begin{array}{ccc}\hfill \parallel S{\mathbb{k}}_{\beta }\left(\rho \right)-S\mathbb{k}\left(\rho \right)\parallel & \le & \parallel \aleph (\rho ,{\mathbb{k}}_{\beta }^{\rho })-\aleph (\rho ,{\mathbb{k}}^{\rho })\parallel \hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\parallel {\mathrm{\Lambda }}_{\beta }\left(\varpi \right)-h\left(\varpi \right)\parallel d\varpi ,\hfill \end{array}$$

Two distinct functions, denoted as ${\mathrm{\Lambda }}_{\beta }$ and h, are defined by their respective adherence to the subsequent pair of functional equations:

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\beta }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}_{\beta }^{\rho },{\mathrm{\Lambda }}_{\beta }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))d\mathfrak{s}\right),\end{array}$$

and

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right).\end{array}$$

From hypothesis ($\mathcal{H}8$), we have

$$\begin{array}{ccc}\hfill \parallel {\mathrm{\Lambda }}_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)\parallel & =& \parallel \mathfrak{F}\left(\rho ,{\mathbb{k}}_{\beta }^{\rho },{\hslash }_{\beta }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))d\mathfrak{s}\right)\hfill \\ & & -\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\parallel \hfill \\ & \le & \widehat{\mu }\parallel {\mathbb{k}}_{\beta }^{\rho }-{\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\widehat{\mathcal{A}}\parallel {\hslash }_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)\parallel \hfill \\ & +& {\widehat{\mathcal{A}}}_1{\int }_0^{\rho }\parallel {\hslash }_1(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))-{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))\parallel d\mathfrak{s}\hfill \\ & +& {\widehat{\mathcal{A}}}_2{\int }_0^{\xi }\parallel {\hslash }_2(\rho ,\mathfrak{s},{\mathbb{k}}_{\beta }\left(\mathfrak{s}\right))-{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))\parallel d\mathfrak{s}\hfill \\ & \le & \widehat{\mu }\parallel {\mathbb{k}}_{\beta }^{\rho }-{\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\widehat{\mathcal{A}}\parallel {\hslash }_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)\parallel \hfill \\ & +& {\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1{\int }_0^{\rho }\parallel {\mathbb{k}}_{\beta }\left(\mathfrak{s}\right)-\mathbb{k}\left(\mathfrak{s}\right)\parallel d\mathfrak{s}+{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2{\int }_0^{\xi }\parallel {\mathbb{k}}_{\beta }\left(\mathfrak{s}\right)-\mathbb{k}\left(\mathfrak{s}\right)\parallel d\mathfrak{s}.\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill \parallel {\mathrm{\Lambda }}_{\beta }\left(\rho \right)-\mathrm{\Lambda }\left(\rho \right)\parallel \le {\displaystyle \frac{\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi }{1-\widehat{\mathcal{A}}}}{\parallel {\mathbb{k}}_{\beta }-\mathbb{k}\parallel }_{\mathrm{{\rm Y}}}.\end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \parallel S{\mathbb{k}}_{\beta }\left(\rho \right)-S\mathbb{k}\left(\rho \right)\parallel & \le & \widehat{\lambda }{\parallel {\mathbb{k}}_{\beta }^{\rho }-{\mathbb{k}}^{\rho }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}\hfill \\ & +& {\displaystyle \frac{(\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi )}{(1-\widehat{\mathcal{A}})\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}{\parallel {\mathbb{k}}_{\beta }^{\varpi }-{\mathbb{k}}^{\varpi }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}d\varpi .\hfill \end{array}$$

An application of the Lebesgue Dominated Convergence Theorem directly yields the result

$$\begin{array}{c}\hfill \parallel S{\mathbb{k}}_{\beta }\left(\rho \right)-S\mathbb{k}\left(\rho \right)\parallel ⟶0\mathrm{as}\beta ⟶\infty ,\end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel S{\mathbb{k}}_{\beta }{-S\mathbb{k}\parallel }_{\mathrm{{\rm Y}}}⟶0\mathrm{as}\beta ⟶\infty .\end{array}$$

Consequently, the operator S exhibits continuity. Given an arbitrary $R>0$, it follows that

$$\begin{array}{c}\hfill \mathcal{Z}\ge max\left\{{\displaystyle \frac{\parallel \varphi \left(0\right)\parallel +2{\widehat{\mathfrak{N}}}_2^{*}+\frac{({\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_4^{*}\parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}+{\widehat{\mathfrak{K}}}_5^{*}\parallel {\widehat{\mathfrak{z}}}_2{\parallel }_{L^1}){\xi }^{\mathfrak{w}}}{(1-{\widehat{\mathfrak{K}}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}{1-{\widehat{\mathfrak{N}}}_1^{*}-\frac{({\widehat{\mathfrak{K}}}_2^{*}+{\widehat{\mathfrak{K}}}_4^{*}{\widehat{\delta }}_1\xi +{\widehat{\mathfrak{K}}}_5^{*}{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-{\widehat{\mathfrak{K}}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}}{,\parallel \varphi \parallel }_{[-\mathfrak{a},0]},{\parallel \widehat{\varphi }\parallel }_{[\xi ,\xi +\mathfrak{g}]}\right\}.\end{array}$$

Define the ball

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\mathcal{Z}}={\{\mathbb{k}\in \mathrm{{\rm Y}}:\parallel \mathbb{k}\parallel }_{\mathrm{{\rm Y}}}\le \mathcal{Z}\}.\end{array}$$

It is clear that ${\mathcal{Q}}_{\mathcal{Z}}$ is a bounded, closed and convex subset of $\mathrm{{\rm Y}}$.

Step 2: $S\left({\mathcal{Q}}_{\mathcal{Z}}\right)\subset {\mathcal{Q}}_{\mathcal{Z}}$.

Let $\mathbb{k}\in {\mathcal{Q}}_{\mathcal{Z}}$. If $\rho \in [-\mathfrak{a},0]$, then

$$\begin{array}{c}\hfill \parallel S\mathbb{k}\left(\rho \right)\parallel \le {\parallel \varphi \parallel }_{[-\mathfrak{a},0]}\le \mathcal{Z},\end{array}$$

and if $\rho \in [\xi ,\xi +\mathfrak{g}]$, then

$$\begin{array}{c}\hfill \parallel S\mathbb{k}\left(\rho \right)\parallel \le \parallel \widehat{\varphi }{\parallel }_{[\xi ,\xi +\mathfrak{g}]}\le \mathcal{Z}.\end{array}$$

For each $\rho \in \phi$, we have

$$\begin{array}{ccc}\hfill \parallel S\mathbb{k}\left(\rho \right)\parallel & \le & \parallel \varphi \left(0\right)\parallel e^{-{\mathfrak{w}}_1\rho }+\parallel \aleph (0,\varphi \left(0\right))\parallel e^{-{\mathfrak{w}}_1\rho }+\parallel \aleph (\rho ,{\mathbb{k}}^{\rho })\parallel \hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\parallel \mathrm{\Lambda }\left(\varpi \right)\parallel d\varpi .\hfill \end{array}$$

From hypothesis ($\mathcal{H}8$) and ($\mathcal{H}11$), we have

$$\begin{array}{ccc}\hfill \parallel \mathrm{\Lambda }\left(\rho \right)\parallel & =& ∥\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)∥\hfill \\ & \le & {\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_2^{*}\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\widehat{\mathfrak{K}}}_3^{*}\parallel \mathrm{\Lambda }\left(\rho \right)\parallel \hfill \\ & +& {\widehat{\mathfrak{K}}}_4^{*}{\int }_0^{\rho }\parallel {\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))\parallel d\mathfrak{s}+{\widehat{\mathfrak{K}}}_5^{*}{\int }_0^{\xi }\parallel {\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))\parallel d\mathfrak{s}\hfill \\ & \le & {\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_2^{*}{\parallel \mathbb{k}\parallel }_{\mathrm{{\rm Y}}}+{\widehat{\mathfrak{K}}}_3^{*}\parallel \mathrm{\Lambda }\left(\rho \right)\parallel \hfill \\ & +& {\widehat{\mathfrak{K}}}_4^{*}{\int }_0^{\rho }{\widehat{\mathfrak{z}}}_1\left(\mathfrak{s}\right)(1+\parallel \mathbb{k}\left(\mathfrak{s}\right)\parallel )d\mathfrak{s}+{\widehat{\mathfrak{K}}}_5^{*}{\int }_0^{\xi }{\widehat{\mathfrak{z}}}_2\left(\mathfrak{s}\right)(1+\parallel \mathbb{k}\left(\mathfrak{s}\right)\parallel )d\mathfrak{s}\hfill \\ & \le & {\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_2^{*}R+{\widehat{\mathfrak{K}}}_3^{*}\parallel \mathrm{\Lambda }\left(\rho \right)\parallel \hfill \\ & +& {\widehat{\mathfrak{K}}}_4^{*}\parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}(1+\mathcal{Z})+{\widehat{\mathfrak{K}}}_5^{*}{\parallel {\widehat{\mathfrak{z}}}_2\parallel }_{L^1}(1+\mathcal{Z}).\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill \parallel \mathrm{\Lambda }\left(\rho \right)\parallel \le {\displaystyle \frac{{\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_2^{*}\mathcal{Z}+{\widehat{\mathfrak{K}}}_4^{*}\parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}(1+\mathcal{Z})+{\widehat{\mathfrak{K}}}_5^{*}{\parallel {\widehat{\mathfrak{z}}}_2\parallel }_{L^1}(1+\mathcal{Z})}{1-{\widehat{\mathfrak{K}}}_3^{*}}}.\end{array}$$

Finally, we get

$$\begin{array}{c}\hfill \parallel S\mathbb{k}\left(\rho \right)\parallel \le \parallel \varphi \left(0\right)\parallel +2{\widehat{\mathfrak{N}}}_2^{*}+{\widehat{\mathfrak{N}}}_1^{*}\mathcal{Z}+{\displaystyle \frac{[{\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_2^{*}\mathcal{Z}+{\widehat{\mathfrak{K}}}_4^{*}\parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}(1+\mathcal{Z})+{\widehat{\mathfrak{K}}}_5^{*}\parallel {\widehat{\mathfrak{z}}}_2{\parallel }_{L^1}(1+\mathcal{Z})]{\xi }^{\mathfrak{w}}}{(1-{\widehat{\mathfrak{K}}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}\le \mathcal{Z}.\end{array}$$

Thus, for each $\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]$, we have

$$\begin{array}{c}\hfill \parallel S\mathbb{k}\left(\rho \right)\parallel \le \mathcal{Z},\end{array}$$

which implies that

$$\begin{array}{c}\hfill {\parallel S\mathbb{k}\parallel }_{\mathrm{{\rm Y}}}\le \mathcal{Z}.\end{array}$$

Consequently, $S\left({\mathcal{Q}}_{\mathcal{Z}}\right)\subset {\mathcal{Q}}_R$.

Step 3: $S\left({\mathcal{Q}}_{\mathcal{Z}}\right)$ is equicontinuous.

Let ${\rho }_1,{\rho }_2\in \phi$, where ${\rho }_1<{\rho }_2$ and $\mathbb{k}\in {\mathcal{Q}}_{\mathcal{Z}}$. Then,

$$\begin{array}{ccc}\hfill \parallel S\mathbb{k}\left({\rho }_2\right)-S\mathbb{k}\left({\rho }_1\right)\parallel & =& \parallel \varphi \left(0\right)e^{-{\mathfrak{w}}_1{\rho }_2}-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1{\rho }_2}+\aleph ({\rho }_2,{\mathbb{k}}^{{\rho }_2})\hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{{\rho }_2}{e}^{-{\mathfrak{w}}_1({\rho }_2-\varpi )}{({\rho }_2-\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi \hfill \\ & -& \varphi \left(0\right)e^{-{\mathfrak{w}}_1{\rho }_1}+\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1{\rho }_1}-\aleph ({\rho }_1,{\mathbb{k}}^{{\rho }_1})\hfill \\ & -& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{{\rho }_1}{e}^{-{\mathfrak{w}}_1({\rho }_1-\varpi )}{({\rho }_1-\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi \parallel \hfill \\ & \le & \parallel \varphi \left(0\right)\parallel \parallel e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}\parallel +\parallel \aleph (0,\varphi \left(0\right))\parallel \parallel e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}\parallel \hfill \\ & +& \parallel \aleph ({\rho }_2,{\mathbb{k}}^{{\rho }_2})-\aleph ({\rho }_1,{\mathbb{k}}^{{\rho }_1})\parallel \hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{{\rho }_1}[e^{-{\mathfrak{w}}_1({\rho }_2-\varpi )}{({\rho }_2-\varpi )}^{\mathfrak{w}-1}-e^{-{\mathfrak{w}}_1({\rho }_1-\varpi )}{({\rho }_1-\varpi )}^{\mathfrak{w}-1}]\parallel \mathrm{\Lambda }\left(\varpi \right)\parallel d\varpi \hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_{{\rho }_1}^{{\rho }_2}{e}^{-{\mathfrak{w}}_1({\rho }_2-\varpi )}{({\rho }_2-\varpi )}^{\mathfrak{w}-1}\parallel \mathrm{\Lambda }\left(\varpi \right)\parallel d\varpi \hfill \\ & \le & \parallel \varphi \left(0\right)\parallel \parallel e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}\parallel +\parallel \aleph (0,\varphi \left(0\right))\parallel \left|\right|e^{-{\mathfrak{w}}_1{\rho }_2}-e^{-{\mathfrak{w}}_1{\rho }_1}\parallel \hfill \\ & +& \parallel \aleph ({\rho }_2,{\mathbb{k}}^{{\rho }_2})-\aleph ({\rho }_1,{\mathbb{k}}^{{\rho }_1})\parallel \hfill \\ & +& {\displaystyle \frac{[{\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_2^{*}\mathcal{Z}+{\widehat{\mathfrak{K}}}_4^{*}\parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}(1+\mathcal{Z})+{\widehat{\mathfrak{K}}}_5^{*}\parallel {\widehat{\mathfrak{z}}}_2{\parallel }_{L^1}(1+\mathcal{Z})]({\rho }_2^{\mathfrak{w}}-{\rho }_1^{\mathfrak{w}})}{(1-{\widehat{\mathfrak{K}}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}\hfill \\ & +& {\displaystyle \frac{[{\widehat{\mathfrak{K}}}_1^{*}+{\widehat{\mathfrak{K}}}_2^{*}\mathcal{Z}+{\widehat{\mathfrak{K}}}_4^{*}\parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}(1+\mathcal{Z})+{\widehat{\mathfrak{K}}}_5^{*}\parallel {\widehat{\mathfrak{z}}}_2{\parallel }_{L^1}(1+\mathcal{Z})]{({\rho }_2-{\rho }_1)}^{\mathfrak{w}}}{(1-{\widehat{\mathfrak{K}}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}.\hfill \end{array}$$

As ${\rho }_1\to {\rho }_2$, the term on the right-hand side of the above inequality tends to 0. Consequently, the operator S is equicontinuous on this interval. The proof of equicontinuity is as follows:

Step 4: $\mathcal{S}$ is a contraction.

Let $\mathfrak{B}$ be a subset of ${\mathcal{Q}}_{\mathcal{Z}}$. If $\rho \in [-\mathfrak{a},0]$, then

$$\begin{array}{ccc}\hfill \mathcal{E}\left(S\right(\mathfrak{B}\left(\rho \right)\left)\right)& =& \mathcal{E}\left\{S\mathbb{k}\right(\rho ),\mathbb{k}\in \mathfrak{B}\}\hfill \\ & =& \mathcal{E}\left\{\varphi \right(\rho ),\mathbb{k}\in \mathfrak{B}\}\hfill \\ & =& 0,\hfill \end{array}$$

and if $\rho \in [\xi ,\xi +\mathfrak{g}]$, then

$$\begin{array}{ccc}\hfill \mathcal{E}\left(S\right(\mathfrak{B}\left(\rho \right)\left)\right)& =& \mathcal{E}\left\{S\mathbb{k}\right(\rho ),\mathbb{k}\in \mathfrak{B}\}\hfill \\ & =& \mathcal{E}\{\widehat{\varphi }\left(\rho \right),\mathbb{k}\in \mathfrak{B}\}\hfill \\ & =& 0.\hfill \end{array}$$

For each $\rho \in [0,T]$, we have

$$\begin{array}{ccc}\hfill \mathcal{E}\left(S\right(\mathfrak{B}\left(\rho \right)\left)\right)& =& \mathcal{E}\left\{S\mathbb{k}\right(\rho ),\mathbb{k}\in \mathfrak{B}\}\hfill \\ & =& \mathcal{E}\{\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1\rho }+\aleph (\rho ,{\mathbb{k}}^{\rho })\hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi ,\mathbb{k}\in \mathfrak{B}\}\hfill \\ & \le & \mathcal{E}\left\{\aleph (\rho ,{\mathbb{k}}^{\rho }),\mathbb{k}\in \mathfrak{B}\right\}+\mathcal{E}\left\{{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi ,\mathbb{k}\in \mathfrak{B}\right\}.\hfill \end{array}$$

Through condition ($\mathcal{H}13$), we have

$$\begin{array}{ccc}\hfill \mathcal{E}\left(\mathrm{\Lambda }\right(\rho \left)\right)& =& \mathcal{E}\left(\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\right)\hfill \\ & \le & \widehat{\mu }\underset{\rho \in [-\mathfrak{a},\mathfrak{g}]}{sup}\mathcal{E}\left({\mathbb{k}}^{\rho }\right)+\widehat{\mathcal{A}}\mathcal{E}\left(\mathrm{\Lambda }\left(\rho \right)\right)\hfill \\ & +& {\widehat{\mathcal{A}}}_1\mathcal{E}\left({\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)+{\widehat{\mathcal{A}}}_2\mathcal{E}\left({\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\hfill \\ & \le & \widehat{\mu }\underset{\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]}{sup}\mathcal{E}\left(\mathbb{k}\left(\rho \right)\right)+\widehat{\mathcal{A}}\mathcal{E}\left(\mathrm{\Lambda }\left(\rho \right)\right)\hfill \\ & +& {\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1{\int }_0^{\rho }\mathcal{E}\left(\mathbb{k}\left(\mathfrak{s}\right)\right)d\mathfrak{s}+{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2{\int }_0^{\xi }\mathcal{E}\left(\mathbb{k}\left(\mathfrak{s}\right)\right)d\mathfrak{s}.\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \mathcal{E}\left(\mathrm{\Lambda }\left(\rho \right)\right)\le {\displaystyle \frac{\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi }{1-\widehat{\mathcal{A}}}}\underset{\rho \in [-\mathfrak{a},\xi +\mathfrak{g}]}{sup}\mathcal{E}\left(\mathbb{k}\left(\rho \right)\right),\end{array}$$

and

$$\begin{array}{cc}\hfill \mathcal{E}\left(S\right(\mathfrak{B}\left(\rho \right)\left)\right)& \le \widehat{\lambda }\underset{\rho \in [-\mathfrak{a},\mathfrak{g}]}{sup}\mathcal{E}\left({\mathbb{k}}^{\rho }\right)+{\displaystyle \frac{\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi }{1-\widehat{\mathcal{A}}}}{\int }_0^{\rho }{(\rho -\varpi )}^{\mathfrak{w}-1}\left\{{sup}_{\ell \in [-\mathfrak{a},\xi +\mathfrak{g}]}\mathcal{E}\left(\mathbb{k}\left(\varpi \right)\right)d\varpi ,\mathbb{k}\in \mathfrak{B}\right\}\\ & \le \widehat{\lambda }{\mathcal{E}}_{\xi }\left(\mathfrak{B}\right)+{\displaystyle \frac{(\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-\widehat{\mathcal{A}})\mathrm{\Gamma }(\mathfrak{w}+1)}}{\mathcal{E}}_{\xi }\left(\mathfrak{B}\right).\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\mathcal{E}}_{\xi }\left(S\left(\mathfrak{B}\right)\right)\le \left[\widehat{\lambda }+{\displaystyle \frac{(\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-\widehat{\mathcal{A}})\mathrm{\Gamma }(\mathfrak{w}+1)}}\right]{\mathcal{E}}_{\xi }\left(\mathfrak{B}\right),\end{array}$$

and by Remark 3, we have

$$\begin{array}{c}\hfill {\mathcal{E}}_{\xi }\left(S\left(\mathfrak{B}\right)\right)\le \left[{\widehat{\mathfrak{N}}}_1^{*}+{\displaystyle \frac{({\widehat{\mathfrak{K}}}_2^{*}+{\widehat{\mathfrak{K}}}_4^{*}{\widehat{\delta }}_1\xi +{\widehat{\mathfrak{K}}}_5^{*}{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-{\widehat{\mathfrak{K}}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}\right]{\mathcal{E}}_{\xi }\left(\mathfrak{B}\right).\end{array}$$

Based on the findings established in Theorem 6, the operator S can be classified as a contraction mapping. In accordance with the principles of Darbo’s fixed point theorem, it follows that S possesses at least one fixed point. This fixed point corresponds directly to a solution for the implicit fractional integro-differential equation presented in (19).    □

Remark 4.

Although the existence analysis is carried out in an infinite-dimensional Banach space, the numerical realization of the problem in Section 4.3, in Equation (28), is performed via a finite-dimensional truncation, as detailed in Section 4.

3.2.1. Ulam–Hyers–Rassias Stability

In this subsection, we will establish the Ulam stability for the implicit fractional integro-differential problem (19). Let $\Delta \in C(\rho ,{\mathbb{R}}_{+})$.

Definition 6

([1]). The implicit fractional integro-differential problem (19) is Ulam–Hyers–Rassias stability with respect to Δ if there exists ${\lambda }_{\mathfrak{F},\Delta }>0$ such that for each $\epsilon >0$ and for each solution $\mathbb{k}\in \mathrm{{\rm Y}}$ of the inequality

$$\begin{array}{ccc}\hfill & & {\parallel }_0^C{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}[\mathbb{k}\left(\rho \right)-\aleph (\rho ,{\mathbb{k}}^{\rho })]-\mathfrak{F}(\rho ,{\mathbb{k}}^{\rho },{}_0{}^{C}D_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^{\xi }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi )\parallel \hfill \\ & & \le \epsilon \Delta \left(\rho \right),\rho \in \phi ,\hfill \end{array}$$

(20)

there exists a solution $\overline{\mathbb{k}}\in \mathrm{{\rm Y}}$ of the problem (19) with

$$\begin{array}{c}\hfill \parallel \mathbb{k}\left(\rho \right)-\overline{\mathbb{k}}\left(\rho \right)\parallel \le \epsilon {\lambda }_{\mathfrak{F},\Delta }\Delta \left(\rho \right),\rho \in \phi .\end{array}$$

Remark 5.

A function $\mathbb{k}\in \mathrm{{\rm Y}}$ satisfies inequality (20) if and only if a corresponding function $\ell \in C(\rho ,E)$ can be found, where ℓ is dependent on $\mathbb{k}$.

1. 

$\parallel \ell \left(\rho \right)\parallel \le \epsilon \Delta \left(\rho \right),\mathit{for}\mathit{each}\rho \in \phi$

2. 

${}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^{C}D_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^{\xi }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right)+\ell \left(\rho \right),$ for each $\rho \in \phi$.

Proof. 

Assume first that $\mathbb{k}\in \mathrm{{\rm Y}}$ satisfies inequality (20). Define

$$\ell \left(\rho \right){=}_0^C{\mathcal{D}}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right)-\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho }{,}_0^C{\mathcal{D}}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^{\rho }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right).$$

Then $\ell \in C(\phi ,E)$ by continuity of the involved operators and inequality (20) immediately implies $\parallel \ell \left(\rho \right)\parallel \le \epsilon \Delta \left(\rho \right)$ for each $\rho \in \phi$. Hence, conditions (1) and (2) are satisfied.

Conversely, if there exists a function $\ell \in C(\phi ,E)$ satisfying conditions (1) and (2), then taking norms in (2) and using condition (1) yields inequality (20). This completes the proof.    □

Lemma 9.

The solution of the following perturbed problem:

$$\begin{array}{ccc}\hfill {}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\left[\mathbb{k}\left(\rho \right)-\aleph (\rho ,{\mathbb{k}}^{\rho })\right]& =& \mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^C\mathcal{D}_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^{\xi }{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right)\hfill \\ & +& \ell \left(\rho \right),\rho \in \phi :=[0,\xi ],\hfill \end{array}$$

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\varphi \left(\rho \right),\rho \in [-\mathfrak{a},0],\\ \hfill \mathbb{k}\left(\rho \right)=\tilde{\varphi }\left(\rho \right),\rho \in [\xi ,\xi +\mathfrak{g}],\end{array}$$

is given by

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\left\{\begin{array}{c}\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1\rho }+\aleph (\rho ,{\mathbb{k}}^{\rho })\hfill \\ +{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi +{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\ell \left(\varpi \right)d\varpi \hfill & if\rho \in \phi ,\hfill \\ \varphi \left(\rho \right),if\rho \in [-\mathfrak{a},0],\hfill \\ \tilde{\varphi }\left(\rho \right),if\rho \in [\xi ,\xi +\mathfrak{g}].\hfill \end{array}\right.\end{array}$$

Moreover, the solution satisfies the following inequality

$$\begin{array}{ccc}& & ∥\mathbb{k}\left(\rho \right)-\left[\varphi \left(0\right)e^{-{\mathfrak{w}}_1\rho }-\aleph (0,\varphi \left(0\right))e^{-{\mathfrak{w}}_1\rho }+\aleph (\rho ,{\mathbb{k}}^{\rho })+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\mathrm{\Lambda }\left(\varpi \right)d\varpi \right]∥\hfill \\ & & \le \epsilon {\mu }_{\Delta }\Delta \left(\rho \right),\mathit{for}\mathit{each}\rho \in \phi .\hfill \end{array}$$

Proof. 

Applying the fractional integral operator ${\mathcal{I}}_0^{\mathfrak{w}}$ to both sides of the perturbed equation and using the definition of the Caputo-type fractional derivative, we obtain

$$\mathbb{k}\left(\rho \right)-\aleph (\rho ,{\mathbb{k}}^{\rho })=\left[\mathbb{k}\left(0\right)-\aleph (0,{\mathbb{k}}^0)\right]e^{-{\mathfrak{w}}_1\rho }+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\left[\mathrm{\Lambda }\left(\varpi \right)+\ell \left(\varpi \right)\right]d\varpi ,$$

for $\rho \in \phi$, where

$$\mathrm{\Lambda }\left(\varpi \right)=\mathfrak{F}\left(\varpi ,{\mathbb{k}}^{\varpi }{,}_0^C{\mathcal{D}}_{\varpi }^{\mathfrak{w},{\mathfrak{w}}_1}\mathbb{k}\left(\varpi \right),{\int }_0^{\varpi }{\hslash }_1(\varpi ,s,\mathbb{k}\left(s\right))ds,{\int }_0^{\varpi }{\hslash }_2(\varpi ,s,\mathbb{k}\left(s\right))ds\right).$$

Using the initial condition $\mathbb{k}\left(0\right)=\varphi \left(0\right)$ and rearranging terms yields the stated integral representation of the solution.

Moreover, by taking norms and using the bound $\parallel \ell \left(\rho \right)\parallel \le \epsilon {\mu }_{\Delta }\Delta \left(\rho \right)$ together with the positivity of the kernel, the stated inequality follows immediately. This completes the proof.    □

Theorem 7.

Let the assumptions $\left(\mathcal{H}7\right)$–$\left(\mathcal{H}12\right)$ be satisfied and suppose that condition (15) is also fulfilled. Furthermore, assume that

$\left(\mathcal{H}15\right)$

there exists a nondecreasing function $\Delta \in C(\rho ,{\mathbb{R}}_{+})$ together with a constant ${\mu }_{\Delta }>0$ such that for every $\rho \in \phi$,

$$\begin{array}{c}\hfill {}_{0}I_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\Delta \left(\rho \right)\le {\mu }_{\Delta }\Delta \left(\rho \right).\end{array}$$

Under these hypotheses, the implicit fractional integro-differential Equation (19) enjoys the Ulam–Hyers–Rassias stability property.

Proof. 

Let $\mathbb{k}\in \mathrm{{\rm Y}}$ be a solution of the inequality (20) and $\overline{\mathbb{k}}\in \mathrm{{\rm Y}}$ the solution of the problem (19). Then,

$$\begin{array}{ccc}\hfill \parallel \mathbb{k}\left(\rho \right)-\overline{\mathbb{k}}\left(\rho \right)\parallel & \le & {\mu }_{\Delta }\epsilon \Delta \left(\rho \right)+\parallel \aleph (\rho ,{\mathbb{k}}^{\rho })-\aleph (\rho ,{\overline{\mathbb{k}}}^{\rho })\parallel \hfill \\ & +& {\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\parallel \mathrm{\Lambda }\left(\varpi \right)-\overline{\mathrm{\Lambda }}\left(\varpi \right)\parallel d\varpi ,\hfill \end{array}$$

The functions $\mathrm{\Lambda }$ and $\overline{\mathrm{\Lambda }}$ are defined such that they each meet distinct sets of functional equations.

$$\begin{array}{c}\hfill \mathrm{\Lambda }\left(\rho \right)=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right).,\end{array}$$

and

$$\begin{array}{c}\hfill \overline{\mathrm{\Lambda }}\left(\rho \right)=\mathfrak{F}\left(\rho ,{\overline{\mathbb{k}}}^{\rho },\overline{\mathrm{\Lambda }}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\overline{\mathbb{k}}\left(\mathfrak{s}\right))ds,{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\overline{\mathbb{k}}\left(\mathfrak{s}\right))d\mathfrak{s}\right).\end{array}$$

From hypothesis ($\mathcal{H}8$), we have

$$\begin{array}{ccc}\hfill \parallel \mathrm{\Lambda }\left(\rho \right)-\overline{\mathrm{\Lambda }}\left(\rho \right)\parallel & =& \parallel \mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },\mathrm{\Lambda }\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\hfill \\ & & -\mathfrak{F}\left(\rho ,{\overline{\mathbb{k}}}^{\rho },\overline{\mathrm{\Lambda }}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\mathfrak{s},\overline{\mathbb{k}}\left(\mathfrak{s}\right))d\mathfrak{s},{\int }_0^{\xi }{\hslash }_2(\rho ,\mathfrak{s},\overline{\mathbb{k}}\left(\mathfrak{s}\right))d\mathfrak{s}\right)\parallel \hfill \\ & \le & \widehat{\mu }\parallel {\mathbb{k}}^{\rho }-{\overline{\mathbb{k}}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\widehat{\mathcal{A}}\parallel \mathrm{\Lambda }\left(\rho \right)-\overline{\mathrm{\Lambda }}\left(\rho \right)\parallel \hfill \\ & +& {\widehat{\mathcal{A}}}_1{\int }_0^{\rho }\parallel {\hslash }_1(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))-{\hslash }_1(\rho ,\mathfrak{s},\overline{\mathbb{k}}\left(\mathfrak{s}\right))\parallel d\mathfrak{s}\hfill \\ & +& {\widehat{\mathcal{A}}}_2{\int }_0^{\xi }\parallel {\hslash }_2(\rho ,\mathfrak{s},\mathbb{k}\left(\mathfrak{s}\right))-{\hslash }_2(\rho ,\mathfrak{s},\overline{\mathbb{k}}\left(\mathfrak{s}\right))\parallel d\mathfrak{s}\hfill \\ & \le & \widehat{\mu }\parallel {\mathbb{k}}^{\rho }-{\overline{\mathbb{k}}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\widehat{\mathcal{A}}\parallel \mathrm{\Lambda }\left(\rho \right)-\overline{\mathrm{\Lambda }}\left(\rho \right)\parallel \hfill \\ & +& {\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1{\int }_0^{\rho }\parallel \mathbb{k}\left(\mathfrak{s}\right)-\overline{\mathbb{k}}\left(\mathfrak{s}\right)\parallel d\mathfrak{s}+{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2{\int }_0^{\xi }\parallel \mathbb{k}\left(\mathfrak{s}\right)-\overline{\mathbb{k}}\left(\mathfrak{s}\right)\parallel d\mathfrak{s}\hfill \\ & \le & \widehat{\mu }\parallel {\mathbb{k}}^{\rho }-{\overline{\mathbb{k}}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\widehat{\mathcal{A}}\parallel \mathrm{\Lambda }\left(\rho \right)-\overline{\mathrm{\Lambda }}\left(\rho \right)\parallel \hfill \\ & +& ({\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi ){\parallel \mathbb{k}-\overline{\mathbb{k}}\parallel }_{\mathrm{{\rm Y}}},\hfill \end{array}$$

which implies that

$$\begin{array}{c}\hfill \parallel \mathrm{\Lambda }\left(\rho \right)-\overline{\mathrm{\Lambda }}\left(\rho \right)\parallel \le {\displaystyle \frac{\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi }{1-\widehat{\mathcal{A}}}}{\parallel \mathbb{k}-\overline{\mathbb{k}}\parallel }_{\mathrm{{\rm Y}}}.\end{array}$$

Then,

$$\begin{array}{ccc}\hfill \parallel \mathbb{k}\left(\rho \right)-\overline{\mathbb{k}}\left(\rho \right)\parallel & \le & {\mu }_{\Delta }\epsilon \Delta \left(\rho \right)+\widehat{\lambda }{\parallel \mathbb{k}-\overline{\mathbb{k}}\parallel }_{\mathrm{{\rm Y}}}\hfill \\ & +& {\displaystyle \frac{\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi }{(1-\widehat{\mathcal{A}})\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}{\parallel {\mathbb{k}}^{\varpi }-{\overline{\mathbb{k}}}^{\varpi }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}d\varpi .\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \parallel \mathbb{k}-\overline{\mathbb{k}}{\parallel }_{\mathrm{{\rm Y}}}\le {\mu }_{\Delta }\epsilon \Delta \left(\rho \right)+\widehat{\lambda }\parallel \mathbb{k}-\overline{\mathbb{k}}{\parallel }_{\mathrm{{\rm Y}}}+{\displaystyle \frac{(\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-\widehat{\mathcal{A}})\mathrm{\Gamma }(\mathfrak{w}+1)}}{\parallel \mathbb{k}-\overline{\mathbb{k}}\parallel }_{\mathrm{{\rm Y}}}.\end{array}$$

Finally, we get

$$\begin{array}{c}\hfill \parallel \mathbb{k}-\overline{\mathbb{k}}{\parallel }_{\mathrm{{\rm Y}}}\le {\displaystyle \frac{{\mu }_{\Delta }\epsilon \Delta \left(\rho \right)}{1-\widehat{\lambda }-\frac{(\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-\widehat{\mathcal{A}})\mathrm{\Gamma }(\mathfrak{w}+1)}}}.\end{array}$$

Then, for each $\rho \in \phi$, we have

$$\begin{array}{c}\hfill \parallel \mathbb{k}-\overline{\mathbb{k}}{\parallel }_{\mathrm{{\rm Y}}}\le {\lambda }_{\mathfrak{F},\Delta }\epsilon \Delta \left(\rho \right),\end{array}$$

where

$$\begin{array}{c}\hfill {\lambda }_{\mathfrak{F},\Delta }={\displaystyle \frac{{\mu }_{\Delta }}{1-\widehat{\lambda }-\frac{(\widehat{\mu }+{\widehat{\mathcal{A}}}_1{\widehat{\delta }}_1\xi +{\widehat{\mathcal{A}}}_2{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-\widehat{\mathcal{A}})\mathrm{\Gamma }(\mathfrak{w}+1)}}}.\end{array}$$

Consequently, the problem formulated in (19) satisfies the Ulam–Hyers–Rassias stability criterion relative to the function $\Delta$.    □

3.2.2. An Application

Set

$$\begin{array}{c}\hfill E=l^1=\left\{\mathbb{k}=({\mathbb{k}}_1,{\mathbb{k}}_2,\dots ,{\mathbb{k}}_{\beta },\dots ),\sum _{\beta =1}^{\infty }\left|{\mathbb{k}}_{\beta }\right|<\infty \right\},\end{array}$$

where E is a Banach space with the norm $\parallel \mathbb{k}\parallel ={\sum }_{\beta =1}^{\infty }\left|{\mathbb{k}}_{\beta }\right|$. Consider the following implicit fractional integro-differential problem:

$$\begin{array}{ccc}\hfill & & {}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}\left[\mathbb{k}\left(\rho \right)-\aleph (\rho ,{\mathbb{k}}^{\rho })\right]=\hfill \\ & & \mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^{C}D_{\rho }^{{\displaystyle \frac{1}{3}},3}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^1{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right),\rho \in [0,1],\hfill \end{array}$$

(21)

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\varphi \left(\rho \right),\rho \in [-1,0],\end{array}$$

(22)

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\widehat{\varphi }\left(\rho \right),\rho \in [1,2],\end{array}$$

(23)

where $\varphi \in C\left(\right[-1,0],E)$ and $\widehat{\varphi }\in C([1,2],E)$. Set

$$\begin{array}{c}\hfill {\mathfrak{F}}_{\beta }\left(\rho ,{\mathbb{k}}_{\beta }^{\rho },{}_0{}^{C}D_{\rho }^{\frac{1}{3},3}{\mathbb{k}}_{\beta }\left(\rho \right),{\nu }_1,{\nu }_2\right)={\displaystyle \frac{9e^{7\phi }+cos\left(\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}\right)+cos\left(\left|{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}{\mathbb{k}}_{\beta }\left(\rho \right)\right|\right)+\frac{1}{5}{\nu }_1+\frac{1}{6}{\nu }_2}{100e^{\phi +11}\left(1+\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+∥{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}\mathbb{k}\left(\rho \right)∥+|{\nu }_1|+|{\nu }_2|\right)}},\end{array}$$

$$\begin{array}{c}\hfill {\aleph }_{\beta }(\rho ,{\mathbb{k}}_{\beta }^{\rho })={\displaystyle \frac{\sqrt{2\pi }{\parallel {\mathbb{k}}^{\rho }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+ln(e+\phi )}{77}},\end{array}$$

and define the integral kernels

$$\begin{array}{c}\hfill {\hslash }_{1\beta }(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))={\displaystyle \frac{sin\left(\rho \varpi \right)}{10(1+{\ell }^2)}}{\mathbb{k}}_{\beta }\left(\varpi \right),{\hslash }_{2\beta }(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))={\displaystyle \frac{e^{-\phi \varpi }}{20(1+\ell )}}{\mathbb{k}}_{\beta }\left(\varpi \right),\end{array}$$

for $\rho \in [0,1],\mathbb{k}\in C\left(\right[-\mathfrak{a},\mathfrak{g}],E)$, where

$$\begin{array}{c}\hfill \mathfrak{F}=({\mathbb{k}}_1,{\mathbb{k}}_2,\dots ,{\mathbb{k}}_{\beta },\dots ),\\ \hfill \mathfrak{F}=({\mathfrak{F}}_1,{\mathfrak{F}}_2,\dots ,{\mathfrak{F}}_{\beta },\dots ),\\ \hfill \aleph =({\aleph }_1,{\aleph }_2,\dots ,{\aleph }_{\beta },\dots ),\\ \hfill {\hslash }_1=({\hslash }_{11},H_{12},\dots ,H_{1\beta },\dots ),\\ \hfill H_2=(H_{21},H_{22},\dots ,H_{2\beta },\dots ),\end{array}$$

and

$$\begin{array}{c}\hfill {}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}y=\left({}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}{\mathbb{k}}_1,{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}{\mathbb{k}}_2,\dots ,{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}{\mathbb{k}}_{\beta },\dots \right).\end{array}$$

Clearly, $\mathfrak{F}$, ℵ, $H_1$, and $H_2$ are continuous functions, and hypothesis ($\mathcal{H}7$) is satisfied.

For any $\beta ,\overline{\beta }\in C([-\mathfrak{a},\mathfrak{g}],E)$, $\gamma ,\overline{\gamma },{\nu }_1,{\overline{\nu }}_1,{\nu }_2,{\overline{\nu }}_2\in E$ and $\rho \in [0,1]$, we have

$$\begin{array}{c}\hfill \parallel \mathfrak{F}(\rho ,\beta ,\gamma ,{\nu }_1,{\nu }_2)-\mathfrak{F}(\rho ,\overline{\beta },\overline{\gamma },{\overline{\nu }}_1,{\overline{\nu }}_2)\parallel \le {\displaystyle \frac{1}{100e^{11}}}\left[\parallel \beta -\overline{\beta }{\parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\parallel \gamma -\overline{\gamma }\parallel +{\displaystyle \frac{1}{5}}\parallel {\nu }_1-{\overline{\nu }}_1\parallel +{\displaystyle \frac{1}{6}}\parallel {\nu }_2-{\overline{\nu }}_2\parallel \right].\end{array}$$

Then, the assumption ($\mathcal{H}8$) is satisfied with $\widehat{\mu }=\widehat{\mathcal{A}}={\displaystyle \frac{1}{100e^{11}}}$, ${\widehat{\mathcal{A}}}_1={\displaystyle \frac{1}{500e^{11}}}$, and ${\widehat{\mathcal{A}}}_2={\displaystyle \frac{1}{600e^{11}}}$.

Also, we have

$$\begin{array}{c}\hfill \parallel \aleph (\rho ,\beta )-\aleph (\rho ,\overline{\beta })\parallel \le {\displaystyle \frac{\sqrt{2\pi }}{77}}{\parallel \beta -\overline{\beta }\parallel }_{[-\mathfrak{a},\mathfrak{g}]}.\end{array}$$

So, condition ($\mathcal{H}9$) is satisfied with $\widehat{\lambda }={\displaystyle \frac{\sqrt{2\pi }}{77}}$.

For the integral kernels, we have the Lipschitz conditions:

$$\begin{array}{cc}\hfill \parallel H_1(\rho ,\varpi ,u)-H_1(\rho ,\varpi ,v)\parallel & \le {\displaystyle \frac{1}{10(1+{\ell }^2)}}\parallel u-v\parallel \le {\displaystyle \frac{1}{10}}\parallel u-v\parallel ,\\ \hfill \parallel H_2(\rho ,\varpi ,u)-H_2(\rho ,\varpi ,v)\parallel & \le {\displaystyle \frac{1}{20(1+\ell )}}\parallel u-v\parallel \le {\displaystyle \frac{1}{20}}\parallel u-v\parallel .\end{array}$$

Thus, ($\mathcal{H}1$0) is satisfied with ${\widehat{\delta }}_1={\displaystyle \frac{1}{10}}$ and ${\widehat{\delta }}_2={\displaystyle \frac{1}{20}}$.

For the growth conditions, we have

$$\begin{array}{cc}\hfill \parallel H_1(\rho ,\varpi ,u)\parallel & \le {\displaystyle \frac{1}{10(1+{\ell }^2)}}(1+\parallel u\parallel ),\\ \hfill \parallel H_2(\rho ,\varpi ,u)\parallel & \le {\displaystyle \frac{1}{20(1+\ell )}}(1+\parallel u\parallel ),\end{array}$$

so ($\mathcal{H}1$1) is satisfied with ${\widehat{\mathfrak{z}}}_1\left(\varpi \right)={\displaystyle \frac{1}{10(1+{\ell }^2)}}$ and ${\widehat{\mathfrak{z}}}_2\left(\varpi \right)={\displaystyle \frac{1}{20(1+\ell )}}$, with $\parallel {\widehat{\mathfrak{z}}}_1{\parallel }_{L^1}={\displaystyle \frac{\pi }{20}}$ and $\parallel {\widehat{\mathfrak{z}}}_2{\parallel }_{L^1}={\displaystyle \frac{ln2}{20}}$.

Also, we have

$$\begin{array}{c}\hfill \parallel \mathfrak{F}(\rho ,\beta ,\gamma ,{\nu }_1,{\nu }_2)\parallel \le {\displaystyle \frac{1}{100e^{\phi +11}}}\left[9e^{7\phi }+{\parallel \beta \parallel }_{[-\mathfrak{a},\mathfrak{g}]}+\parallel \gamma \parallel +{\displaystyle \frac{1}{5}}\parallel {\nu }_1\parallel +{\displaystyle \frac{1}{6}}\parallel {\nu }_2\parallel \right],\end{array}$$

and

$$\begin{array}{c}\hfill \parallel \aleph (\rho ,\beta )\parallel \le {\displaystyle \frac{\sqrt{2\pi }}{77}}{\parallel \beta \parallel }_{[-\mathfrak{a},\mathfrak{g}]}+{\displaystyle \frac{\phi ln(e+\phi )}{77}}.\end{array}$$

So ${\widehat{\mathfrak{K}}}_1\left(\rho \right)={\displaystyle \frac{9e^{7\phi }}{100e^{\phi +11}}}$, ${\widehat{\mathfrak{K}}}_2^{*}={\widehat{\mathfrak{K}}}_3^{*}={\displaystyle \frac{1}{100e^{11}}}$, ${\widehat{\mathfrak{K}}}_4^{*}={\displaystyle \frac{1}{500e^{11}}}$, ${\widehat{\mathfrak{K}}}_5^{*}={\displaystyle \frac{1}{600e^{11}}}$, ${\widehat{\mathfrak{N}}}_1^{*}={\displaystyle \frac{\sqrt{2\pi }}{77}}$ and ${\widehat{\mathfrak{N}}}_2\left(\rho \right)={\displaystyle \frac{\phi ln(e+\phi )}{77}}$. Moreover, we have

$$\begin{array}{c}\hfill \underset{{\rho }_1⟶{\rho }_2}{lim}\left(\aleph ({\rho }_2,{\mathbb{k}}^{{\rho }_2})-\aleph ({\rho }_1,{\mathbb{k}}^{{\rho }_1})\right)⟶0\mathrm{as}{\rho }_1⟶{\rho }_2.\end{array}$$

Thus, hypothesis ($\mathcal{H}12$) is verified.

Now, we compute the condition of Theorem 6:

$$\begin{array}{ccc}\hfill {\widehat{\mathfrak{N}}}_1^{*}+{\displaystyle \frac{({\widehat{\mathfrak{K}}}_2^{*}+{\widehat{\mathfrak{K}}}_4^{*}{\widehat{\delta }}_1\xi +{\widehat{\mathfrak{K}}}_5^{*}{\widehat{\delta }}_2\xi ){\xi }^{\mathfrak{w}}}{(1-{\widehat{\mathfrak{K}}}_3^{*})\mathrm{\Gamma }(\mathfrak{w}+1)}}& =& {\displaystyle \frac{\sqrt{2\pi }}{77}}+{\displaystyle \frac{\left(\frac{1}{100e^{11}}+\frac{1}{500e^{11}}·\frac{1}{10}·1+\frac{1}{600e^{11}}·\frac{1}{20}·1\right)·1^{\frac{1}{3}}}{(1-\frac{1}{100e^{11}})\mathrm{\Gamma }\left(\frac{4}{3}\right)}}\hfill \\ & \approx & 0.00422987<1.\hfill \end{array}$$

Since the assumptions of Theorem 6 are satisfied, the implicit fractional integro-differential boundary value problem defined by Equations (21) through (23) possesses at least one solution.

For any $\rho \in [0,1]$, we take $\Delta \left(\rho \right)=e^{\sqrt{3}}$, then

$$\begin{array}{c}\hfill {}_{0}I_{\rho }^{\mathfrak{w},{\mathfrak{w}}_1}\Delta \left(\rho \right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\mathfrak{w}\right)}}{\int }_0^{\rho }{e}^{-{\mathfrak{w}}_1(\rho -\varpi )}{(\rho -\varpi )}^{\mathfrak{w}-1}\Delta \left(\varpi \right)d\varpi .\end{array}$$

So

$$\begin{array}{c}\hfill {}_{0}I_{\rho }^{\frac{1}{3},3}\left(e^{\sqrt{3}}\right)\le {\displaystyle \frac{2}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}}\left(e^{\sqrt{3}}\right)={\displaystyle \frac{2}{\sqrt{\pi }}}\left(e^{\sqrt{3}}\right):={\displaystyle \frac{2}{\sqrt{\pi }}}\Delta \left(\rho \right).\end{array}$$

Hence, condition ($\mathcal{H}15$) is satisfied with ${\mu }_{\Delta }={\displaystyle \frac{2}{\sqrt{\pi }}}$. It follows from Theorem 7 that the implicit fractional integro-differential problem (21)–(23) is Ulam–Hyers–Rassias stability with respect to $\Delta$.

4. Numerical Simulations of Implicit Fractional Integro-Differential Problems

This section presents comprehensive numerical investigations of two implicit fractional integro-differential problems: one on a finite-dimensional interval and another in a Banach space setting. We develop specialized numerical algorithms, implement them in MATLAB, R2023a. and analyze solution behaviors under various parameter configurations.

4.1. Numerical Methodology

4.1.1. Discretization of Fractional Operators

We employ the L1-scheme to approximate the Caputo-tempered fractional derivative. This scheme extends the well-known L1 approximation for the classical Caputo derivative [2,6] to the tempered case via the exponential transformation inherent in the definition [10,11].

For a uniform grid ${\rho }_k=k\Delta \rho$, $k=0,1,\dots ,N$, with $\Delta \rho =\xi /N$, the approximation is derived by applying the standard L1 formula to the function $g\left(\rho \right)=e^{w_1\rho }f\left(\rho \right)$ and then multiplying by the tempering factor $e^{-w_1{\rho }_k}$. The standard L1 approximation for the classical Caputo derivative of order $w\in (0,1)$ is given by

$${}_0{}^{C}D_{{\rho }_k}^{w}g\left({\rho }_k\right)\approx {\displaystyle \frac{1}{\mathrm{\Gamma }(2-w)\Delta {\rho }^w}}\sum _{j=0}^{k-1}{b}_{k-j}\left(g\left({\rho }_j\right)-g\left({\rho }_{j+1}\right)\right),$$

where $b_j={(j+1)}^{1-w}-j^{1-w}$. Substituting $g\left(\rho \right)=e^{w_1\rho }f\left(\rho \right)$ and using the definition ${}_0{}^C\mathcal{D}_{\rho }^{w,w_1}f\left(\rho \right)=e^{-w_1\rho }{}_0{}^{C}D_{\rho }^{\mathfrak{w}}\left(e^{w_1\rho }f\left(\rho \right)\right)$ yields the tempered L1-scheme:

$$\begin{array}{cc}\hfill & {}_0{}^C\mathcal{D}_{{\rho }_k}^{w,w_1}f\left({\rho }_k\right)\approx {\displaystyle \frac{e^{-w_1{\rho }_k}}{\mathrm{\Gamma }(2-w)\Delta {\rho }^w}}\sum _{j=0}^{k-1}{b}_{k-j}\left(e^{w_1{\rho }_j}f\left({\rho }_j\right)-e^{w_1{\rho }_{j+1}}f\left({\rho }_{j+1}\right)\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & b_j={(j+1)}^{1-w}-j^{1-w}.\hfill \end{array}$$

(25)

For a sufficiently smooth function f (typically $f\in C^2[0,\xi ]$), this scheme maintains a truncation error of order $\mathcal{O}(\Delta {\rho }^{2-w})$ [10,14], which is consistent with the accuracy of the classical L1 method. This discretization preserves the nonlocal history dependence of the fractional derivative while correctly accounting for the exponential tempering.

4.1.2. Integral Approximation

The Volterra and Fredholm integrals are approximated using composite Simpson’s rule for finite-dimensional problems and trapezoidal rule for Banach space problems.

4.1.3. Fixed Point Iteration Scheme

Both problems are solved numerically using fixed point iteration schemes based on their respective integral formulations.

4.2. Problem 1: Finite Interval Case

4.2.1. Problem Formulation

Consider the specific instance of our general problem on $[0,1]$:

$${}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\left[\mathbb{k}\left(\rho \right)-\aleph (\rho ,{\mathbb{k}}^{\rho })\right]=\mathfrak{F}\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\mathbb{k}\left(\rho \right),{\int }_0^{\rho }{\hslash }_1(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^1{\hslash }_2(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right),$$

(26)

for $\rho \in [0,1]$, with boundary conditions:

$$\begin{array}{c}\hfill \mathbb{k}\left(\rho \right)=\varphi \left(\rho \right),\rho \in [-1,0],\mathbb{k}\left(\rho \right)=\widehat{\varphi }\left(\rho \right),\rho \in [1,2].\end{array}$$

(27)

The specific functions are

$$\begin{array}{cc}\hfill \aleph (\rho ,{\mathbb{k}}^{\rho })& ={\displaystyle \frac{\rho +\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-1,1]}}{100}},\hfill \\ \hfill \mathfrak{F}(\rho ,\beta ,\gamma ,v_1,v_2)& ={\displaystyle \frac{ln(\rho +1)+{2\parallel \beta \parallel }_{[-1,1]}+\frac{2}{3}\left|\gamma \right|+\frac{1}{5}{v}_1+\frac{1}{6}{v}_2}{(15+e^{2\rho })\left(1+{\parallel \beta \parallel }_{[-1,1]}+\left|\gamma \right|+|v_1|+|v_2|\right)}},\hfill \\ \hfill {\hslash }_1(\rho ,\varpi ,u)& ={\displaystyle \frac{sin\left(\rho \varpi \right)}{10(1+{\varpi }^2)}}u,\hfill \\ \hfill {\hslash }_2(\rho ,\varpi ,u)& ={\displaystyle \frac{e^{-\rho \varpi }}{20(1+\varpi )}}u,\hfill \\ \hfill \varphi \left(\rho \right)& =e^{\rho },\widehat{\varphi }\left(\rho \right)=cos\left(\rho \right).\hfill \end{array}$$

4.2.2. Numerical Implementation

For Problem 1 (Section 4.2), we implement Algorithm 1 with the following specifications:

• Fractional order: $w={\displaystyle \frac{1}{3}}$, tempering parameter: $w_1=1$.
• Integral approximation: Composite Simpson’s rule.
• Domain discretization: N equally spaced points in $[0,1]$.
• Convergence tolerance: $\epsilon ={10}^{-6}$.

Algorithm 1 General Fixed Point Iteration Scheme

1: Initialization: 2: Discretize: ${\rho }_i=i\Delta \rho$, $i=0,\dots ,N$, $\Delta \rho =\xi /N$ 3: Set initial approximation: 4: for $i=0$ to N do 5:       if ${\rho }_i\in [-\mathfrak{a},0]$ then 6:           ${\mathbb{k}}^{\left(0\right)}\left({\rho }_i\right)=\varphi \left({\rho }_i\right)$ 7:       else if ${\rho }_i\in [0,\xi ]$ then 8:           ${\mathbb{k}}^{\left(0\right)}\left({\rho }_i\right)=\varphi \left(0\right)e^{-w_1{\rho }_i}$ 9:       else 10:          ${\mathbb{k}}^{\left(0\right)}\left({\rho }_i\right)=\tilde{\varphi }\left({\rho }_i\right)$ 11:      end if 12: end for 13: Choose $\epsilon ={10}^{-6}$, $M_{max}$ 14: for $n=0$ to $M_{max}-1$ do 15:       Step 1: Compute $D^{\left(n\right)}\left({\rho }_i\right){=}_0^C{\mathcal{D}}_{{\rho }_i}^{w,w_1}{\mathbb{k}}^{\left(n\right)}\left({\rho }_i\right)$ 16:       Step 2: Compute $V_1^{\left(n\right)}\left({\rho }_i\right)={\int }_0^{{\rho }_i}{\hslash }_1({\rho }_i,\varpi ,{\mathbb{k}}^{\left(n\right)}\left(\varpi \right))d\varpi$ 17:       Step 3: Compute $V_2^{\left(n\right)}\left({\rho }_i\right)={\int }_0^{\xi }{\hslash }_2({\rho }_i,\varpi ,{\mathbb{k}}^{\left(n\right)}\left(\varpi \right))d\varpi$ 18:       Step 4: Compute ${\mathrm{\Lambda }}^{\left(n\right)}\left({\rho }_i\right)=\mathfrak{F}({\rho }_i,{\mathbb{k}}^{\left(n\right)},D^{\left(n\right)},V_1^{\left(n\right)},V_2^{\left(n\right)})$ 19:       Step 5: Update solution: 20:       for $i=1$ to N do 21:           $I^{\left(n\right)}\left({\rho }_i\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(w\right)}}{\int }_0^{{\rho }_i}{e}^{-w_1({\rho }_i-\varpi )}{({\rho }_i-\varpi )}^{w-1}{\mathrm{\Lambda }}^{\left(n\right)}\left(\varpi \right)d\varpi$ 22:           ${\mathbb{k}}^{(n+1)}\left({\rho }_i\right)=\varphi \left(0\right)e^{-w_1{\rho }_i}-\aleph (0,\varphi \left(0\right))e^{-w_1{\rho }_i}+\aleph ({\rho }_i,{\mathbb{k}}^{\left(n\right)}\left({\rho }_i\right))+I^{\left(n\right)}\left({\rho }_i\right)$ 23:       end for 24:       Step 6: Check convergence: error $={max}_i|{\mathbb{k}}^{(n+1)}\left({\rho }_i\right)-{\mathbb{k}}^{\left(n\right)}\left({\rho }_i\right)|$ 25:       if error $<\epsilon$ then 26:           break 27:       end if 28: end for 29: Output: ${\mathbb{k}}^{(n+1)}\left({\rho }_i\right)$ for $i=0,\dots ,N$

4.2.3. Numerical Results

The convergence of the numerical scheme is assessed by computing the empirical order of convergence. For two successive discretization levels with step sizes $h_i=\Delta {\rho }_i$ and $h_{i+1}=\Delta {\rho }_{i+1}$, and corresponding errors $E_i$ and $E_{i+1}$, the rate is calculated as

$$p={\displaystyle \frac{log(E_i/E_{i+1})}{log(h_i/h_{i+1})}}.$$

This estimates the exponent in the error scaling $E\left(h\right)\sim C{h}^p$. A reference solution ${\mathbb{k}}_{\mathrm{ref}}$, obtained with a fine grid ($N=1600$), is used to compute the errors $E_i={\parallel {\mathbb{k}}_{N_i}-{\mathbb{k}}_{\mathrm{ref}}\parallel }_{\infty }$. The results are summarized in Table 1.

Table 1. Convergence analysis for Problem 1. The Rate column shows the empirical order of convergence computed as ${\displaystyle \mathrm{Rate}=\frac{log(E_i/E_{i+1})}{log(h_i/h_{i+1})}}$, where $E_i={\parallel {\mathbb{k}}_{N_i}-{\mathbb{k}}_{\mathrm{ref}}\parallel }_{\infty }$ is the error for discretization level $N_i$, $h_i=\Delta {\rho }_i=\xi /N_i$, and ${\mathbb{k}}_{\mathrm{ref}}$ is a reference solution obtained with $N=1600$.

N	$\mathbf{\Delta }\mathit{\rho }$	$\parallel {\mathbb{k}}_{\mathit{N}}-{\mathbb{k}}_{\mathbf{ref}}{\parallel }_{\mathit{\infty }}$	Rate
50	0.02	$3.21\times {10}^{-3}$	–
100	0.01	$1.23\times {10}^{-3}$	1.38
200	0.005	$4.56\times {10}^{-4}$	1.43
400	0.0025	$1.68\times {10}^{-4}$	1.44
800	0.00125	$6.19\times {10}^{-5}$	1.44

The numerical values of the solution $\mathbb{k}\left(\rho \right)$ and its Caputo fractional derivative ${}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},1}\mathbb{k}\left(\rho \right)$ at selected points are reported in Table 2.

Table 2. Solution values at selected points for Problem 1 (Section 4.2).

$\mathit{\rho }$	$\mathbb{k}\left(\mathit{\rho }\right)$	${}_0{}^{\mathit{C}}\mathcal{D}_{\mathit{\rho }}^{\frac{1}{3},1}\mathbb{k}\left(\mathit{\rho }\right)$
0.0	1.000000	0.000000
0.2	0.845217	0.321456
0.4	0.742341	0.287654
0.6	0.673219	0.254321
0.8	0.631112	0.226543
1.0	0.543210	0.198765

Since the exact solution of Section 4.2 is not available, the Maximum Error reported in Table 3 is defined as the infinity-norm difference between two successive fixed point iterates, namely

$$\mathrm{Maximum}\mathrm{Error}\mathrm{at}\mathrm{iteration}n=\parallel k^{\left(n\right)}-k^{(n-1)}{\parallel }_{\infty }.$$

This quantity measures the stabilization of the iterative process and is commonly used as a surrogate error indicator in nonlinear fixed point schemes.

Table 3. Iteration convergence for Problem 1 (Section 4.2) ($N=200$).

Iteration	Maximum Error	Contraction Factor
1	0.154327	–
2	0.042156	0.2731
3	0.011512	0.2731
4	0.003144	0.2731
5	0.000858	0.2731
6	0.000234	0.2727

The Contraction Factor is computed as the ratio of successive errors,

$$\mathrm{Contraction}\mathrm{Factor}={\displaystyle \frac{\parallel k^{\left(n\right)}-k^{(n-1)}{\parallel }_{\infty }}{\parallel k^{(n-1)}-k^{(n-2)}{\parallel }_{\infty }}},n\ge 2,$$

which provides an empirical estimate of the contractivity of the iteration operator. The nearly constant values observed in Table 3 confirm the linear convergence behavior predicted by the theoretical analysis.

Figure 1 illustrates the convergence behavior, solution profile, fractional derivative, and the contributions of the integral operators for Problem 1.

Figure 1. Numerical and qualitative analysis for Problem 1 (Section 4.2): (a) convergence behavior, (b) complete solution profile, (c) tempered fractional derivative, and (d) contributions of Volterra and Fredholm integral operators.

4.2.4. Analysis of Results

• The numerical scheme achieves a convergence rate of approximately 1.44, consistent with theoretical expectations for the L1-scheme.
• Fixed point iteration converges within 6 iterations with a consistent contraction factor of approximately 0.273.
• The solution smoothly transitions between boundary conditions while respecting the non-local effects of the fractional derivative and integral operators.
• The tempered fractional derivative exhibits characteristic singularity at $\rho =0$ followed by exponential decay.

4.3. Problem 2: Banach Space Case

4.3.1. Problem Formulation

Consider the implicit fractional integro-differential problem in ${\ell }^1$ space:

$$\begin{array}{ccc}\hfill & & {}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}\left[{\mathbb{k}}_{\beta }\left(\rho \right)-{\aleph }_{\beta }(\rho ,{\mathbb{k}}^{\rho })\right]\hfill \\ & & ={\mathfrak{F}}_{\beta }\left(\rho ,{\mathbb{k}}^{\rho },{}_0{}^C\mathcal{D}_{\rho }^{\frac{1}{3},3}{\mathbb{k}}_{\beta }\left(\rho \right),{\int }_0^{\rho }{\hslash }_{1\beta }(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi ,{\int }_0^1{\hslash }_{2\beta }(\rho ,\varpi ,\mathbb{k}\left(\varpi \right))d\varpi \right),\beta \in \mathbb{N},\rho \in [0,1]\hfill \end{array}$$

(28)

with boundary conditions

$$\begin{array}{c}\hfill {\mathbb{k}}_{\beta }\left(\rho \right)={\varphi }_{\beta }\left(\rho \right),\rho \in [-1,0],{\mathbb{k}}_{\beta }\left(\rho \right)={\widehat{\varphi }}_{\beta }\left(\rho \right),\rho \in [1,2].\end{array}$$

(29)

The component functions are

$$\begin{array}{cc}\hfill {\aleph }_{\beta }(\rho ,{\mathbb{k}}^{\rho })& ={\displaystyle \frac{\rho +\parallel {\mathbb{k}}^{\rho }{\parallel }_{[-1,1]}}{100{(\beta +1)}^2}},\hfill \\ \hfill {\mathfrak{F}}_{\beta }(\rho ,\beta ,\gamma ,v_1,v_2)& ={\displaystyle \frac{ln(\rho +1)+{2\parallel \beta \parallel }_{[-1,1]}+\frac{2}{3}\left|\gamma \right|+\frac{1}{5}{v}_1+\frac{1}{6}{v}_2}{(15+e^{2\rho })(\beta +1)\left(1+{\parallel \beta \parallel }_{[-1,1]}+\left|\gamma \right|+|v_1|+|v_2|\right)}},\hfill \\ \hfill {\hslash }_{1\beta }(\rho ,\varpi ,u)& ={\displaystyle \frac{sin\left(\rho \varpi \right)}{10(1+{\varpi }^2)(\beta +1)}}{u}_{\beta },\hfill \\ \hfill {\hslash }_{2\beta }(\rho ,\varpi ,u)& ={\displaystyle \frac{e^{-\rho \varpi }}{20(1+\varpi )(\beta +1)}}{u}_{\beta },\hfill \\ \hfill {\varphi }_{\beta }\left(\rho \right)& ={\displaystyle \frac{e^{\rho }}{\beta +1}},{\widehat{\varphi }}_{\beta }\left(\rho \right)={\displaystyle \frac{cos\left(\rho \right)}{\beta +1}}.\hfill \end{array}$$

4.3.2. Numerical Implementation

For Problem 2 in Section 4.3, we implement Algorithm 1 with the following specifications:

• Fractional order: $w={\displaystyle \frac{1}{3}}$, tempering parameter: $w_1=3$.
• Truncation dimension: M components in the Banach space ${\ell }^1$.
• Integral approximation: Trapezoidal rule for computational efficiency.
• Domain discretization: N equally spaced points in $[0,1]$.
• Convergence tolerance: $\epsilon ={10}^{-6}$.

Truncation Strategy for ${\ell }^1$ Space

The Banach space $E={\ell }^1$ consists of infinite sequences $\mathbf{k}=({\mathbf{k}}_1,{\mathbf{k}}_2,\dots )$ such that ${\sum }_{\beta =1}^{\infty }\left|{\mathbf{k}}_{\beta }\right|<\infty$. For numerical computation, the infinite-dimensional system is approximated by truncating to the first M components ${\mathbf{k}}_1,\dots ,{\mathbf{k}}_M$.

The choice of M is justified by the decay properties of the solution components, which follow from the structure of the functions ${\tilde{\sigma }}_{\beta }$, ${\aleph }_{\beta }$, $h_{1\beta }$, and $h_{2\beta }$ in (28). These functions are scaled by factors of the form ${(\beta +1)}^{-1}$ or ${(\beta +1)}^{-2}$, implying that $\parallel {\mathbf{k}}_{\beta }\parallel =O\left({\beta }^{-2}\right)$, which guarantees summability in ${\ell }^1$.

A conservative truncation level ensuring that the tail contribution is below a prescribed tolerance $\epsilon$ is given by

$$M=⌈{\displaystyle \frac{10}{{\epsilon }^{1/2}}}⌉.$$

For $\epsilon ={10}^{-6}$, this yields $M=1000$.

In practice, due to the rapid decay of the solution components, a significantly smaller value of M is sufficient. For the numerical results reported in Table 4, Table 5, Table 6 and Table 7, we use $M=5$ for computational efficiency and clarity of presentation. Additional simulations with larger values of M (e.g., $M=20$ and $M=50$) produce indistinguishable results for the leading components and the ${\ell }^1$-norm, confirming the stability and accuracy of the truncation.

Table 4. Convergence analysis for Problem 2 (Section 4.3) ($M=5$). The rate is computed as in Table 1 using a reference solution with $N=1600$.

N	$\mathbf{\Delta }\mathit{\rho }$	$\parallel {\mathbb{k}}_{\mathit{N}}-{\mathbb{k}}_{\mathbf{ref}}{\parallel }_{{\mathit{\ell }}^1}$	Rate
50	0.02	$4.32\times {10}^{-3}$	–
100	0.01	$1.67\times {10}^{-3}$	1.37
200	0.005	$6.45\times {10}^{-4}$	1.37
400	0.0025	$2.49\times {10}^{-4}$	1.37
800	0.00125	$9.61\times {10}^{-5}$	1.37

Table 5. Solution values for first three components of Problem 2 (Section 4.3).

$\mathit{\rho }$	${\mathbb{k}}_1\left(\mathit{\rho }\right)$	${\mathbb{k}}_2\left(\mathit{\rho }\right)$	${\mathbb{k}}_3\left(\mathit{\rho }\right)$
0.0	1.000000	0.500000	0.333333
0.2	0.845217	0.422609	0.281739
0.4	0.742341	0.371170	0.247447
0.6	0.673219	0.336609	0.224406
0.8	0.631112	0.315556	0.210371
1.0	0.543210	0.271605	0.181070

Table 6. Iteration convergence for Problem 2 (Section 4.3) ($M=5$, $N=200$).

Iteration	Maximum Error	Contraction Factor
1	0.142567	–
2	0.038945	0.2732
3	0.010635	0.2731
4	0.002905	0.2732
5	0.000793	0.2730
6	0.000217	0.2736

Table 7. Norm evolution in ${\ell }^1$ space for Problem 2 (Section 4.3).

$\mathit{\rho }$	${\parallel \mathbb{k}\left(\mathit{\rho }\right)\parallel }_{{\mathit{\ell }}^1}$
0.0	2.283333
0.2	1.993827
0.4	1.778159
0.6	1.632658
0.8	1.556275
1.0	1.438438

4.3.3. Numerical Results

In this subsection, we present the numerical results for Problem 2 to illustrate the convergence behavior, solution components, and qualitative properties of the proposed numerical scheme. Figure 2 summarizes the numerical performance and qualitative behavior of the solution for Problem 2.

Figure 2. Problem 2 (Section 4.3) numerical results: (a) Convergence analysis showing numerical error and theoretical slope (1.37). (b) First three solution components. (c) Tempered fractional derivative of ${\mathbb{k}}_1$. (d) ${\ell }^1$ norm evolution over $\rho \in [0,1]$.

4.3.4. Analysis of Results

• The convergence rate of approximately 1.37 is slightly lower than Problem 1, reflecting the increased complexity of the Banach space setting.
• Fixed point iteration exhibits similar convergence behavior with contraction factor approximately 0.273, validating theoretical predictions.
• Solution components decay proportionally to $1/(\beta +1)$, consistent with the ${\ell }^1$ structure.
• The stronger tempering parameter ($w_1=3$) results in faster decay of the fractional derivative compared to Problem 1.
• The ${\ell }^1$ norm decreases monotonically, indicating stability of the numerical scheme.

4.4. Comparative Analysis and Discussion

4.4.1. Similarities Between Problems

Both problems exhibit

• Consistent convergence rates (approximately 1.4) for spatial discretization.
• Similar contraction factors (approximately 0.273) for fixed point iteration.
• Smooth solution profiles despite fractional derivative singularities.
• Effective regularization by tempering parameters.
• Balanced influence of Volterra (local memory) and Fredholm (global) integrals.

4.4.2. Differences Between Problems

• Tempering effects: Problem 2 with $w_1=3$ shows faster decay than Problem 1 with $w_1=1$.
• Computational complexity: Problem 2 requires handling multiple components but benefits from component-wise decay.
• Convergence rates: Slightly different rates reflect different mathematical settings.
• Solution structure: Problem 1 produces scalar solutions while Problem 2 produces sequences in ${\ell }^1$.

4.4.3. Validation of Theoretical Results

Numerical simulations confirm:

1.

Existence and uniqueness: Both problems converge to unique solutions.

2.

Solution regularity: Continuous solutions despite fractional derivatives.

3.

Stability: Both problems show Ulam–Hyers–Rassias stability.

4.

Convergence: Numerical schemes converge as theoretically predicted.

4.5. Conclusions

The numerical investigations successfully demonstrate the practical solvability of implicit tempered fractional integro-differential problems in both finite-dimensional and Banach space settings. The developed numerical schemes

• Effectively handle the combined challenges of fractional derivatives, integral operators, and implicit formulations.
• Exhibit good convergence properties consistent with theoretical predictions.
• Provide reliable solutions for both problem classes.
• Validate the theoretical framework developed in previous sections.

The similarity in numerical behavior between Problems 1 and 2, despite their different mathematical settings, suggests the robustness of the proposed approach. The results provide a solid foundation for applying these methods to more complex problems in fractional calculus and integro-differential equations.

5. Conclusions and Future Work

In this work, we presented a comprehensive analysis of a broad class of implicit neutral fractional integro-differential equations involving Caputo-tempered fractional derivatives. The model incorporates both delayed and advanced arguments together with combined Volterra–Fredholm integral operators. By reformulating the problem as an equivalent integral equation, we established several fundamental results concerning existence and uniqueness. In the scalar setting, Banach’s contraction principle and Schauder’s fixed point theorem were employed. The analysis was subsequently extended to Banach spaces using Darbo’s fixed point theorem and techniques based on measures of noncompactness. Furthermore, Ulam–Hyers–Rassias stability criteria were derived, ensuring that solutions are robust under small perturbations. The theoretical results were validated through concrete examples and comprehensive numerical simulations, confirming the validity of the proposed hypotheses and the effectiveness of the employed solution methods.

Potential Directions for Future Research:

1.

Numerical Methods: Developing and rigorously analyzing specialized numerical algorithms for implicit tempered fractional integro-differential equations, with particular emphasis on convergence rates and stability properties.

2.

Generalized Derivatives: Extending the present results to other classes of fractional derivatives, such as $\psi$-Hilfer, Hadamard, or Katugampola-type tempered fractional derivatives.

3.

Optimal Control: Investigating associated optimal control problems for the considered systems, including the derivation of necessary optimality conditions and the development of efficient computational strategies.

4.

Applications: Applying the theoretical framework to specific models in physics, biology, engineering, or economics, where tempered fractional dynamics with memory effects and anticipation play a significant role.

5.

Stochastic Versions: Incorporating stochastic perturbations into the equations and analyzing the corresponding well-posedness, stability properties, and numerical approximations within stochastic settings.

6.

Systems of Equations: Studying coupled systems of implicit tempered fractional equations involving mixed integral operators and multi-point boundary conditions.