On a Nonlinear Proportional Fractional Integro-Differential Equation with Functional Boundary Conditions: Existence, Uniqueness, and Ulam–Hyers Stability

Abstract

This work introduces a new category of proportional fractional integro-differential equations (PFIDEs) governed by functional boundary conditions. We derive verifiable sufficient criteria that guarantee the Ulam–Hyers Stability, existence and uniqueness of solutions to this problem. Our analytical approach leverages Babenko’s method to construct an inverse operator, which allows us to reformulate the differential problem into an equivalent integral equation. The analysis is then conducted using key mathematical tools, including contraction mapping principle of Banach, the Leray–Schauder alternative, and properties of multivariate Mittag–Leffler functions. The Ulam–Hyers Stability is rigorously examined to assess the system’s resilience to small perturbations. The applicability and effectiveness of the established theoretical results are demonstrated through two illustrative examples. This research provides a unified and adaptable framework that advances the analysis of complex fractional-order dynamical systems subject to nonlocal constraints.

1. Introduction

Let $m,l$ be two positive integers and consider constants ${\rho }_i\in \mathbb{R}$ for $1\le i\le m$. For $1\le j\le l$, assume we are given functions ${{\rm Y}}_j,\varphi :[0,1]\times \mathbb{R}\to \mathbb{R}$ and a functional $\xi :C[0,1]\to \mathbb{R}$. Building upon recent developments in [1], this paper is devoted to analyzing the FIDEs with a nonlocal boundary condition defined, for $1<\alpha \le 2$, by

$$\left\{\begin{array}{c}{\displaystyle {}^{C}D_0^{\alpha ,\lambda }\left[𝓌\left(x\right)-\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }𝓌\left(x\right)-\sum _{j=1}^l{I}_0^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\right]=\varphi \left(x,𝓌\left(x\right)\right),x\in (0,1),}\hfill \\ {\displaystyle 𝓌\left(0\right)=\zeta ,𝓌\left(1\right)=\xi \left(𝓌\right),}\hfill \end{array}\right.$$

(1)

where $\zeta \in \mathbb{R}$ is a constant, and ${\rho }_i\in \mathbb{R},{\beta }_i>0,{\alpha }_j>0$ for all $i=1,\dots ,m$ and $j=1,\dots ,l.$ Our central aim is to establish criteria governing the existence and uniqueness of solutions to this problem, as well as its Ulam–Hyers stability.

To the best of our knowledge, the specific structure of the Boundary Value Problem (BVP) (1) is new. The principal impetus for this investigation stems from the effective use of inverse operator theory applied to fractional integrals which is bounded in spaces of Banach. This technique yields a coherent and systematic framework for examining a broad class of PFIDEs subject to functional boundary conditions. Consequently, our work not only extends the applicability of existing methodologies but also presents a consolidated approach for probing fundamental properties like stability, existence and uniqueness in nonlocal BVP.

The scope of boundary value problems has significantly broadened from classical formulations to encompass fractional differential equations, integro-differential systems, and models with impulsive effects [2,3,4,5]. These advanced frameworks are particularly adept at capturing real-world phenomena characterized by memory effects and discontinuous transitions [6,7,8,9]. This expansion has spurred sustained investigative efforts into fundamental questions of existence, uniqueness, and stability of solutions, alongside the development of numerical methods [10,11,12,13]. Illustrative of this progress, Adigüzel et al. [14] recast a nonlinear fractional BVP involving Riemann–Liouville derivatives into an equivalent fixed-point problem, thereby deriving new criteria for solution existence. In a related vein, Li et al. [1] utilized fixed-point theory combined with Babenko’s technique to prove the Ulam–Hyers stability, existence and uniqueness for a specific category of FIDEs. Meanwhile, Kumar and Malik [15] contributed to these advancements by studying FIDEs with non-instantaneous impulses under periodic boundary conditions on time scales, thus extending classical conclusions to more dynamic settings. In [16], the authors investigated existence results for fractional delta-nabla difference equations with mixed boundary conditions, employing the contraction principle of Banach and the fixed point theorem of Schauder. Their model, however, incorporates integral terms that apply only to nonlinear functions of the unknown solution.

This reveals a clear gap in the literature: the analysis of equations where the fractional differential operator itself is proportionally coupled with a linear integral operator acting directly on the unknown function. Such a structure is not merely a technical extension but a substantive generalization that models systems where the rate of change depends on a tempered difference between the current state and its weighted past history. The presence of the linear integral term ${\sum }_{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }𝓌\left(x\right)$ inside the derivative creates a composite operator whose inversion is fundamentally more challenging than the cases studied in [1,14]. Consequently, existing analytical frameworks cannot be directly applied.

The principal impetus for this investigation is to bridge this gap. We develop a novel methodology that extends Babenko’s technique to deconstruct and invert this composite operator, thereby establishing a unified framework for a new category of nonlinear proportional fractional integro-differential equations subject to functional boundary conditions. Our central aim is to derive verifiable sufficient criteria that guarantee the Hyers-Ulam stability, uniqueness and existence of solutions to this more complex and previously unaddressed problem.

The organization of this paper is as follows: Section 2 provides the necessary definitions and preliminary results required for the subsequent analysis. In Section 3, we transform the original problem presented in (1) into an equivalent integral equation using inverse operator theory and analyze the uniqueness of the solutions using the fixed point of Banach. Existence is established via the fixed-point theorem of Leray–Schauder. Section 4 focuses on the stability in the sense of Hyers–Ulam of the system, deriving sufficient conditions for stability. Finally, Section 5 showcases the theoretical results through two examples.

2. Preliminarily

Definition 1

([17]). Let $𝓌\in {\mathcal{L}}^1[a,b]$, $\lambda \ge 0$ and $\alpha >0$. The fractional integral in the sense of tempered Riemann-Liouville is given by

$$\begin{array}{ccc}{\displaystyle I_a^{\alpha ,\lambda }𝓌\left(x\right)}\hfill & =\hfill & {\displaystyle exp(-\lambda x)I_a^{\alpha }\left(exp\left(\lambda x\right)𝓌\left(x\right)\right)}\hfill \\ & =\hfill & {\displaystyle \frac{exp(-\lambda x)}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_a^x{\left(x-s\right)}^{\alpha -1}exp\left(\lambda s\right)𝓌\left(s\right)ds}\hfill \\ & =\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_a^x{\left(x-s\right)}^{\alpha -1}exp\left(-\lambda \left(x-s\right)\right)𝓌\left(s\right)ds,}\hfill \end{array}$$

where $I_a^{\alpha }𝓌\left(x\right)$ denotes the Riemann-Liouville fractional integral of ordre $\alpha .$

Definition 2

([17]). Let $𝓌\in {\mathcal{L}}^1[a,b]$, $\lambda \ge 0$ and $n-1<\alpha <n,n\in {\mathbb{N}}^{*}.$ The fractional derivative in the sense of tempered Riemann- Liouville is given by

$$\begin{array}{ccc}{\displaystyle D_a^{\alpha ,\lambda }𝓌\left(x\right)}\hfill & =\hfill & {\displaystyle exp(-\lambda x)D_a^{\alpha }\left(exp\left(\lambda x\right)𝓌\left(x\right)\right)}\hfill \\ & =\hfill & {\displaystyle \frac{exp(-\lambda x)}{\mathsf{\Gamma }(n-\alpha )}\frac{d^n}{d{x}^n}{\int }_a^x{\left(x-s\right)}^{n-\alpha -1}exp\left(\lambda s\right)𝓌\left(s\right)ds}\hfill \\ & =\hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(n-\alpha )}\frac{d^n}{d{x}^n}{\int }_a^x{\left(x-s\right)}^{n-\alpha -1}exp\left(-\lambda \left(x-s\right)\right)𝓌\left(s\right)ds,}\hfill \end{array}$$

where $D_a^{\alpha }𝓌\left(x\right)$ denotes the Riemann-Liouville fractional derivative of ordre $\alpha .$

Definition 3

([17]). Let $\lambda \ge 0,$ $n\in {\mathbb{N}}^{*}$, $n-1<\alpha <n$ and $𝓌\left(x\right)\in A{C}^n[a,b].$ The fractional derivative in the sense of tempered Caputo is given by

$$\begin{array}{ccc}{\displaystyle {}^{C}D_a^{\alpha ,\lambda }𝓌\left(x\right)}\hfill & =\hfill & {\displaystyle exp{(-\lambda x)}^C{D}_a^{\alpha }\left(exp\left(\lambda x\right)𝓌\left(x\right)\right)}\hfill \\ & =\hfill & {\displaystyle \frac{exp(-\lambda x)}{\mathsf{\Gamma }(n-\alpha )}{\int }_a^x{\left(x-s\right)}^{n-\alpha -1}\frac{d^n}{d{s}^n}\left(exp\left(\lambda s\right)𝓌\left(s\right)\right)ds,}\hfill \end{array}$$

where ${}^{C}D_a^{\alpha }𝓌\left(x\right)$ denotes the Caputo fractional derivative of ordre $\alpha .$

Definition 4

([18]). The multivariate Mittag–Leffler function is given by the following expression:

$$\begin{array}{cc}& {\mathcal{E}}_{({\delta }_1,{\delta }_2,\dots ,{\delta }_p),\gamma }\left(v_1,v_2,\dots ,v_p\right)\hfill \\ & {\displaystyle =\sum _{n=0}^{\infty }\sum _{n_1+n_2+\dots +n_p=n}\left(\begin{array}{c}n\\ n_1,n_2,\dots ,n_p\end{array}\right){\displaystyle \frac{v_1^{n_1}{v}_2^{n_2}\dots v_p^{n_p}}{\mathsf{\Gamma }\left({\delta }_1{n}_1+{\delta }_2{n}_2+\dots +{\delta }_p{n}_p+\gamma \right)}},}\hfill \end{array}$$

for $v=\left(v_1,v_2,\dots ,v_p\right)\in {\mathbb{C}}^p$ and parameters ${\delta }_i,\gamma >0$ where $i=1,2,\dots ,p$. The multinomial coefficient in the series is defined as

$$\left(\begin{array}{c}n\\ n_1,n_2,\dots ,n_p\end{array}\right)={\displaystyle \frac{n!}{n_1!n_2!\dots n_p!}}.$$

Theorem 1.

Let $(X,\parallel ·\parallel )$ be a Banach space, $0<k<1$ and let $T:X\to X$ such that

$$∥T\left(x\right)-T\left(y\right)∥\le k∥x-y∥,\forall x,y\in X.$$

Thus, T possesses exactly one fixed point.

Theorem 2.

Let X be a Banach space and let $T:X\to X$ be a completely continuous mapping. Define

$$\mathsf{\Lambda }=\{\mathcal{U}\in X:=\lambda T(\mathcal{U}),\lambda \in [0,1\left]\right\}.$$

If the collection Λ is bounded, then T possesses a fixed point in X.

3. Existence of Solutions

In this part, we provide sufficient conditions for the uniqueness and existence of solutions to our problem (1), based on the fixed-point theorem of Banach. Moreover, we establish the existence of solutions based on the fixed-point theorem of Schaefer. In the rest of the paper, we set $𝓜(s,\rho )=\left(\begin{array}{c}s\\ s_1,s_2,\dots ,s_m\end{array}\right){\rho }_1^{s_1}\dots {\rho }_m^{s_m}$ and $𝓜(s,\left|\rho \right|)=\left(\begin{array}{c}s\\ s_1,s_2,\dots ,s_m\end{array}\right){\left|{\rho }_1\right|}^{s_1}\dots {\left|{\rho }_m\right|}^{s_m}$

Lemma 1.

Suppose that $m,l$ two positive integers, ${{\rm Y}}_j,\varphi :[0,1]\times \mathbb{R}\to \mathbb{R}$ for $1\le j\le l$ are bounded and continuous, and $\xi :C[0,1]\to \mathbb{R}$ is bounded. Consequently, the BVP (1) can be reformulated as the implicit integral equation given by:

$$\begin{array}{ccc}𝓌\left(x\right)\hfill & =\hfill & {\displaystyle \sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\zeta exp(-\lambda x)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \left[\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}-\frac{x^{{\beta }_1^{s_1}+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right]}\hfill \\ & \hfill & {\displaystyle +\xi \left(𝓌\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -exp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -exp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}.}\hfill \end{array}$$

Moreover, if

$$\varrho =1-exp\left(\lambda \right){\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{i=1}^m{\displaystyle \frac{\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}}>0,$$

(2)

then,

$$\begin{array}{ccc}∥𝓌∥\hfill & \le \hfill & {\displaystyle \frac{1}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\frac{1}{\varrho }\sum _{j=1}^l{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),{\alpha }_j+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥{{\rm Y}}_j\left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\frac{\left|\zeta \right|}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+\frac{\left|\xi \left(𝓌\right)\right|exp\left(\lambda \right)}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & \hfill & {\displaystyle +\frac{exp\left(\lambda \right)}{\varrho \mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\frac{exp\left(\lambda \right)}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{j=1}^l\frac{∥{{\rm Y}}_j\left(.,.\right)∥}{\mathsf{\Gamma }({\alpha }_j+1)}<\infty .}\hfill \end{array}$$

Remark 1.

The inverse operator of

$$\mathcal{A}:=\left(I-\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }\right)$$

is rigorously defined via the Neumann series

$${\mathcal{A}}^{-1}=\sum _{s=0}^{\infty }{\left(\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }\right)}^s,$$

provided the series converges in the operator norm. Under this framework, the inverse operator ${\mathcal{A}}^{-1}$ is therefore well-defined and bounded on the space $C[0,1]$, ensuring its applicability to all functions $u\in C[0,1]$.

Remark 2.

The application of Babenko’s method and the resulting series representation rely on the invertibility of the linear operator $(1-\sum {\rho }_i{I}^{{\beta }_i,\lambda })$ via its Neumann series. This is justified because the operator $\sum {\rho }_i{I}^{{\beta }_i,\lambda }$ is a contraction on $C\left(\right[0,1\left]\right)$ under a condition related to (2). Specifically, the standard estimate $\parallel I^{\gamma ,\lambda }w\parallel \le \parallel w\parallel /\mathsf{\Gamma }(\gamma +1)$ implies that $\sum {\rho }_i{I}^{{\beta }_i,\lambda }$ has operator norm at most $\kappa =\sum |{\rho }_i|/\mathsf{\Gamma }({\beta }_i+1)$. The stronger Condition (2), ensures not only that κ is sufficiently small to guarantee convergence of the geometric series, but also controls the additional exponential terms arising from the boundary conditions, thereby validating the well-posedness of the series representation in Lemma 1.

Proof. 

We start by applying the integral operator $I_a^{\alpha ,\lambda }$ to the first equation in (1) we find

$$𝓌\left(x\right)=c_{0}exp(-\lambda x)+c_{1}xexp(-\lambda x)+\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }𝓌\left(x\right)+\sum _{j=1}^l{I}_0^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)+I_0^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right).$$

The boundary conditions imply that:

$$\left\{\begin{array}{ccc}{c}_0\hfill & =\hfill & \zeta ,\hfill \\ c_1\hfill & =\hfill & {\displaystyle -exp\left(\lambda \right)I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)-exp\left(\lambda \right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)}\hfill \\ & \hfill & {\displaystyle -exp\left(\lambda \right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)+exp\left(\lambda \right)\xi \left(𝓌\right)-\zeta ,}\hfill \end{array}\right.$$

it implies that,

$$\begin{array}{ccc}𝓌\left(x\right)\hfill & =\hfill & {\displaystyle I_0^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)+\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }𝓌\left(x\right)+\sum _{j=1}^l{I}_0^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)+\zeta exp(-\lambda x)(1-x)}\hfill \\ & \hfill & {\displaystyle +xexp\left(\lambda (1-x)\right)\xi \left(𝓌\right)-xexp\left(\lambda (1-x)\right)I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle -xexp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)-xexp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \end{array}$$

Therefore, we have:

$$\begin{array}{cc}\hfill & {\displaystyle \left(1-\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }\right)𝓌\left(x\right)}\hfill \\ & {\displaystyle =I_0^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)+\sum _{j=1}^l{I}_0^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)+\zeta exp(-\lambda x)(1-x)}\hfill \\ & {\displaystyle +xexp\left(\lambda (1-x)\right)\xi \left(𝓌\right)-xexp\left(\lambda (1-x)\right)I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)}\hfill \\ & {\displaystyle -xexp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)}\hfill \\ & {\displaystyle -xexp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \end{array}$$

The existence of the inverse operator $\left(1-{\sum }_{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }\right)$ in $C[0,1]$ allows us to apply Babenko’s approach [19]. This yields directly:

$$\begin{array}{ccc}𝓌\left(x\right)\hfill & =\hfill & {\displaystyle {\left(1-{\sum }_{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }\right)}^{-1}[I_0^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)+\sum _{j=1}^l{I}_0^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\zeta exp(-\lambda x)(1-x)+xexp\left(\lambda (1-x)\right)\xi \left(𝓌\right)}\hfill \\ & \hfill & {\displaystyle -xexp\left(\lambda (1-x)\right)I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)-xexp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)}\hfill \\ & \hfill & {\displaystyle -xexp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)]}\hfill \\ & =\hfill & {\displaystyle \sum _{s=0}^{\infty }{\left(\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }\right)}^s[I_0^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)+\sum _{j=1}^l{I}_0^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\zeta exp(-\lambda x)(1-x)+xexp\left(\lambda (1-x)\right)\xi \left(𝓌\right)}\hfill \\ & \hfill & {\displaystyle -xexp\left(\lambda (1-x)\right)I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)-xexp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)}\hfill \\ & \hfill & {\displaystyle -xexp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)]}\hfill \end{array}$$

As a consequence,

$$\begin{array}{ccc}𝓌\left(x\right)\hfill & =\hfill & {\displaystyle \sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\zeta \sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m,\lambda }exp(-\lambda x)(1-x)}\hfill \\ & \hfill & {\displaystyle +\xi \left(𝓌\right)\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m,\lambda }xexp\left(\lambda (1-x)\right)}\hfill \\ & \hfill & {\displaystyle -I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m,\lambda }xexp\left(\lambda (1-x)\right)}\hfill \\ & \hfill & {\displaystyle -\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m,\lambda }xexp\left(\lambda (1-x)\right)}\hfill \\ & \hfill & {\displaystyle -\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m,\lambda }xexp\left(\lambda (1-x)\right).}\hfill \end{array}$$

By using the fact that, for all $\alpha >0:$

$$\left\{\begin{array}{c}{\displaystyle I_0^{\alpha ,\lambda }exp\left(-\lambda x\right)=\frac{exp\left(-\lambda x\right)}{\mathsf{\Gamma }(\alpha +1)}{x}^{\alpha },}\hfill \\ {\displaystyle I_0^{\alpha ,\lambda }xexp\left(-\lambda x\right)=\frac{exp\left(-\lambda x\right)}{\mathsf{\Gamma }(\alpha +2)}{x}^{\alpha +1},}\hfill \end{array}\right.$$

we get,

$$\begin{array}{ccc}𝓌\left(x\right)\hfill & =\hfill & {\displaystyle \sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\zeta exp(-\lambda x)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \left[\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}-\frac{x^{{\beta }_1^{s_1}+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right]}\hfill \\ & \hfill & {\displaystyle +\xi \left(𝓌\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -exp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -exp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )}\hfill \\ & \hfill & {\displaystyle \times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}.}\hfill \end{array}$$

This integral form is therefore equivalent to the original problem (1).

By using the fact that, for all $\alpha >0:$

$$\begin{array}{ccc}{\displaystyle I_0^{\alpha ,\lambda }1}\hfill & =\hfill & {\displaystyle \frac{exp\left(-\lambda x\right)}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^x{\left(x-s\right)}^{\alpha -1}exp\left(\lambda s\right)ds}\hfill \\ & \le \hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^x{\left(x-s\right)}^{\alpha -1}ds}\hfill \\ & \le \hfill & {\displaystyle \frac{1}{\mathsf{\Gamma }(\alpha +1)}{x}^{\alpha }}\hfill \end{array}$$

we get,

$$\begin{array}{ccc}\left|𝓌\left(x\right)\right|\hfill & \le \hfill & {\displaystyle \sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{∥\varphi \left(.,.\right)∥}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha +1\right)}}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|){\displaystyle \frac{∥{{\rm Y}}_j\left(.,.\right)∥}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j+1\right)}}}\hfill \\ & \hfill & {\displaystyle +\left|\zeta \right|\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}}\hfill \\ & \hfill & {\displaystyle +\left|\xi \left(𝓌\right)\right|exp\left(\lambda \right)\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle +\frac{exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{∥\varphi \left(.,.\right)∥}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right)∥𝓌\left(x\right)∥\sum _{i=1}^m{\displaystyle \frac{\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}}\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right)\sum _{j=1}^l\frac{∥{{\rm Y}}_j\left(.,.\right)∥}{\mathsf{\Gamma }({\alpha }_j+1)}\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}.}\hfill \end{array}$$

Employing the definition of the multivariate Mittag–Leffler function, we derive:

$$\begin{array}{ccc}\left|𝓌\left(x\right)\right|\hfill & \le \hfill & {\displaystyle {\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),{\alpha }_j+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥{{\rm Y}}_j\left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\left|\zeta \right|{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & \hfill & {\displaystyle +\left|\xi \left(𝓌\right)\right|exp\left(\lambda \right){\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & \hfill & {\displaystyle +\frac{exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right)∥𝓌\left(x\right)∥{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{i=1}^m{\displaystyle \frac{\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}}}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right){\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{j=1}^l\frac{∥{{\rm Y}}_j\left(.,.\right)∥}{\mathsf{\Gamma }({\alpha }_j+1)}.}\hfill \end{array}$$

It follows from (2) that

$$\begin{array}{ccc}∥𝓌∥\hfill & \le \hfill & {\displaystyle \frac{1}{\varrho }{\mathcal{E}}_{({\beta }_1,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\frac{1}{\varrho }\sum _{j=1}^l{\mathcal{E}}_{({\beta }_1,\dots ,{\beta }_m),{\alpha }_j+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥{{\rm Y}}_j\left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\frac{\left|\zeta \right|}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & \hfill & {\displaystyle +\frac{\left|\xi \left(𝓌\right)\right|exp\left(\lambda \right)}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & \hfill & {\displaystyle +\frac{exp\left(\lambda \right)}{\varrho \mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & \hfill & {\displaystyle +\frac{exp\left(\lambda \right)}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{j=1}^l\frac{∥{{\rm Y}}_j\left(.,.\right)∥}{\mathsf{\Gamma }({\alpha }_j+1)}<\infty .}\hfill \end{array}$$

□

Theorem 3.

Suppose that $m,l$ two positive integers, ${{\rm Y}}_j,\varphi :[0,1]\times \mathbb{R}\to \mathbb{R}$ for $1\le j\le l$ are bounded and continuous, and $\xi :C[0,1]\to \mathbb{R}$ is a functional, such that, $\forall x\in [0,1],y_1,y_2\in \mathbb{R},$ and $\forall {𝓌}_1,{𝓌}_2\in C[0,1]$:

$${\displaystyle \left|\varphi \left(x,y_1\right)-\varphi \left(x,y_2\right)\right|\le \mathcal{L}\left|y_1-y_2\right|,}$$

(3)

$${\displaystyle \left|{{\rm Y}}_j\left(x,y_1\right)-{{\rm Y}}_j\left(x,y_2\right)\right|\le {\mathcal{L}}_j\left|y_1-y_2\right|,}$$

(4)

and

$${\displaystyle \left|\xi \left({𝓌}_1\right)-\xi \left({𝓌}_2\right)\right|\le \nu ∥{𝓌}_1-{𝓌}_2∥,}$$

(5)

with, $\mathcal{L}\ge 0,$ ${\mathcal{L}}_j\ge 0,$ $\nu \ge 0$ and

$$\begin{array}{ccc}\varpi \hfill & =\hfill & [\mathcal{L}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+\sum _{j=1}^l{\mathcal{L}}_i{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),{\alpha }_i+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\hfill \\ & \hfill & {\displaystyle +\left(\nu exp\left(\lambda \right)+\frac{\mathcal{L}exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}+\sum _{i=1}^m\frac{exp\left(\lambda \right)\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}+\sum _{j=1}^l\frac{{\mathcal{L}}_{i}exp\left(\lambda \right)}{\mathsf{\Gamma }({\alpha }_j+1)}\right)}\hfill \\ & \hfill & {\displaystyle \times {\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)]<1}\hfill \end{array}$$

(6)

Therefore, there exists precisely one function $C[0,1]$ that satisfies the BVP given in (1).

Proof. 

Let $\mathcal{A}:C[0,1]\to C[0,1]$, with

$$\begin{array}{ccc}\left(\mathcal{A}𝓌\right)\left(x\right)\hfill & =\hfill & {\displaystyle \sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)}\hfill \\ & \hfill & {\displaystyle +\zeta exp(-\lambda x)\sum _{s=0}^{\infty }𝓜(s,\rho )\left[\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}-\frac{x^{{\beta }_1^{s_1}+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right]}\hfill \\ & \hfill & {\displaystyle +\xi \left(𝓌\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -exp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & \hfill & {\displaystyle -exp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}.}\hfill \end{array}$$

The contraction property of $\mathcal{A}$ will be demonstrated as follows. Let ${𝓌}_1\left(x\right)$ and ${𝓌}_2\left(x\right)$ be any functions in $C[0,1]$. Then:

$$\begin{array}{ccc}\hfill & \hfill & \left|\left(\mathcal{A}{𝓌}_1\right)\left(x\right)-\left(\mathcal{A}{𝓌}_2\right)\left(x\right)\right|\hfill \\ & \hfill & {\displaystyle \le \sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\left|I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\left(\varphi \left(x,{𝓌}_1\left(x\right)\right)-\varphi \left(x,{𝓌}_2\left(x\right)\right)\right)\right|}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\left|I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }\left({{\rm Y}}_j\left(x,{𝓌}_1\left(x\right)\right)-{{\rm Y}}_j\left(x,{𝓌}_2\left(x\right)\right)\right)\right|}\hfill \\ & \hfill & {\displaystyle +[exp\left(\lambda \right)\left|\xi \left({𝓌}_1\right)-\xi \left({𝓌}_2\right)\right|+exp\left(\lambda \right)\left|I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,{𝓌}_1\left(x\right)\right)-I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,{𝓌}_2\left(x\right)\right)\right|}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right)\sum _{i=1}^m\left|{\rho }_i\right|\left|I_{0,x=1}^{{\beta }_i,\lambda }{𝓌}_1\left(x\right)-I_{0,x=1}^{{\beta }_i,\lambda }{𝓌}_2\left(x\right)\right|}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right)\sum _{j=1}^l\left|I_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,{𝓌}_1\left(x\right)\right)-I_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,{𝓌}_2\left(x\right)\right)\right|]}\hfill \\ & \hfill & {\displaystyle \times \sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}.}\hfill \end{array}$$

From inequalities (3)–(5), we obtain:

$$\begin{array}{cc}\hfill & ∥\left(\mathcal{A}{𝓌}_1\right)-\left(\mathcal{A}{𝓌}_2\right)∥\hfill \\ & {\displaystyle \le [\mathcal{L}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+\sum _{j=1}^l{\mathcal{L}}_i{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),{\alpha }_j+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & {\displaystyle +\left(\nu exp\left(\lambda \right)+\frac{\mathcal{L}exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}+\sum _{i=1}^m\frac{exp\left(\lambda \right)\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}+\sum _{j=1}^l\frac{{\mathcal{L}}_{i}exp\left(\lambda \right)}{\mathsf{\Gamma }({\alpha }_j+1)}\right)}\hfill \\ & {\displaystyle \times {\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)]∥{𝓌}_1-{𝓌}_2∥.}\hfill \end{array}$$

It follows from (6) that, $\mathcal{A}$ is a contraction. Therefore, $\mathcal{A}$ admits a unique fixed point, which corresponds to a solution of the problem (1). □

Theorem 4.

Suppose that $m,l$ two positive integers, ${{\rm Y}}_j,\varphi :[0,1]\times \mathbb{R}\to \mathbb{R}$ for $1\le j\le l$ are bounded and continuous and $\xi :C[0,1]\to \mathbb{R}$ is a functional. Furthermore, satisfaction of Condition (2) ensures the existence of a solution to BVP (1).

Proof. 

We apply Leray–Schauder’s fixed point theorem to demonstrate that $\mathcal{A}$ possesses a fixed point, organizing the proof as follows.

• Step 1: Continuity of $\mathcal{A}$.
• Let ${𝓌}_n$ be a sequence such that ${𝓌}_n\to 𝓌$ in $C[0,1]$. Then for each $x\in [0,1]$ we have

  $$\begin{array}{ccc}\hfill & \hfill & ∥\left(\mathcal{A}{𝓌}_n\right)-\left(\mathcal{A}𝓌\right)∥\hfill \\ & \hfill & {\displaystyle \le {\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\underset{x\in [0,1]}{sup}\left|\varphi \left(x,{𝓌}_n\left(x\right)\right)-\varphi \left(x,𝓌\left(x\right)\right)\right|}\hfill \\ & \hfill & {\displaystyle +\sum _{j=1}^l{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),{\alpha }_j+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\underset{x\in [0,1]}{sup}\left|{{\rm Y}}_j\left(x,{𝓌}_n\left(x\right)\right)-{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\right|}\hfill \\ & \hfill & {\displaystyle +[exp\left(\lambda \right)\left|\xi \left({𝓌}_n\right)-\xi \left(𝓌\right)\right|+\frac{exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}\underset{x\in [0,1]}{sup}\left|\varphi \left(x,{𝓌}_n\left(x\right)\right)-\varphi \left(x,𝓌\left(x\right)\right)\right|}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right)\underset{x\in [0,1]}{sup}\left|{𝓌}_n\left(x\right)-𝓌\left(x\right)\right|\sum _{i=1}^m{\displaystyle \frac{\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}}}\hfill \\ & \hfill & {\displaystyle +exp\left(\lambda \right)\underset{x\in [0,1]}{sup}\left|{{\rm Y}}_j\left(x,{𝓌}_n\left(x\right)\right)-{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\right|\sum _{j=1}^l\frac{1}{\mathsf{\Gamma }({\alpha }_j+1)}]}\hfill \\ & \hfill & {\displaystyle \times {\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right).}\hfill \end{array}$$
• The continuity of $\mathcal{A}$ follows directly from the continuity of the functions $\varphi$, ${{\rm Y}}_j$, and the functional $\xi$.
• Step 2:$\mathcal{A}$ preserves boundedness in $C[0,1]$.
• This result is obtained by applying the methodology used in the Lemma 1.
• Step 3: We establish the equicontinuity property.
• Let $x_1,x_2\in [0,1],$ and let $𝓌\in {\mathcal{B}}_{\mathcal{N}}=\{𝓌\in C[0,1],\mathrm{with}∥𝓌∥\le \mathcal{N}\}$, therefore,

  $$\begin{array}{cc}\hfill & \left|\left(\mathcal{A}𝓌\right)\left(x_1\right)-\left(\mathcal{A}𝓌\right)\left(x_2\right)\right|\hfill \\ & {\displaystyle \le \sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\left|I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\varphi \left(x_2,𝓌\left(x_2\right)\right)-I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\varphi \left(x_1,𝓌\left(x_1\right)\right)\right|}\hfill \\ & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\left|I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }{{\rm Y}}_j\left(x_2,𝓌\left(x_2\right)\right)-I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }{{\rm Y}}_j\left(x_1,𝓌\left(x_1\right)\right)\right|}\hfill \\ & {\displaystyle +\left|\zeta \right|\left|exp(-\lambda x_2)\sum _{s=0}^{\infty }𝓜(s,\rho )\left[\frac{x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}-\frac{x_2^{{\beta }_1^{s_1}+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right]\right.}\hfill \\ & {\displaystyle \left.-exp(-\lambda x_1)\sum _{s=0}^{\infty }𝓜(s,\rho )\left[\frac{x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}-\frac{x_1^{{\beta }_1^{s_1}+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right]\right|}\hfill \\ & {\displaystyle +\left|\xi \left(𝓌\right)\right|\left|exp\left(\lambda (1-x_2)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right.}\hfill \\ & {\displaystyle \left.-exp\left(\lambda (1-x_1)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right|}\hfill \\ & {\displaystyle +\left|I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)\right|\left|exp\left(\lambda (1-x_2)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right.}\hfill \\ & {\displaystyle \left.-exp\left(\lambda (1-x_1)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right|}\hfill \\ & {\displaystyle +\left|\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)\right|\left|exp\left(\lambda (1-x_2)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right.}\hfill \\ & {\displaystyle \left.-exp\left(\lambda (1-x_1)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right|}\hfill \\ & {\displaystyle +\left|\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\right|\left|exp\left(\lambda (1-x_2)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right.}\hfill \\ & {\displaystyle \left.-exp\left(\lambda (1-x_1)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right|.}\hfill \end{array}$$

  it imlies that

  $$\begin{array}{cc}\hfill & \left|\left(\mathcal{A}𝓌\right)\left(x_1\right)-\left(\mathcal{A}𝓌\right)\left(x_2\right)\right|\hfill \\ & {\displaystyle \le \sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha \right)}}\hfill \\ & {\displaystyle \times \left|{\int }_0^{x_2}{\left(x_2-s\right)}^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha -1}\varphi \left(s,𝓌\left(s\right)\right)ds\right.}\hfill \\ & {\displaystyle \left.-{\int }_0^{x_1}{\left(x_1-s\right)}^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha -1}\varphi \left(s,𝓌\left(s\right)\right)ds\right|}\hfill \\ & {\displaystyle +\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j\right)}}\hfill \\ & {\displaystyle \times \left|{\int }_0^{x_2}{\left(x_2-s\right)}^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j-1}{{\rm Y}}_j\left(s,𝓌\left(s\right)\right)ds\right.}\hfill \\ & {\displaystyle \left.-{\int }_0^{x_1}{\left(x_1-s\right)}^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j-1}{{\rm Y}}_j\left(s,𝓌\left(s\right)\right)ds\right|}\hfill \\ & {\displaystyle +\left|\zeta \right|\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)}\hfill \\ & {\displaystyle \times \left[\frac{\left|x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}-x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}\right|}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}+\frac{\left|x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}-x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}\right|}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right]}\hfill \\ & {\displaystyle +2\lambda \left|\zeta \right|\left|x_2-x_1\right|\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}}\hfill \\ & {\displaystyle +[exp\left(\lambda \right)\left|\xi \left(𝓌\right)\right|+exp\left(\lambda \right)\left|I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)\right|+exp\left(\lambda \right)\sum _{i=1}^m\left|{\rho }_i\right|\left|I_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)\right|}\hfill \\ & {\displaystyle +exp\left(\lambda \right)\sum _{j=1}^l\left|I_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\right|]\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{\left|x_2^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}-x_1^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}\right|}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & {\displaystyle +\lambda \left|x_2-x_1\right|[exp\left(\lambda \right)\left|\xi \left(𝓌\right)\right|+exp\left(\lambda \right)\left|I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,𝓌\left(x\right)\right)\right|+exp\left(\lambda \right)\sum _{i=1}^m\left|{\rho }_i\right|\left|I_{0,x=1}^{{\beta }_i,\lambda }𝓌\left(x\right)\right|}\hfill \\ & {\displaystyle +exp\left(\lambda \right)\sum _{j=1}^l\left|I_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,𝓌\left(x\right)\right)\right|]\sum _{s=0}^{\infty }𝓜(s,\left|\rho \right|)\frac{1}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \end{array}$$
• Now, usig the fact that, for all $\alpha >0,x_1,x_2\in (0,1)$ with $x_1\le x_2$ and for any function h which is bounded on $(0,1)$:

  $${\displaystyle \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}\left|{\int }_0^{x_2}(x_2-s)h\left(s\right)-{\int }_0^{x_1}(x_1-s)h\left(s\right)ds\right|\le \frac{{sup}_{s\in [0,1]}\left|h\left(s\right)\right|}{\mathsf{\Gamma }(\alpha +1)}\left(2{(x_2-x_1)}^{\alpha }+\left(x_2^{\alpha }-x_1^{\alpha }\right)\right),}$$

  we deduce that, the above equation tends to zero as $x_2\to x_1.$
• Step 3: A priori bounds.
• To complete the proof, we must establish that the following set

  $$\mathcal{E}=\left\{\mathcal{U}\right(x)\in C[0,1\left]\mathrm{such}\mathrm{that},\mathcal{U}\right(x)=\theta \mathcal{A}\mathcal{U}(x)\mathrm{for}0<\theta <1\}.$$

  is bounded.
• Consider $\mathcal{U}\left(x\right)\in \mathcal{E}$. By adapting the methodology from Lemma 1, we obtain:

  $$\begin{array}{cc}∥𝓌∥\le \hfill & {\displaystyle {\mathcal{E}}_{({\beta }_1,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & {\displaystyle +\sum _{j=1}^l{\mathcal{E}}_{({\beta }_1,\dots ,{\beta }_m),{\alpha }_j+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥{{\rm Y}}_j\left(.,.\right)∥}\hfill \\ & {\displaystyle +\left|\zeta \right|{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+exp\left(\lambda \right)\left|\xi \left(𝓌\right)\right|{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & {\displaystyle +\frac{exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & {\displaystyle +exp\left(\lambda \right)∥𝓌∥{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{i=1}^m{\displaystyle \frac{\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}}}\hfill \\ & {\displaystyle +exp\left(\lambda \right){\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{j=1}^l\frac{∥{{\rm Y}}_j\left(.,.\right)∥}{\mathsf{\Gamma }({\alpha }_j+1)}.}\hfill \end{array}$$

  From (2), we obtain,

  $$\begin{array}{cc}∥𝓌∥\le \hfill & {\displaystyle \frac{1}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & {\displaystyle +\frac{1}{\varrho }\sum _{j=1}^l{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),{\alpha }_j+1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥{{\rm Y}}_j\left(.,.\right)∥}\hfill \\ & {\displaystyle +\frac{\left|\zeta \right|}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+\frac{\left|\xi \left(𝓌\right)\right|exp\left(\lambda \right)}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)}\hfill \\ & {\displaystyle +\frac{exp\left(\lambda \right)}{\varrho \mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)∥\varphi \left(.,.\right)∥}\hfill \\ & {\displaystyle +\frac{exp\left(\lambda \right)}{\varrho }{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\sum _{j=1}^l\frac{∥{{\rm Y}}_j\left(.,.\right)∥}{\mathsf{\Gamma }({\alpha }_j+1)}<\infty .}\hfill \end{array}$$

  Therefore, Schaefer’s Fixed-Point Theorem guarantees the existence of a solution of (1). □

4. Ulam Stability

We consider the following inequality for all $(x,y)\in J$ and for all $ϵ>0$:

$${\displaystyle \left|{}^{C}D_0^{\alpha ,\lambda }\left[\mathcal{U}\left(x\right)-\sum _{i=1}^m{\rho }_i{I}_0^{{\beta }_i,\lambda }\mathcal{U}\left(x\right)-\sum _{j=1}^l{I}_0^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,\mathcal{U}\left(x\right)\right)\right]-\varphi \left(x,\mathcal{U}\left(x\right)\right)\right|\le ϵ.}$$

(7)

Definition 5.

The BVP (1) is called Ulam–Hyers stable provided there is a constant $𝓌>0$ with for any $ϵ>0$ and for any function $\mathcal{U}\left(x\right)$ satisfying the inequality (7) under the conditions $\mathcal{U}\left(0\right)=\zeta ,\mathcal{U}\left(1\right)=\xi \left(\mathcal{U}\right)$ there exists a solution $𝓌\left(x\right)$ of (1) such that

$$\left|𝓌\left(x\right)-\mathcal{U}\left(x\right)\right|\le 𝓌ϵ.$$

Remark 3.

If $\mathcal{U}\left(x\right)$ is a solution of (7) then $\mathcal{U}\left(x\right)$ satisfies the following integral inequality

$$\begin{array}{cc}\hfill & {\displaystyle |\mathcal{U}\left(x\right)-\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+\alpha ,\lambda }\varphi \left(x,\mathcal{U}\left(x\right)\right)}\hfill \\ & {\displaystyle -\sum _{j=1}^l\sum _{s=0}^{\infty }𝓜(s,\rho )I_0^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,\mathcal{U}\left(x\right)\right)}\hfill \\ & {\displaystyle -\zeta exp(-\lambda x)\sum _{s=0}^{\infty }𝓜(s,\rho )\left[\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1\right)}-\frac{x^{{\beta }_1^{s_1}+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}\right]}\hfill \\ & {\displaystyle -\xi \left(\mathcal{U}\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & {\displaystyle +I_{0,x=1}^{\alpha ,\lambda }\varphi \left(x,\mathcal{U}\left(x\right)\right)exp\left(\lambda (1-x)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & {\displaystyle +exp\left(\lambda (1-x)\right)\sum _{i=1}^m{\rho }_i{I}_{0,x=1}^{{\beta }_i,\lambda }\mathcal{U}\left(x\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}}\hfill \\ & {\displaystyle +exp\left(\lambda (1-x)\right)\sum _{j=1}^l{I}_{0,x=1}^{{\alpha }_j,\lambda }{{\rm Y}}_j\left(x,\mathcal{U}\left(x\right)\right)\sum _{s=0}^{\infty }𝓜(s,\rho )\times \frac{x^{{\beta }_1{s}_1+\dots +{\beta }_m{s}_m+1}}{\mathsf{\Gamma }\left({\beta }_1{s}_1+\dots +{\beta }_m{s}_m+2\right)}|}\hfill \\ & {\displaystyle \le \left[{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+\frac{exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\right]ϵ.}\hfill \end{array}$$

Theorem 5.

Suppose that $m,l$ two positive integers, ${{\rm Y}}_j,\varphi :[0,1]\times \mathbb{R}\to \mathbb{R}$ for $1\le j\le l$ are bounded and continuous, and $\xi :C[0,1]\to \mathbb{R}$ is a functional, with (3)–(6) hold. Therefore, the BVP (1) is Ulam–Hyers stable.

Proof. 

Consider a solution $\mathcal{U}\left(x\right)$ of inequality (7) and let $𝓌\left(x\right)$ denote the unique solution to the BVP (1). Following an argument analogous to the one presented in Theorem 3 and utilizing the result from Remark 3, we derive the following estimate:

$$\begin{array}{cc}\hfill & ∥\mathcal{U}-𝓌∥\hfill \\ & {\displaystyle \le \left[{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+\frac{exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\right]ϵ}\hfill \\ & {\displaystyle +\varpi ∥\mathcal{U}-𝓌∥,}\hfill \end{array}$$

it imples that,

$$∥\mathcal{U}-𝓌∥\le {\displaystyle \frac{\left[{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),\alpha +1}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)+\frac{exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}{\mathcal{E}}_{({\beta }_1,{\beta }_2,\dots ,{\beta }_m),2}\left(\left|{\rho }_1\right|,\dots ,\left|{\rho }_m\right|\right)\right]}{1-\varpi }}ϵ.$$

Hence, the BVP (1) is Ulam–Hyers stable. □

5. Two Examples

In this section, we present two examples that show how our theoretical results work in practice.

Example 1.

Let the following BVP:

$$\left\{\begin{array}{cc}{\displaystyle {}^{C}D_0^{1.7,1.1}[𝓌\left(x\right)+\frac{1}{11}{I}_0^{0.7,1.1}𝓌\left(x\right)+\frac{1}{13}{I}_0^{2.5,1.1}𝓌\left(x\right)-\frac{1}{43}{I}_0^{1.3,1.1}tanh\left(x𝓌\left(x\right)\right)}\hfill & \hfill \\ {\displaystyle -\frac{1}{31}{I}_0^{3.2,1.1}\left(\frac{x}{\sqrt{{𝓌}^2\left(x\right)+1}}\right)]=\frac{1}{59}sin\left(𝓌\left(x\right)\right),}\hfill & x\in (0,1),\hfill \\ {\displaystyle 𝓌\left(0\right)=1,𝓌\left(1\right)=\frac{1}{41}\left(exp\left(-𝓌\left(0.25\right)\right)+exp\left(-𝓌\left(0.5\right)\right)\right),}\hfill \end{array}\right.$$

(8)

We have, ${\displaystyle \varphi \left(x,𝓌\right)=\frac{1}{59}sin\left(𝓌\right),{{\rm Y}}_1\left(x,𝓌\right)=\frac{1}{43}tanh\left(x𝓌\right),{{\rm Y}}_2\left(x,𝓌\right)=\frac{1}{31}\left(\frac{x}{\sqrt{{𝓌}^2+1}}\right)}$ and ${\displaystyle \xi \left(𝓌\right)=\frac{1}{41}\left(exp\left(-𝓌\left(0.25\right)\right)+exp\left(-𝓌\left(0.5\right)\right)\right).}$ The functions ϕ, ${{\rm Y}}_1$, and ${{\rm Y}}_2$ are continuous and bounded on the domain $[0,1]\times \mathbb{R}$, and the functional ξ exhibits continuity and boundedness on $C[0,1]$. Furthermore, an application of the Mean Value Theorem yields, for any $x\in [0,1]$ and all ${𝓌}_1,{𝓌}_2\in \mathbb{R}$, the following inequalities:

$$\begin{array}{ccc}{\displaystyle |\varphi \left(x,{𝓌}_1\right)-\varphi \left(x,{𝓌}_2\right)|}\hfill & =\hfill & {\displaystyle \frac{1}{59}|sin\left({𝓌}_1\right)-sin\left({𝓌}_1\right)|}\hfill \\ \hfill & \le \hfill & {\displaystyle \frac{1}{59}|{𝓌}_1-{𝓌}_2|,}\hfill \end{array}$$

$$\begin{array}{ccc}{\displaystyle |{{\rm Y}}_1\left(x,{𝓌}_1\right)-{{\rm Y}}_1\left(x,{𝓌}_2\right)|}\hfill & =\hfill & {\displaystyle \frac{1}{43}|tanh\left(x{𝓌}_1\right)-tanh\left(x{𝓌}_2\right)|}\hfill \\ \hfill & \le \hfill & {\displaystyle \frac{1}{43}|{𝓌}_1-{𝓌}_2|}\hfill \end{array}$$

since, ${\displaystyle \left|\frac{d}{d𝓌}tanh\left(𝓌\right)\right|=\frac{1}{{cosh}^2\left(𝓌\right)}\le 1,}$

$$\begin{array}{ccc}{\displaystyle |{{\rm Y}}_2\left(x,{𝓌}_1\right)-{{\rm Y}}_2\left(x,{𝓌}_2\right)|}\hfill & =\hfill & {\displaystyle \frac{1}{31}\left|\left(\frac{x}{\sqrt{{𝓌}_1^2+1}}\right)-\left(\frac{x}{\sqrt{{𝓌}_2^2+1}}\right)\right|}\hfill \\ \hfill & \le \hfill & {\displaystyle \frac{1}{31}|{𝓌}_1-{𝓌}_2|}\hfill \end{array}$$

since, ${\displaystyle \left|\frac{d}{d𝓌}\frac{1}{\sqrt{{𝓌}^2+1}}\right|=\left|\frac{-𝓌}{{\left({𝓌}^2+1\right)}^{\frac{3}{2}}}\right|\le 1,}$ and for all ${𝓌}_1,{𝓌}_2\in C[0,1]:$

$$\begin{array}{ccc}{\displaystyle |\xi \left({𝓌}_1\right)-\xi \left({𝓌}_2\right)|}\hfill & \le \hfill & {\displaystyle \frac{1}{41}\left(|{𝓌}_1\left(0.25\right)-{𝓌}_2\left(0.25\right)|+|{𝓌}_1\left(0.5\right)-{𝓌}_2\left(0.5\right)|\right)}\hfill \\ \hfill & \le \hfill & {\displaystyle \frac{2}{41}∥{𝓌}_1-{𝓌}_2∥}\hfill \end{array}$$

From the above, we have

$${\displaystyle \alpha =1.7,\lambda =1.1,{\beta }_1=0.7,{\beta }_2=2.5,\left|{\rho }_1\right|=\frac{1}{11},\left|{\rho }_2\right|=\frac{1}{13},\mathcal{L}=\frac{1}{59},{\mathcal{L}}_1=\frac{1}{43},{\mathcal{L}}_2=\frac{1}{31},\nu =\frac{2}{41}}$$

So,

$$\begin{array}{ccc}\varpi \hfill & =\hfill & [\mathcal{L}{\mathcal{E}}_{({\beta }_1,{\beta }_2),\alpha +1}\left(\left|{\rho }_1\right|,\left|{\rho }_2\right|\right)+\sum _{j=1}^2{\mathcal{L}}_i{\mathcal{E}}_{({\beta }_1,{\beta }_2,),{\alpha }_i+1}\left(\left|{\rho }_1\right|,\left|{\rho }_2\right|\right)\hfill \\ & \hfill & {\displaystyle +\left(\nu +\frac{\mathcal{L}exp\left(\lambda \right)}{\mathsf{\Gamma }(\alpha +1)}+\sum _{i=1}^2\frac{exp\left(\lambda \right)\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}+\sum _{j=1}^2\frac{{\mathcal{L}}_{i}exp\left(\lambda \right)}{\mathsf{\Gamma }({\alpha }_j+1)}\right){\mathcal{E}}_{({\beta }_1,{\beta }_2),2}\left(\left|{\rho }_1\right|,\left|{\rho }_2\right|\right)]}\hfill \\ {\displaystyle }\hfill & :=\hfill & 0.597301<1.\hfill \end{array}$$

Therefore, all the assumptions of Theorem 3 and Theorem 5 are fulfilled. Consequently, the BVP (8) admits a unique solution, and this solution is stable.

Example 2.

Consider the following BVP:

$$\left\{\begin{array}{cc}{\displaystyle {}^{C}D_0^{1.5,1.7}[𝓌\left(x\right)-\frac{1}{19}{I}_0^{1.7,1.7}𝓌\left(x\right)+\frac{1}{29}{I}_0^{2.7,1.7}𝓌\left(x\right)-I_0^{3.3,1.7}arctan\left(x𝓌\left(x\right)\right)}\hfill & \hfill \\ {\displaystyle -I_0^{2.1,1.7}sin\left(\frac{\pi }{2\left({𝓌}^2\left(x\right)+1\right)}\right)]=ln\left(x+\frac{1}{\sqrt{1+{𝓌}^2\left(x\right)}}\right),}\hfill & x\in (0,1),\hfill \\ {\displaystyle 𝓌\left(0\right)=1,𝓌\left(1\right)=cos\left(-𝓌\left(0.5\right)\right),}\hfill \end{array}\right.$$

(9)

In this case, the functions are defined by ${\displaystyle \varphi \left(x,𝓌\right)=ln\left(x+\frac{1}{\sqrt{1+{𝓌}^2\left(x\right)}}\right)}$, ${{\rm Y}}_1\left(x,𝓌\right)=arctan\left(x𝓌\left(x\right)\right)$, ${{\rm Y}}_2\left(x,𝓌\right)=sin\left({\displaystyle \frac{\pi }{2\left({𝓌}^2\left(x\right)+1\right)}}\right)$, and the functional by ${\displaystyle \xi \left(𝓌\right)=cos\left(-𝓌\left(0.5\right)\right).}$ These definitions ensure that ϕ, ${{\rm Y}}_1$, and ${{\rm Y}}_2$ are continuous and bounded on $[0,1]\times \mathbb{R}$, while ξ is continuous and bounded on $C[0,1]$. Consequently, we identify the parameters as follows:

$${\displaystyle \alpha =1.5,\lambda =1.7,{\beta }_1=1.7,{\beta }_2=2.7,\left|{\rho }_1\right|=\frac{1}{19},\left|{\rho }_2\right|=\frac{1}{29}.}$$

So,

$$\varrho =1-exp\left(\lambda \right){\mathcal{E}}_{({\beta }_1,{\beta }_2),2}\left(\left|{\rho }_1\right|,\left|{\rho }_2\right|\right)\sum _{i=1}^2{\displaystyle \frac{\left|{\rho }_i\right|}{\mathsf{\Gamma }({\beta }_i+1)}}=0.764769>0.$$

Therefore, all the assumptions of Theorem 4 are fulfilled. As a consequence, the BVP (9) admits at least one solution on $[0,1]$.

6. Conclusions

This research establishes a comprehensive theoretical framework for a new category of PFIDEs governed by functional boundary conditions. The methodology is centered on Babenko’s technique, which is employed to build an inverse operator that transforms the original BVP into an equivalent integral formulation. This reformulation then permits a direct and efficient investigation using foundational results from fixed-point theory.

Our main results provide verifiable sufficient conditions that ensure the existence of at least one solution, the guarantee of a unique solution, and the Ulam–Hyers Stability of the system. The criteria for uniqueness and stability were derived via Banach’s contraction principle, while an existence result was established under milder conditions using the Leray–Schauder alternative. The practical relevance of our theoretical work was confirmed through two detailed examples, which demonstrated the applicability of our criteria to specific, non-trivial problems.

This research contributes significantly to the field by offering a generalized approach for handling complex fractional-order dynamics with integral terms and nonlocal constraints. The framework presented here is not only theoretically sound but also versatile, opening avenues for future research. Potential extensions of this work include the development of numerical schemes for approximation, exploration of optimal control problems governed by such equations, and the generalization of the framework to coupled systems or equations involving variable-order fractional operators.