Existence, Uniqueness and Stability Analysis for Generalized Φ-Caputo Fractional Boundary Value Problems

Abstract

This study investigates solutions of a class of boundary value problems involving the $\Phi$-Caputo fractional derivative and the p-Laplacian operator. Through the application of fixed-point theory, we confirm the existence and uniqueness of solutions to nonlinear $\Phi$-Caputo fractional differential equations with the p-Laplacian operator. Moreover, we have demonstrated that this problem is stable in the framework of Ulam–Hyers stability. Our findings enhance the theoretical understanding of fractional differential equations and have potential applications in various scientific and engineering fields. In addition, an illustrative example is provided to support the key insights derived from this research.

1. Introduction

In recent years, $\Phi$-Caputo fractional derivatives [1,2,3,4,5,6,7] have emerged as essential tools in fractional calculus, accurately representing complex dynamics with non-local memory characteristics. When these derivatives are coupled with the p-Laplacian operator [8,9,10], they enable a deeper exploration of systems governed by both fractional derivatives [11,12,13] and nonlinear mechanisms.

Fractional-order models excel at modeling anomalous diffusion in heterogeneous media, surpassing traditional models in terms of accuracy. In engineering, they are used to evaluate viscoelastic materials [14], capturing behavior tied to memory effects and dynamic loads. These models also improve biological studies, where they account for memory-based interactions, aiding in research on population dynamics and disease transmission [15,16,17].

The concept of symmetry plays an important role in the analysis of fractional differential equations, especially in proving the existence and uniqueness of solutions. Symmetry helps simplify the structure of equations and highlight invariant properties. Fixed-point theorems, such as Banach’s fixed-point theorem, are commonly employed in this context, particularly when the fractional derivative operator satisfies Lipschitz conditions. Additionally, the symmetric nature of certain operators or boundary conditions supports the effective use of fixed-point techniques, ensuring that solutions reflect the inherent structure of the problem.

In functional equation theory, stability analysis has emerged as a critical and engaging field of study. Stability, as a foundational property in mathematical analysis, is of significant importance across a wide range of disciplines in engineering and science. The concept of Ulam–Hyers stability for differential equations was first introduced in the 1940s by Ulam and Hyers. Broadly speaking, the Ulam–Hyers stability for a differential equation addresses whether an exact solution exists in close proximity to an approximate solution to the equation. This concept is therefore crucial for the analysis of numerical and approximate solutions, as well as for practical applications involving differential equations. Consequently, numerous researchers have examined different facets of Ulam–Hyers stability, focusing on problems involving fractional integrals and fractional differential equations, utilizing a variety of techniques as outlined in [18,19,20,21] and related references. However, studies specifically investigating the stability of fractional differential equations remain relatively scarce.

This work focuses on the existence, uniqueness, and stability of solutions for fractional boundary value problems of $\Phi$-Caputo with the p-Laplacian operator. The parameter $\Phi$ in the $\Phi$-Caputo derivative allows refined modeling of memory effects, while its integration with the p-Laplacian operator improves the ability to address real-world complexities.

The primary goal of this work is to systematically demonstrate the existence, uniqueness, and stability of solutions for the fractional boundary value problems considered. This research involves a detailed theoretical investigation, in which advanced analytical and computational methods are used to examine the intrinsic behavior of the system. Building on previous research into p-Laplacian $\Phi$-Caputo fractional boundary problems with fractional derivatives and power function-based integrals, this study examines uniqueness using Green’s functions, targeting a mixed-boundary problem that includes $\Phi$-Caputo fractional derivatives and integrals.

Wang and Bai in [10] studied:

$$\begin{array}{c}\begin{array}{c}{}^{C}D_{1^{-}}^{\gamma }\left({\varphi }_p\left(D_{0^{+}}^{\delta }z\left(s\right)\right)\right)=g(s,z\left(s\right),D_{0^{+}}^{\delta }z\left(s\right)),0<s<1,\hfill \\ z\left(0\right)=0,z\left(1\right)=r_{1}z\left(\mu \right),\hfill \\ D_{0^{+}}^{\delta }z\left(1\right)=0,{\varphi }_p\left(D_{0^{+}}^{\delta }z\left(0\right)\right)=r_2{\varphi }_p\left(D_{0^{+}}^{\delta }z\left(\eta \right)\right),\hfill \end{array}\hfill \end{array}$$

(1)

where $0<\mu ,\eta <1$, $1<\gamma ,\delta \le 2$, $0\le r_1<{\displaystyle \frac{1}{{\mu }^{\delta -1}}}$, $0\le r_2<{\displaystyle \frac{1}{1-\eta }}$, ${\varphi }_p$ is the p-Laplacian operator, ${}^{C}D_{1^{-}}^{\gamma }$ is the right Caputo fractional derivative, and $D_{0^{+}}^{\delta }$ is the left Riemann–Liouville fractional derivative and the function $g\in C([0,1]\times {\mathbb{R}}^2,\mathbb{R})$.

In [5], Mfadel, Melliani and Elomari investigated

$$\begin{array}{c}{\left({\varphi }_p{(}^C{D}_{0^{+}}^{\alpha ,\psi }z\left(s\right))\right)}^{\prime }=f(s,z\left(s\right)),s\in \mathsf{\Delta }=[0,T],\hfill \\ z\left(0\right)={\sigma }_{1}z\left(T\right),z^{\prime }\left(0\right)={\sigma }_2{z}^{\prime }\left(T\right),\hfill \end{array}$$

(2)

where $D_{0^{+}}^{\alpha ,\psi }$ is the $\psi$-Caputo fractional derivative and ${\varphi }_p$ is a p-Laplacian operator, i.e., ${\varphi }_p\left(s\right)=s^{p-1}$, $p-1>0$.

In [8], Alsaedi, Alghanmi, Ahmad and Alharbi discussed

$$\begin{array}{c}{}^{\rho }D_{1^{-}}^{\alpha }\left({\varphi }_p{(}^{\rho }{D}_{0^{+}}^{\beta }z\left(s\right))\right)={\nu }_{1}f(s,z\left(s\right){,}^{\rho }{D}_{0^{+}}^{\beta }z\left(s\right))+{\nu }_2^{\rho }{I}_{0^{+}}^{\zeta }g(s,z\left(s\right){,}^{\rho }{D}_{0^{+}}^{\beta }z\left(s\right)),\hfill \\ z\left(0\right)=0,z\left(1\right)={\lambda }_{1}z\left(\mu \right),\hfill \\ {}^{\rho }D_{0^{+}}^{\beta }z\left(1\right)=0,{\varphi }_p{(}^{\rho }{D}_{0^{+}}^{\beta }z\left(0\right))={\lambda }_2{\varphi }_p{(}^{\rho }{D}_{0^{+}}^{\beta }z\left(\eta \right)),\hfill \end{array}$$

(3)

where ${\varphi }_p\left(s\right)$ is the p Laplacian operator, $1<\alpha ,\beta \le 2$, $\rho ,\zeta >0$, $0<\mu ,\eta <1$, $0\le {\lambda }_1<{\displaystyle \frac{1}{{\mu }^{\rho (\beta -1)}}}$, $0\le {\lambda }_2<{\displaystyle \frac{1}{(1-{\mu }^{\rho })\alpha -1}}$, ${}^{\rho }D_{1^{-}}^{\alpha }$ and ${}^{\rho }D_{0^{+}}^{\beta }$, respectively, are the right end left fractional derivatives of orders $\alpha$ and $\beta$ with respect to the power function, and ${}^{\rho }I_{0^{+}}^{\zeta }$ is the fractional integral of order $\zeta$ with respect to a power function.

In [22], Batit Ozen studied

$$\begin{array}{c}{}^{C}D_{b^{-}}^{\beta ,\aleph }\left({\varphi }_p{(}^C{D}_{b^{-}}^{\alpha ,\aleph }z\left(s\right))\right)=f(s,z\left(s\right)),s\in [a,b],\hfill \\ z\left(a\right)=D_{b^{-}}^{\alpha ,\aleph }z\left(a\right)=0,z\left(b\right)=D_{b^{-}}^{\alpha ,\aleph }z\left(b\right)=0,\hfill \end{array}$$

(4)

where $f:[a,b]\times \mathbb{R}\to \mathbb{R}$ is a continuous function and ${\varphi }_p$ is a p-Laplacian operator, and $\alpha ,\beta \in (1,2]$, $D_{b^{-}}^{\alpha ,\aleph }$, $D_{b^{-}}^{\beta ,\aleph }$ are right-sided ℵ-Caputo fractional derivatives of orders $\alpha$ and $\beta$, respectively.

In studies numbered [8,10], the authors investigate differential equations involving the Caputo fractional derivative defined at the endpoints of the domain. On the other hand, studies [5,22] consider a differential equation involving the generalized Caputo fractional derivative. Inspired by these works, the problem we address provides a further generalization by incorporating the generalized Caputo derivative operator at the endpoints, and we focus on establishing existence results for the p-Laplacian $\Phi$-Caputo fractional boundary value problem $\left(CFBVP\right)$, as described below:

$$\begin{array}{c}{}^{C}D_{b^{-}}^{\beta ,\Phi }\left[{\varphi }_p{(}^C{D}_{a^{+}}^{\alpha ,\Phi }y\left(t\right))\right]=F(t,y\left(t\right)),a\le t\le b,\hfill \\ y\left(a\right)=D_{a^{+}}^{\alpha ,\Phi }y\left(a\right)=0,y\left(b\right)=D_{a^{+}}^{\alpha ,\Phi }y\left(b\right)=0,\hfill \end{array}$$

(5)

here, ${\varphi }_p$ represents the p-Laplacian operator, defined as ${\varphi }_p\left(t\right)={\left|t\right|}^{p-2}t$, where p and q satisfy ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$ with $p,q\in (1,\infty )$. For orders $\alpha$ and $\beta$ in the range $1<\alpha ,\beta \le 2$, ${}^{C}D_{a^{+}}^{\alpha ,\Phi }$ and ${}^{C}D_{b^{-}}^{\beta ,\Phi }$ denote the left- and right-sided $\Phi$-Caputo fractional derivatives, respectively. The function $F:[a,b]\times \mathbb{R}\to \mathbb{R}$ is assumed to be continuous.

The problem addressed in this study distinguishes itself from existing literature by incorporating both the p-Laplacian operator and the generalized Caputo fractional derivative defined at the boundaries. This combination significantly enhances the modeling of complex physical phenomena, capturing intricate behaviors that are not adequately represented by traditional operators. By integrating these advanced mathematical tools, our approach offers a more nuanced understanding of systems where nonlinearity and memory effects play pivotal roles.

The paper is organized as follows. Section 2 introduces the fundamental definitions, lemmas, and theorems that form the mathematical foundation of the study. These preliminaries are essential for understanding the more advanced results presented in later sections. In Section 3, we derive the corresponding Green’s functions, which are instrumental in formulating and solving the main problems. We present our principal existence results using fixed-point theorems such as Banach’s, Schaefer’s, and Brouwer’s theorems. Rigorous proofs are provided to justify the applicability of these theorems. Additionally, an illustrative example is included to demonstrate the practical use of the theoretical results. Section 4 is devoted to the stability analysis of the proposed problem. We discuss various types of stability and their relevance, aiming to understand the long-term behavior of solutions and the robustness of the model under different conditions. Finally, Section 5 concludes the paper with a summary of the main findings and outlines possible directions for future research. This section emphasizes the significance of the obtained results and suggests extensions that could further develop the theoretical framework introduced in this study.

2. Preliminaries

This section presents key definitions, notations, lemmas, and results related to the $\Phi$-fractional derivative and integral, which will be referenced throughout the paper. For some special cases of $\Phi$, we consider $\Phi \left(t\right)=t$ or $\Phi \left(t\right)=lnt$, we obtain the Riemann–Liouville and Hadamard fractional operators. In this case, we define $m=\left[\alpha \right]+1$, with $\left[\alpha \right]$ denoting the integer part of the real number $\alpha$ and $\Gamma \left(\alpha \right)$ is a gamma function. Let $\alpha >0$ and suppose $I=[a,b]$ is either a finite or infinite interval.

Definition 1 ([12]).

Consider $\alpha >0$, and let $h:[a,b]\to \mathbb{R}$ be an integrable function defined on the interval $[a,b]$. Also, let $\Phi \in C^1([a,b],\mathbb{R})$ be a differentiable, increasing function with the property that ${\Phi }^{\prime }\left(t\right)\ne 0$, $\forall t\in [a,b]$. The α-th order right-sided and left-sided $\Phi$-Riemann–Liouville fractional integrals of h are given by the following expression:

$$I_{a^{+}}^{\alpha ,\Phi }h\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{a}{\overset{t}{\int }}{\Phi }^{\prime }\left(s\right){(\Phi (t)-\Phi (s\left)\right)}^{\alpha -1}h\left(s\right)ds,$$

and

$$I_{b^{-}}^{\alpha ,\Phi }h\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\underset{t}{\overset{b}{\int }}{\Phi }^{\prime }\left(s\right){(\Phi (s)-\Phi (t\left)\right)}^{\alpha -1}h\left(s\right)ds.$$

Definition 2 ([12]).

Consider $\alpha >0$, with $h:[a,b]\to \mathbb{R}$ being an integrable function, and $\Phi \in C^m([a,b],\mathbb{R})$ an increasing function such that ${\Phi }^{\prime }\left(t\right)\ne 0,$$\forall t\in [a,b]$. The left-sided and right-sided Φ-Riemann–Liouville fractional derivatives of h of order α are defined by

$$\begin{array}{c}{D}_{a^{+}}^{\alpha ,\Phi }h\left(t\right)={\left[{\displaystyle \frac{1}{{\Phi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right]}^m{I}_{a^{+}}^{m-\alpha ,\Phi }h\left(t\right),\hfill \end{array}$$

and

$$\begin{array}{c}{D}_{b^{-}}^{\alpha ,\Phi }h\left(t\right)={\left[-{\displaystyle \frac{1}{{\Phi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right]}^m{I}_{b^{-}}^{m-\alpha ,\Phi }h\left(t\right).\hfill \end{array}$$

Definition 3 ([4]).

Consider $\alpha >0$, and assume that $\Phi ,h\in C^m([a,b],\mathbb{R})$ are two functions, with Φ increasing and ${\Phi }^{\prime }\left(t\right)\ne 0$ for every $t\in [a,b]$. The left-sided and right-sided Φ-Caputo fractional derivatives of h of order α are given by

$$\begin{array}{c}{}^{C}D_{a^{+}}^{\alpha ,\Phi }h\left(t\right)=I_{a^{+}}^{m-\alpha ,\Phi }{\left[{\displaystyle \frac{1}{{\Phi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right]}^{m}h\left(t\right),\hfill \end{array}$$

and

$$\begin{array}{c}{}^{C}D_{b^{-}}^{\alpha ,\Phi }h\left(t\right)=I_{b^{-}}^{m-\alpha ,\Phi }{\left[-{\displaystyle \frac{1}{{\Phi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right]}^{m}h\left(t\right).\hfill \end{array}$$

Theorem 1 ([4]).

Let $h_1,h_2\in C^m([a,b],\mathbb{R})$ and $\alpha >0$. Then,

$$\begin{array}{c}{}^{C}D_{a^{+}}^{\alpha ,\Phi }{h}_1\left(t\right){=}^C{D}_{a^{+}}^{\alpha ,\Phi }{h}_2\left(t\right)⟷h_1\left(t\right)=h_2\left(t\right)+\sum _{k=0}^{m-1}{c}_k{(\Phi \left(t\right)-\Phi \left(a\right))}^k,\hfill \end{array}$$

and

$$\begin{array}{c}{}^{C}D_{b^{-}}^{\alpha ,\Phi }{h}_1\left(t\right){=}^C{D}_{b^{-}}^{\alpha ,\Phi }{h}_2\left(t\right)⟷h_1\left(t\right)=h_2\left(t\right)+\sum _{k=0}^{m-1}{d}_k{(\Phi \left(b\right)-\Phi \left(t\right))}^k,\hfill \end{array}$$

where $c_k$ and $d_k$ are arbitrary constants.

Lemma 1 ([4]).

Let α and β be positive numbers, and let $h:[a,b]\to \mathbb{R}$ be an integrable function defined on the interval $[a,b]$ and $\Phi \in C^1([a,b],\mathbb{R})$. Then,

(1) 

$I_{a^{+}}^{\alpha ,\Phi }{(\Phi \left(t\right)-\Phi \left(a\right))}^{\beta -1}={\displaystyle \frac{\Gamma \left(\beta \right)}{\Gamma (\alpha +\beta )}}{(\Phi \left(t\right)-\Phi \left(a\right))}^{\alpha +\beta -1}$,

(2) 

$I_{a^{+}}^{\alpha ,\Phi }{I}_{a^{+}}^{\beta ,\Phi }h\left(t\right)=I_{a^{+}}^{\alpha +\beta ,\Phi }h\left(t\right)$.

To better understand the contributions of the generalized $\Phi$-Caputo fractional derivative in this study, consider the following examples:

Enhancing memory effects with $\Phi \left(t\right)=e^{kt}$: Choosing $\Phi \left(t\right)=e^{kt}$ (with $k>0$) allows for parametric control over memory effects. For instance, in modeling the behavior of capacitors and inductors in electrical circuits, energy accumulation and discharge processes can be analyzed by adjusting k to influence the impact of past states on current behavior.

Modeling slowing processes with $\Phi \left(t\right)=log(1+t)$: This selection is suitable for modeling dynamics that slow over time. For example, analyzing growth trends in social media usage can be approached by using this model to capture rapid initial growth followed by a slowdown as saturation levels are reached.

Modeling periodic effects with $\Phi \left(t\right)=sin\left(kt\right)$: This choice facilitates modeling periodic phenomena. For instance, modeling seasonal variations in water consumption or energy demand can be effectively achieved using this functional form.

Lemma 2 ([10]).

The Laplacian operator satisfies the following relationships:

(1) 

There are $p\in (1,2]$, $|k_1|,|k_2|\ge n>0$, and $k_1,k_2>0$, such that

$$\begin{array}{c}|{\varphi }_p\left(k_2\right)-{\varphi }_p\left(k_1\right)|\le (p-1)n^{p-2}|k_2-k_1|,\hfill \end{array}$$

(6)

(2) 

There are $p\in (2,\infty )$ and $|k_1|,|k_2|\le N$, such that

$$\begin{array}{c}|{\varphi }_p\left(k_2\right)-{\varphi }_p\left(k_1\right)|\le (p-1)N^{p-2}|k_2-k_1|.\hfill \end{array}$$

(7)

Below, the fixed-point theorems of Schaefer Banach and Brouwer are given in order.

Theorem 2 ([23]).

Let X be a Banach space and $A:X\to X$ be continuous and compact operator. Assume further that the set $\{x\in X:x=\lambda Ax\}$, for some $\lambda \in (0,1)$ is bounded. Then, the operator A has a fixed point in X.

Theorem 3 ([24]).

Let A be a contraction mapping from a closed subset X of a Banach space Y into X. Then, there exists a unique fixed point $x\in X$, such that $A\left(x\right)=x$.

Theorem 4 ([25]).

Let X be a nonempty compact (closed and bounded) covex set in a Banach space while $A:X\to X$ is a continuous self mapping. Then, A has at least one fixed point in X.

We observe that Schaefer’s and Brouwer’s fixed-point theorems are helpful to derive the existence of the solution and the Banach fixed-point theorem leads to the uniqueness of the solution.

3. Main Results

Throughout this section, we derive Green’s function to demonstrate the uniqueness of solutions for the CFBVP (5). To accomplish this, we present an important result that provides the solution to the CFBVP (5). Consider the Banach space $Y=\mathcal{C}[a,b]$ with maximum norm ${\displaystyle \parallel y\parallel =\underset{t\in [a,b]}{max}\left|y\left(t\right)\right|}$ for $y\in Y$.

We now analyze the following boundary value problem

$$\begin{array}{c}{}^{C}D_{b^{-}}^{\beta ,\Phi }\left[{\varphi }_p{(}^C{D}_{a^{+}}^{\alpha ,\Phi }y\left(t\right))\right]=H\left(t\right),\hfill \\ y\left(a\right)=D_{a^{+}}^{\alpha ,\Phi }y\left(a\right)=0,y\left(b\right)=D_{a^{+}}^{\alpha ,\Phi }y\left(b\right)=0.\hfill \end{array}$$

(8)

Lemma 3.

Assume that $1<\alpha , \beta \le 2$ and $H\in C\left(\right[a,b],\mathbb{R})$. Then, $y=y\left(t\right)$ is a solution if and only if

$$\begin{array}{c}y\left(t\right)={\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )H\left(\tau \right)d\tau \right)ds,\hfill \end{array}$$

(9)

where the Green’s functions ${\mathcal{K}}_{*}^1(t,s)$ and ${\mathcal{K}}_{*}^2(t,s)$ given by

$$\begin{array}{c}{\mathcal{K}}_{*}^1(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{\Phi \left(b\right)-\Phi \left(t\right)}{\Phi \left(b\right)-\Phi \left(a\right)}}{(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1},\hfill & s<t,\hfill \\ \\ {\displaystyle \frac{\Phi \left(b\right)-\Phi \left(t\right)}{\Phi \left(b\right)-\Phi \left(a\right)}}{(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}-{(\Phi \left(s\right)-\Phi \left(t\right))}^{\beta -1},\hfill & s\ge t,\hfill \end{array}\right.\hfill \end{array}$$

(10)

and

$$\begin{array}{c}{\mathcal{K}}_{*}^2(t,s)=\left\{\begin{array}{cc}{\displaystyle \frac{\Phi \left(t\right)-\Phi \left(a\right)}{\Phi \left(b\right)-\Phi \left(a\right)}}{(\Phi \left(b\right)-\Phi \left(s\right))}^{\alpha -1},\hfill & s>t,\hfill \\ \\ {\displaystyle \frac{\Phi \left(t\right)-\Phi \left(a\right)}{\Phi \left(b\right)-\Phi \left(a\right)}}{(\Phi \left(b\right)-\Phi \left(s\right))}^{\alpha -1}-{(\Phi \left(t\right)-\Phi \left(s\right))}^{\alpha -1},\hfill & s\le t.\hfill \end{array}\right.\hfill \end{array}$$

(11)

Proof. 

Let $-{\varphi }_p\left(D_{a^{+}}^{\alpha ,\Phi }y\left(t\right)\right):=v\left(t\right)$, and then the boundary value problem (8) changes to

$$\begin{array}{c}\begin{array}{c}-D_{b^{-}}^{\beta ,\Phi }v\left(t\right)=H\left(t\right),v\left(a\right)=0,v\left(b\right)=0,\hfill \end{array}\hfill \end{array}$$

and

$$\begin{array}{c}\begin{array}{c}{D}_{a^{+}}^{\alpha ,\Phi }y\left(t\right)=-{\varphi }_{q}v\left(t\right):=k\left(t\right),y\left(a\right)=0,y\left(b\right)=0.\hfill \end{array}\hfill \end{array}$$

(12)

Solving the equation $D_{b^{-}}^{\beta ,\Phi }v\left(t\right)=-H\left(t\right)$, we obtain

$$\begin{array}{c}v\left(t\right)=-{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_t^b{\Phi }^{\prime }\left(s\right){(\Phi \left(s\right)-\Phi \left(t\right))}^{\beta -1}H\left(s\right)ds+c_0+c_1(\Phi (b)-\Phi (t\left)\right),\hfill \end{array}$$

where $c_0$ and $c_1$ are constants. Using the second boundary condition $v\left(b\right)=0$ yields $c_0=0$. Since the first condition $v\left(a\right)=0$, then

$$\begin{array}{c}-{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}H\left(s\right)ds+c_1(\Phi (b)-\Phi (a\left)\right)=0,\hfill \\ \hfill c_1={\displaystyle \frac{1}{\Gamma \left(\beta \right)(\Phi (b)-\Phi (a\left)\right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}H\left(s\right)ds.\end{array}$$

So,

$$\begin{array}{ccc}\hfill v\left(t\right)& =& {\displaystyle \frac{\Phi \left(b\right)-\Phi \left(t\right)}{\Gamma \left(\beta \right)(\Phi (b)-\Phi (a\left)\right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}H\left(s\right)ds\hfill \\ & & -{\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_t^b{\Phi }^{\prime }\left(s\right){(\Phi \left(s\right)-\Phi \left(t\right))}^{\beta -1}H\left(s\right)ds\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^1(t,s)H\left(s\right)ds,\hfill \end{array}$$

where ${\mathcal{K}}_{*}^1(t,s)$ is given in (10), and applying the integral $I_{a^{+}}^{\alpha ,\Phi }$ to both sides of the differential equation in (12), we have

$$\begin{array}{c}y\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\Phi }^{\prime }\left(s\right){(\Phi \left(t\right)-\Phi \left(s\right))}^{\alpha -1}k\left(s\right)ds+d_0+d_1(\Phi (t)-\Phi (a\left)\right),\hfill \end{array}$$

where $d_0$ and $d_1$ are constants. Using the boundary conditions in (12), we obtain $d_0=0$ and

$$\begin{array}{c}{d}_1={\displaystyle \frac{-1}{\Gamma \left(\alpha \right)(\Phi (b)-\Phi (a\left)\right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){(\Phi \left(b\right)-\Phi \left(s\right))}^{\alpha -1}k\left(s\right)ds.\hfill \end{array}$$

We determine that

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^t{\Phi }^{\prime }\left(s\right){(\Phi \left(t\right)-\Phi \left(s\right))}^{\alpha -1}k\left(s\right)ds\hfill \\ & & -{\displaystyle \frac{\Phi \left(t\right)-\Phi \left(a\right)}{\Gamma \left(\alpha \right)(\Phi (b)-\Phi (a\left)\right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){(\Phi \left(b\right)-\Phi \left(s\right))}^{\alpha -1}k\left(s\right)ds\hfill \\ & =& {\displaystyle \frac{-1}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s)k\left(s\right)ds.\hfill \end{array}$$

□

Hence, we find that

$$\begin{array}{ccc}\hfill y\left(t\right)& =& {\displaystyle \frac{-1}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s)(-{\varphi }_{q}v\left(s\right))ds\hfill \\ & =& {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )H\left(\tau \right)d\tau \right)ds\hfill \\ & =& {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )H\left(\tau \right)d\tau \right)ds.\hfill \end{array}$$

Lemma 4.

The following attributes are inherent to ${\mathcal{K}}_{*}^1(t,s)$ and ${\mathcal{K}}_{*}^2(t,s)$ on $[a,b]\times [a,b]$:

$\left(i\right)$ ${\mathcal{K}}_{*}^1(t,s)$ and ${\mathcal{K}}_{*}^2(t,s)$ are continuous functions;

$\left(ii\right)$ ${\mathcal{K}}_{*}^1(t,s)$ are non-negative function.

Proof. 

$\left(i\right)$ Given that $\Phi$ is continuous function on $[a,b]$, it can be inferred that ${\mathcal{K}}_{*}^1(t,s)$ and ${\mathcal{K}}_{*}^2(t,s)$ are continuous functions.

$\left(ii\right)$ For $s\ge t$, we have

$$\begin{array}{ccc}\hfill {\mathcal{K}}_{*}^1(t,s)& =& {\displaystyle \frac{\Phi \left(b\right)-\Phi \left(t\right)}{\Phi \left(b\right)-\Phi \left(a\right)}}{(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}-{(\Phi \left(s\right)-\Phi \left(t\right))}^{\beta -1}\hfill \\ & =& {\displaystyle \frac{\Phi \left(b\right)-\Phi \left(t\right)}{\Phi \left(b\right)-\Phi \left(a\right)}}{(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}-{(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}{\left[1-{\displaystyle \frac{\Phi \left(t\right)-\Phi \left(a\right)}{\Phi \left(s\right)-\Phi \left(a\right)}}\right]}^{\beta -1}\hfill \\ & =& {(\Phi \left(s\right)-\Phi \left(a\right))}^{\beta -1}\left[{\displaystyle \frac{\Phi \left(b\right)-\Phi \left(t\right)}{\Phi \left(b\right)-\Phi \left(a\right)}}-{\left(1-{\displaystyle \frac{\Phi \left(t\right)-\Phi \left(a\right)}{\Phi \left(s\right)-\Phi \left(a\right)}}\right)}^{\beta -1}\right]\ge 0.\hfill \end{array}$$

□

Let us define the operator $A:Y⟶Y$ with

$$\begin{array}{c}Ay\left(t\right)={\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right)ds,t\in [a,b].\hfill \end{array}$$

(13)

It is easy to see that the operator A is well defined.

Given number $R>0$, denote

$$\begin{array}{c}{D}_R=\left\{(t,y):a\le t\le b,\left|y\right|\le {\displaystyle \frac{R^{q-1}{M}_1^{q-1}{M}_2}{\Gamma \left(\alpha \right){(\Gamma \left(\beta \right))}^{q-1}}}\right\},\hfill \end{array}$$

(14)

and by a closed ball $B[O,R]$ in the space of continuous functions $C[a,b]$.

Lemma 5.

For given $F(t,y(t\left)\right)\in C\left(\right[a,b]\times \mathbb{R},\mathbb{R})$, set

$$\begin{array}{c}{\displaystyle w\left(s\right)={\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right),y\left(t\right)=\frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s)w\left(s\right)ds,}\hfill \\ {\displaystyle M_1:={\int }_a^b{\Phi }^{\prime }\left(s\right)\underset{t\in [a,b]}{max}{\mathcal{K}}_{*}^1(t,s)ds,M_2:={\int }_a^b{\Phi }^{\prime }\left(s\right)\underset{t\in [a,b]}{max}\left|{\mathcal{K}}_{*}^2(t,s)\right|ds.}\hfill \end{array}$$

Then

$$\begin{array}{c}\begin{array}{c}{\displaystyle \parallel w\parallel \le M_1^{q-1}{\parallel F\parallel }^{q-1},\parallel y\parallel \le \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}{M}_1^{q-1}{M}_2{\parallel F\parallel }^{q-1}.}\hfill \end{array}\hfill \end{array}$$

Proof. 

Since ${\varphi }_q$ is increasing, then

$$\begin{array}{ccc}\hfill \left|w\right(s\left)\right|& \le & {\displaystyle {\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right)\left|{\mathcal{K}}_{*}^1(s,\tau )\right|\left|F(\tau ,y\left(\tau \right))\right|d\tau \right)}\hfill \\ & \le & {\displaystyle {\parallel F\parallel }^{q-1}{\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right)\underset{s\in [a,b]}{max}{\mathcal{K}}_{*}^1(s,\tau )d\tau \right)}\hfill \\ & =& {\parallel F\parallel }^{q-1}{M}_1^{q-1}.\hfill \end{array}$$

So, $\parallel w\parallel \le M_1^{q-1}{\parallel F\parallel }^{q-1}$. Similarly, ${\displaystyle \parallel y\parallel \le \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}{M}_1^{q-1}{M}_2{\parallel F\parallel }^{q-1}}$ holds. □

Theorem 5.

Assume that $1<p\le 2$, and that there exist some numbers $R,Q>0$ such that the following conditions hold:

$\left(H_1\right)$ $\left|F\right(t,y\left(t\right)\left)\right|\le R$ for $(t,y)\in D_R$,

$\left(H_2\right)$ $|F(t,y_2)-F(t,y_1)|\le Q|y_2-y_1|$ for $(t,y_1),(t,y_2)\in D_R$,

$\left(H_3\right)$ ${\displaystyle L:=\frac{(q-1)R^{q-2}{M}_1^{q-1}{M}_{2}Q}{\Gamma \left(\alpha \right){(\Gamma \left(\beta \right))}^{q-1}}<1}$.

Then, the CFBVP (5) has a unique solution.

Proof. 

First of all, define an operator

$$\begin{array}{c}A:C[a,b]⟶C[a,b]\hfill \end{array}$$

by

$$\begin{array}{c}\left(Ay\right)\left(t\right)=F\left(t,{\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right)ds\right).\hfill \end{array}$$

From the continuity of ${\mathcal{K}}_{*}^1(s,\tau ),{\mathcal{K}}_{*}^2(t,s)$ and $F(t,y(t\left)\right)$, it is not hard to see that the operator A is a continuous operator. According to Lemma 3, it is easy to draw the following conclusion that if the CFBVP (5) has a solution $y\left(t\right)$, then $F(t,y\left(t\right)){=}^C{D}_{b^{-}}^{\beta ,\Phi }\left({\varphi }_p{(}^C{D}_{a^{+}}^{\alpha ,\Phi }y\left(t\right))\right)$ is the fixed point of the operator A. On the contrary, if $F(t,y(t\left)\right)$ is a fixed point of the operator A, then

$$\begin{array}{c}y\left(t\right)={\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right)ds.\hfill \end{array}$$

is also the solution to the CFBVP (5).

Next, what we need to prove is that the operator A maps $B[O,R]$ into itself. Given $F(t,y(t\left)\right)\in B[O,R]$, according to Lemma 5, we have $\left|w\left(s\right)\right|\le M_1^{q-1}{R}^{q-1}$, ${\displaystyle \left|y\left(t\right)\right|\le \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}{M}_1^{q-1}{M}_2{R}^{q-1}}$. Consequently, for any $t\in [a,b]$, we have $(t,y\left(t\right))\in D_R$. So, based on $\left(H_1\right)$, we can conclude that $\left|\right(Ay\left)\right(t\left)\right|=\left|F\right(t,y\left(t\right)\left)\right|\le R$, therefore $\left(Ay\right)\left(t\right)\in B[O,R]$. Namely, the operator A is an operator that maps $B[O,R]$ into itself. Finally, the operator

$$\begin{array}{c}A:B[O,R]⟶B[O,R]\hfill \end{array}$$

is proven to be a contraction mapping. Obviously, $B[O,R]$ is proven to be a complete distance space on account of the fact that $B[O,R]$ is a subspace of $C\left(\right[a,b],\parallel .\parallel )$. From $\left(H_2\right)$, Lemmas 2 and 5, we have

$$\begin{array}{c}{\displaystyle |{\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau |\le R{M}_1:=N,}\hfill \end{array}$$

for each $y_1\left(t\right),y_2\left(t\right)\in B[O,R]$ and $1<p\le 2$; that is, $q\ge 2$, we gain

$|\left(A{y}_2\right)\left(t\right)-\left(A{y}_1\right)\left(t\right)|$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}(q-1)N^{q-2}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right)|{\mathcal{K}}_{*}^2(t,s)\left|\right|{\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )(F(\tau ,y_2\left(\tau \right))-F(\tau ,y_1\left(\tau \right)))d\tau |ds\hfill \\ & \le & {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}(q-1)N^{q-2}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right)\left|{\mathcal{K}}_{*}^2(t,s)\right|\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )Q\parallel y_2-y_1\parallel d\tau \right)ds\hfill \\ & \le & {\displaystyle \frac{(q-1){\left(R{M}_1\right)}^{q-2}Q{M}_1{M}_2}{\Gamma \left(\alpha \right)\Gamma {\left(\beta \right)}^{q-1}}}\parallel y_2-y_1\parallel \hfill \\ & =& L\parallel y_2-y_1\parallel .\hfill \end{array}$$

Therefore, the operator $A:B[O,R]⟶B[O,R]$ is a contraction mapping. Hence, based on the Banach contraction mapping principle, it follows that the operator A has a unique fixed point which is the unique solution of problem CFBVP (5) on $[a,b]$. This completes the proof. □

Theorem 6.

Assume that $p>1$, $F:[a,b]\times \mathbb{R}⟶\mathbb{R}$ is a continuous function and there exists a constant $M>0$ such that $\left|F\right(t,y\left(t\right)\left)\right|\le M$ for each $t\in [a,b]$ and $y\in C[a,b]$. Then, the CFBVP (5) has a solution on $[a,b]$.

Proof. 

To prove that the operator A defined by (13) has at least one fixed point, we will use Schaefer’s fixed-point theorem. So, we separate the proof into four steps.

Step 1. Let $y_n$ be a sequence of function such that $y_n⟶y$ on $[a,b]$. Given that F is a continuous function on $[a,b]$, we have $F(t,y_n\left(t\right))⟶F(t,y\left(t\right))$. Therefore, we obtain

$|\left(A{y}_n\right)\left(t\right)-\left(Ay\right)\left(t\right)|$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{{\int }_a^b{\Phi }^{\prime }\left(s\right)\left|{\mathcal{K}}_{*}^2(t,s)\right||{\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y_n\left(\tau \right))d\tau \right)-{\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right)|ds}{\Gamma \left(\alpha \right)\Gamma {\left(\beta \right)}^{q-1}}},\hfill \end{array}$$

and ${\varphi }_q$ is continuous function which implies that $\parallel A{y}_n-Ay\parallel ⟶0$ as $n⟶\infty$. Hence, the operator A is continuous.

Step 2. Choosing $r>0$ with ${\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}{M}^{q-1}{M}_1^{q-1}{M}_2}{\Gamma \left(\alpha \right)}}\le r$, where $M_1$ and $M_2$ are given in Lemma 5, we define a bounded ball $B_r$ as $B_r=\{y\in C[a,b]:\parallel y\parallel \le r\}$. Then, for any $y\in B_r$, we have

$$\begin{array}{ccc}\hfill \left|\right(Ay\left)\right(t\left)\right|& =& \left|{\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right)ds\right|\hfill \\ & \le & {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right)\left|{\mathcal{K}}_{*}^2(t,s)\right|{\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )\left|F(\tau ,y\left(\tau \right))\right|d\tau \right)ds\hfill \\ & \le & {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}{M}^{q-1}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right)\left|{\mathcal{K}}_{*}^2(t,s)\right|{\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right)\underset{s\in [a,b]}{max}{\mathcal{K}}_{*}^1(s,\tau )d\tau \right)ds\hfill \\ & \le & {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}{M}^{q-1}{M}_1^{q-1}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right)\underset{t\in [a,b]}{max}\left|{\mathcal{K}}_{*}^2(t,s)\right|ds\hfill \\ & =& {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}{M}^{q-1}{M}_1^{q-1}{M}_2}{\Gamma \left(\alpha \right)}}.\hfill \end{array}$$

This means that $\parallel Ay\parallel \le r$. Therefore, the set $A\left(B_r\right)$ is uniformly bounded.

Step 3. Let $t_1,t_2\in [a,b]$ with $t_1<t_2$ be two points and $B_r$ be a bounded ball in $C[a,b]$. Then, for any $y\in B_r$, we get

$|\left(Ay\right)\left(t_2\right)-\left(Ay\right)\left(t_1\right)|$

$$\begin{array}{ccc}& =& {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}\left|{\int }_a^b{\Phi }^{\prime }\left(s\right)\left[{\mathcal{K}}_{*}^2(t_2,s)-{\mathcal{K}}_{*}^2(t_1,s)\right]{\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right)ds\right|\hfill \\ & \le & {\displaystyle \frac{\Gamma {\left(\beta \right)}^{1-q}}{\Gamma \left(\alpha \right)}}{\int }_a^b{\Phi }^{\prime }\left(s\right)|{\mathcal{K}}_{*}^2(t_2,s)-{\mathcal{K}}_{*}^2(t_1,s)|\left|{\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,y\left(\tau \right))d\tau \right)\right|ds.\hfill \end{array}$$

As $t_2⟶t_1$, the right-hand side of the above inequality tends to zero. This shows that the set $A\left(B_r\right)$ is an equicontinuous set. Therefore, both uniform boundedness and equicontinuity of the family $\{Ay:y\in B_r\}\subset C([a,b];\mathbb{R})$ are rigorously justified, allowing us to conclude the relative compactness of $A\left(B_r\right)$ in $C\left(\right[a,b];\mathbb{R})$ via the Arzela–Ascoli theorem. So, we deduce that the operator $A:C[a,b]⟶C[a,b]$ is completely continuous.

Step 4. Finally, we show that the set $E=\{y\in C[a,b]:y=\lambda Ay\mathrm{for}\mathrm{some}\lambda \in (0,1\left)\right\}$ is bounded. Let $y\in E$ be a solution to the problem CFBVP (5). Then, $y\left(t\right)=\lambda \left(Ay\right)\left(t\right),\lambda \in (0,1)$. Hence, for each $t\in [a,b]$, by the method of computation in Step 2, we obtain $\parallel Ay\parallel \le r$. This reveals that the set E is bounded. By applying Schaefer’s fixed-point theory, we find that A has at least one fixed point which is a solution of the CFBVP (5) on $[a,b]$. The proof is completed. □

Corollary 1.

Under the conditions of Theorem 6, the CFBVP (5) has at least one solution.

Proof. 

Consider $B_r=\{y\in C[a,b]:\parallel y\parallel \le r\mathrm{with}{\displaystyle \frac{M^{q-1}{M}_1^{q-1}{M}_2}{\Gamma \left(\alpha \right)\Gamma {\left(\beta \right)}^{q-1}}}\le r\}$. Clearly, $B_r$ is a non-empty compact convex subset of $C[a,b]$. Let A be an operator as defined in (13). It is clear that A is a continuous operator. Similarly, via Step 2 in the proof of Theorem 6, we get $\parallel Ay\parallel \le r$ and thus $A:B_r⟶B_r.$ It follows that when using the Brouwer fixed-point theorem, there exists a fixed point such that $\parallel y\parallel \le r.$ The proof is completed. □

Example 1.

Consider the following boundary value problem:

$$\begin{array}{c}\begin{array}{c}{D}_{1^{-}}^{{\displaystyle \frac{3}{2}},t}\left({\varphi }_p\left(D_{0^{+}}^{{\displaystyle \frac{3}{2}},t}y\left(t\right)\right)\right)={\displaystyle \frac{sin(1+y)}{{10}^2+t^2}},t\in [0,1],\hfill \\ y\left(0\right)=D_{0^{+}}^{{\displaystyle \frac{3}{2}},t}y\left(0\right)=0,y\left(1\right)=D_{0^{+}}^{{\displaystyle \frac{3}{2}},t}y\left(1\right)=0.\hfill \end{array}\hfill \end{array}$$

(15)

Note that (15) is a particular case of (5) with $\Phi \left(t\right)=t$, $F(t,y)={\displaystyle \frac{sin(1+y)}{{10}^2+t^2}}$ and $a=0,b=1,\alpha =\beta ={\displaystyle \frac{3}{2}},p=2$.

From the given data, we have

$${\mathcal{K}}_{*}^1(t,s)=\{\begin{array}{cc}(1-t)\sqrt{s},\hfill & s\le t,\hfill \\ (1-t)\sqrt{s}-\sqrt{s-t},\hfill & s\ge t,\end{array}$$

$${\mathcal{K}}_{*}^2(t,s)=\{\begin{array}{cc}t\sqrt{1-s},\hfill & s\ge t,\hfill \\ t\sqrt{1-s}-\sqrt{t-s},\hfill & s\le t.\end{array}$$

First, we will show that the conditions $\left(H_1\right)$–$\left(H_3\right)$ in Theorem 5 are satisfied.

$\left(H_1\right)$ $F(t,y)={\displaystyle \frac{sin(1+y)}{{10}^2+t^2}}$, $\forall (t,y)\in [0,1]\times \mathbb{R}$ is continuous and $\left|f(t,y)\right|=|{\displaystyle \frac{sin(1+y)}{{10}^2+t^2}}|\le {\displaystyle \frac{1}{100}}=R$,

$\left(H_2\right)$ $|F(t,y_2)-F(t,y_1)|=|{\displaystyle \frac{sin(1+y_2)}{{10}^2+t^2}}-{\displaystyle \frac{sin(1+y_1)}{{10}^2+t^2}}|\le {\displaystyle \frac{|y_2-y_1|}{{10}^2+t^2}}\le {\displaystyle \frac{\parallel y_2-y_1\parallel }{100}}$, $Q={\displaystyle \frac{1}{100}}$.

We can easily calculate

$$\begin{array}{ccc}\hfill M_1& =& {\int }_0^1\underset{t\in [0,1]}{max}{\mathcal{K}}_{*}^1(t,s)ds={\int }_0^1\sqrt{s}ds={\displaystyle \frac{2}{3}},\hfill \\ \hfill M_2& =& {\int }_0^1\underset{t\in [0,1]}{max}{\mathcal{K}}_{*}^2(t,s)ds={\int }_0^1\sqrt{1-s}ds={\displaystyle \frac{2}{3}}.\hfill \end{array}$$

$\left(H_3\right)$$L:={\displaystyle \frac{\frac{1}{{10}^2}\frac{2^4}{3^2}}{\Gamma {\left(\frac{3}{2}\right)}^2}}={\displaystyle \frac{2^4}{{10}^{2}9\pi }}=0,005658843<1.$

According to Theorem 5, (15) has a unique solution, such that $\parallel y\parallel \le {\displaystyle \frac{16}{900\pi }}$.

4. Generalized Ulam Stabilities

The stability of the solution plays a crucial role in qualitative theory, as finding an approximate solution is often challenging for most nonlinear problems. By utilizing Ulam–Hyers stability, we can establish the stability of the solution for fractional-order problems. In this section, we present a theorem showing that the CFBVP (5) admits both Ulam–Hyers and generalized Ulam–Hyers stabilities. By using the

$$\begin{array}{c}x\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right){(\Gamma \left(\beta \right))}^{q-1}}}{\int }_a^b{\Phi }^{\prime }\left(s\right){\mathcal{K}}_{*}^2(t,s){\varphi }_q\left({\int }_a^b{\Phi }^{\prime }\left(\tau \right){\mathcal{K}}_{*}^1(s,\tau )F(\tau ,x\left(\tau \right))d\tau \right)ds.\hfill \end{array}$$

(16)

we discuss the Ulam Stability, here $x\left(t\right)\in C\left(\right[a,b],\mathbb{R})$ and $F:[a,b]\times \mathbb{R}\to \mathbb{R}$ is continuous function. Then, we define the nonlinear continuous operator

$$\begin{array}{c}P:C([a,b],\mathbb{R})\to C([a,b],\mathbb{R})\hfill \\ Px\left(t\right){=}^C{D}_{b^{-}}^{\beta ,\Phi }\left({\varphi }_p{(}^C{D}_{a^{+}}^{\alpha ,\Phi }x\left(t\right))\right)-F(t,x\left(t\right)).\hfill \end{array}$$

(17)

Definition 4.

For each $\epsilon >0$ and for each solution $x\left(t\right)$ of (5), such that

$$\begin{array}{c}∥Px∥\le \epsilon ,\hfill \end{array}$$

(18)

the problem (5), is said to have Ulam–Hyers stability if we can find a positive real number ν and a solution $y\left(t\right)\in C\left(\right[a,b],\mathbb{R})$ of (5) satisfying the inequality

$$\begin{array}{c}\left|y-x\right|\le \nu {\epsilon }^{*},\hfill \end{array}$$

where ${\epsilon }^{*}$ is a positive real number depending on ε.

Definition 5.

Let $m\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$, such that

$$\begin{array}{c}\left|y\left(t\right)-x\left(t\right)\right|\le m\left(\epsilon \right),t\in [a,b].\hfill \end{array}$$

Then, the problem (5) is said to be generalized Ulam–Hyers stable.

Theorem 7.

Under assumptions (H1)–(H3), the problem (5) is both Ulam–Hyers and generalized Ulam–Hyers stable.

Proof. 

Consider $y\in C\left(\right[a,b],\mathbb{R})$ to be a solution of (5) in the sense of Theorem 5 and let x be any solution that satisfies (16). Let the operator P be defined as in Equation (17) and A as in Equation (13). Then, Lemma 3 establishes the equivalence between the operators P and $A-Id$ (where Id is the identity operator) for every solution $x\in C\left(\right[a,b],\mathbb{R})$ of (5). This relation is used in the proof to connect the fixed-point formulation and the stability analysis framework.

Therefore, we deduce by the fixed-point property of the operator A that

$$\begin{array}{cc}\hfill \left|y\right(t)-x(t\left)\right|& =\left|y\right(t)-Ax(t)+Ax(t)-x(t\left)\right|\hfill \\ & =\left|x\right(t)-Ax(t)+Ax(t)-Ay(t\left)\right|\hfill \\ & \le \left|Ax\right(t)-Ay(t\left)\right|+\left|Ax\right(t)-x(t\left)\right|\hfill \\ & \le ∥Ax-Ay∥+∥Ax-x∥\hfill \\ & <L∥x-y∥+∥(A-I)x∥\hfill \\ & =L∥x-y∥+∥Px∥\hfill \\ & \le L∥x-y∥+\epsilon .\hfill \end{array}$$

Because $L<1$ and $\epsilon >0$, we find

$$\begin{array}{c}\left|x-y\right|\le {\displaystyle \frac{\epsilon }{1-L}}=m\left(\epsilon \right).\hfill \end{array}$$

The generalized Ulam–Hyers stability follows by taking $m\left(\epsilon \right)={\displaystyle \frac{\epsilon }{1-L}}$. In addition, fixing ${\epsilon }^{*}={\displaystyle \frac{\epsilon }{1-L}}$ and $\nu =1$. □

5. Conclusions

This study investigates the $\Phi$-Caputo fractional boundary value problem with the p-Laplacian operator, focusing on existence, uniqueness, and stability. We established the existence and uniqueness of solutions using fixed-point theorems, ensuring a unique solution for given initial conditions and boundary constraints. Stability was analyzed through the generalized Ulam–Hyers method, demonstrating that the solutions are stable under certain conditions. The stability of a fractional boundary value problem such as Ulam–Hyers or generalized Ulam–Hyers stability is closely linked to certain parameters involved in the formulation of the problem, particularly the Lipschitz constant L and error tolerance $\epsilon$. These parameters have effects on the stability of the system, influencing the behavior of the solution and its sensitivity to changes in initial conditions or perturbations. The results have significant implications for modeling real-world phenomena such as viscoelastic materials, porous media, and biological processes, where nonlocal and nonlinear behaviors are involved. Overall, this study enhances the understanding of fractional boundary value problems with p-Laplacian dynamics and provides a foundation for future research on fractional calculus and its applications in nonlinear systems.