Hyers–Ulam Stability Results of Solutions for a Multi-Point φ-Riemann-Liouville Fractional Boundary Value Problem

Abstract

In this study, we investigate the existence, uniqueness, and Hyers–Ulam stability of a multi-term boundary value problem involving generalized $\phi$-Riemann–Liouville operators. The uniqueness of the solution is demonstrated using Banach’s fixed-point theorem, while the existence is established through the application of classical fixed-point theorems by Krasnoselskii. We then delve into the Hyers–Ulam stability of the solutions, an aspect that has garnered significant attention from various researchers. By adapting certain sufficient conditions, we achieve stability results for the Hyers–Ulam (HU) type. Finally, we illustrate the theoretical findings with examples to enhance understanding.

1. Introduction

The field of fractional calculus has garnered significant attention from researchers due to its extensive applications in addressing practical problems across various domains such as viscoelasticity, biological sciences, ecology, and aerodynamics. Numerous studies have shown that fractional-order differential equations offer versatile methods to tackle intricate issues in statistical physics and environmental science. For example, the evolution of fractional calculus is elaborated in [1], while [2] highlights some groundbreaking applications in the field. Specific applications can be found in [3,4] and the related references. The emergence of fractional-order differential equations with boundary value problems has been notable, with multi-point boundary conditions and integral boundary conditions becoming focal points of research. The investigations presented in [5,6] are particularly commendable. Despite this, the majority of researchers tend to focus exclusively on either integral conditions or multi-point conditions. The study of non-linear fractional-order differential equations has been extensively explored due to their applications in the modeling of viscoelasticity [7]. Various methods have been developed to address damping in mechanical systems using fractional operators [8]. The fractional calculus approach has also been applied to protein dynamics [9] and continuum mechanics [10]. In addition, significant contributions have been made in the theory of fixed points [11] and in the relaxation of polymers [12].

Recent studies have investigated numerical solutions for multi-term fractional boundary value problems [13], including those with $\phi$-Riemann–Liouville operators [14]. The existence and uniqueness of solutions for multi-point boundary value problems have also been well documented [15,16]. Several works have focused on the positive solutions of fractional differential equations [17] and the solvability of specific classes of impulsive fractional differential equations [18]. In particular, Zhao et al. have contributed significantly to the study of integral boundary value problems for higher-order nonlinear fractional differential equations [6,19]. The eigenvalue problems for singular higher-order fractional differential equations have also been explored [20].

Moreover, the existence results for fractional differential equations with integral and multi-point boundary conditions have been established [21]. The study of higher-order nonlinear fractional differential equations with changing sign measures has provided valuable information in the field [22], and multiple positive solutions for singular fractional differential equations have been found [23]. In recent decades, many mathematicians have investigated the existence and stability of solutions to their proposed FBVPs; among them, Rezapour et al. [13] studied the following multi-term fractional BVP in 2021:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{\varrho ;\phi }u\left(z\right)=h\left(z,u\left(z\right),{\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }u\left(z\right),{\mathcal{D}}_{0^{+}}^{{\kappa }_2;\phi }u\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_n;\phi }u\left(z\right)\right),\hfill \\ u\left(0\right)=0,\hfill \\ u\left(1\right)=p{I}_{0^{+}}^{\mu ;\phi }{k}_1\left(\xi ,u\left(\xi \right)\right)+q{I}_{0^{+}}^{\nu ;\phi }{k}_2\left(\eta ,u\left(\eta \right)\right),\hfill \end{array}\right.$$

where $0\le z\le 1$, $1<\varrho <2$, $0<{\kappa }_1<{\kappa }_2<\cdots <{\kappa }_n<1$, $\varrho >{\kappa }_n+1$, $h:[0,1]\times {\mathbb{R}}^{n+1}\to \mathbb{R}$, and $k_j:[0,1]\times \mathbb{R}\to \mathbb{R}(j=1,2)$ are continuous functions; ${\mathcal{D}}_{0^{+}}^{\varrho ;\phi }$, ${\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }$,…, ${\mathcal{D}}_{0^{+}}^{{\kappa }_n;\phi }$ are the $\phi$-RL-derivative, depending on an increasing function $\phi$ of order $\varrho$, ${\kappa }_1$,…,${\kappa }_n$ respectively; and $I_{0^{+}}^{\gamma ;\phi }$ is the $\phi$-RL-integral, depending on the special function $\phi$ of order $\gamma \in \{\mu ,\nu \}$, with $\mu ,\nu ,p,q>0$, $0<\xi ,\eta \le 1$, they discuss the existence and uniqueness of the last $\phi$-FBVP, utilizing the conception of Banach’s principle fixed-point theorem, and they applied both of the numerical techniques DGJIM and ADM to find approximate solutions for the BVP and compare the results of these methods. For very recent papers that study the existence, uniqueness, and stability of both nabla and delta fractional difference problems, please see [24,25,26,27].

Jia, Gu, and Zhang [6] studied the following higher fractional differential equation with fractional multi-point boundary conditions using the reduced order method as follows:

$$\left\{\begin{array}{c}-{\mathcal{D}}^{\gamma }x\left(z\right)=f\left(z,x\left(z\right),{\mathcal{D}}^{{\mu }_1}x\left(z\right),\cdots ,{\mathcal{D}}^{{\mu }_{n-1}}x\left(z\right)\right),0<z<1,\hfill \\ {\displaystyle x\left(0\right)=0,{\mathcal{D}}^{{\mu }_i}x\left(0\right)=0,{\mathcal{D}}^{\mu }x\left(1\right)=\sum _{j=1}^{p-2}{a}_j{\mathcal{D}}^{\mu }x\left({\xi }_j\right),1\le i\le n-1,}\hfill \end{array}\right.$$

where $n\ge 3$, $n\in \mathbb{N}$, $n-1<\gamma <n$, and $n-l-1<\gamma -{\mu }_l<n-l$ for all $l=1,2,\cdots ,n-2$, and $\mu -{\mu }_{n-1}>0$, $\gamma -{\mu }_{n-1}\le 2$, $\gamma -\mu >1$, $a_j\in [0;+\infty )$, $0<{\xi }_1<{\xi }_2<\cdots <{\xi }_{p-2}<1$, and ${\displaystyle \sum _{j=1}^{p-2}}{a}_j{\xi }_j^{\gamma -\mu -1}\ne 1$. ${\mathcal{D}}^{\gamma }$ is the standard RL-derivative, and $f:[0,1]\times {\mathbb{R}}^n\to \mathbb{R}$ is a continuous function. They investigated the existence and uniqueness of a nontrivial solution by applying the Leray–Schauder nonlinear alternative and Schauder’s fixed-point theorem.

In [14], the authors used an iterative method to find an approximate solution to the following proposed problem:

$$\left\{\begin{array}{c}{\displaystyle {\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }\left({\mathcal{D}}_{0^{+}}^{{\kappa }_2;\phi }u\right)\left(z\right)=h\left(z,u\left(z\right)\right),z\in \mathcal{O}=[0,1],}\hfill \\ u\left(0\right)=0,\hfill \\ u\left(1\right)=\lambda I_{0^{+}}^{{\kappa }_3;\phi }g\left(\xi ,u\left(\xi \right)\right),\hfill \end{array}\right.$$

(1)

where $0<{\kappa }_1,{\kappa }_2,\xi <1$, ${\kappa }_1+{\kappa }_2>1$, and $h,g:[0,1]\times \mathbb{R}\to \mathbb{R}$ are continuous functions, ${\mathcal{D}}_{0^{+}}^{\kappa ;\phi }$ is the $\phi$-RL-derivative operator of order $\kappa$, and $I_{0^{+}}^{{\kappa }_3;\phi }$ is the $\phi$-RL-integral operator of order ${\kappa }_3>0$ and $\lambda >0$. To apply the numerical method, they transform the BVP (1) into an equivalent $\phi$-fractional integral equation by using Banach’s principal fixed-point theorem. A unique solution to the problem was established. For more details on the reasons for considering the $\varphi$-Riemann–Liouville derivative in this boundary value problem and the physical or engineering significance of its results, please see [28,29].

All the results mentioned above motivate us to dedicate our energy to solving the following generalized $\phi$-FBVP of the multi-point differential equation:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{\kappa ;\phi }x\left(z\right)=f\left(z,x\left(z\right),{\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right),z\in \mathcal{O}=[0,1],\hfill \\ x\left(0\right)=0,{\mathcal{D}}_{0^{+}}^{{\kappa }_i;\phi }x\left(0\right)=0,1\le i\le n-2,\hfill \\ {\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)={\displaystyle \sum _{j=1}^{m-2}}{p}_j{\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left({\eta }_j\right)+x_0,\hfill \end{array}\right.$$

(2)

where

$$n\ge 3,n-1<\kappa \le n,\nu >0,p_j\in (0,\infty ),x_0\in \mathbb{R}$$

$$0<{\eta }_1<{\eta }_2<\cdots <{\eta }_{m-2}<1,{\kappa }_1<{\kappa }_2<\cdots <{\kappa }_{n-2},$$

and $f:[0,1]\times {\mathbb{R}}^{n-1}\to \mathbb{R}$ is a continuous function. ${\mathcal{D}}_{0^{+}}^{\gamma ;\phi },I_{0^{+}}^{\beta ;\phi }$ are, respectively, the fractional derivative and integral of order $\gamma$ and $\beta$ in the Riemann–Liouville sense, with respect to the increasing function $\phi$. In the former, we provide the $\phi$-integral equation, which is equivalent to the multi-term $\phi$-RLFBVP (2); then, we study the existence and uniqueness of the solution by means of Krasnoselskii’s and Banach’s fixed-point theorems; and after that, we discuss the stability of our proposed problem.

Our work is organized as follows: In Section 1, we introduce the significance of fractional calculus and its wide-ranging applications in various fields. This section sets the stage for our study by discussing the motivation, relevance, and objectives of our research. Section 2 deals with the necessary mathematical background and basic tools required for our analysis. Section 3 presents the main results on the existence and uniqueness of solutions for the multi-term FBVP. In Section 4, we focus on the Hyers–Ulam stability of the solutions for the multi-point $\phi$-Riemann–Liouville fractional boundary value problem. We explore the concept of Hyers–Ulam stability and its significance in the context of differential equations. This section includes sufficient conditions for the stability of solutions.

2. Preliminaries and Some Basic Tools

Definition 1

([30]). Let $\Phi :[a,b]\to \mathbb{R}$ be integrable and $\phi \in C^n[a,b]$ be an increasing mapping, such that ${\phi }^{\prime }\left(z\right)\ne 0$ for all $z\in [a,b]$. The φ-Riemann–Liouville integral of order $\varrho >0$ of the function Φ is represented as

$$I_{a^{+}}^{\varrho ;\phi }\Phi \left(z\right)={\displaystyle \frac{1}{\Gamma \left(\varrho \right)}}{\int }_a^z{\phi }^{\prime }\left(s\right){\left(\phi \left(z\right)-\phi \left(s\right)\right)}^{\varrho -1}\Phi \left(s\right)ds,$$

and the φ-RL-derivative of order $\varrho >0$ for the same function is defined by

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{a^{+}}^{\varrho ;\phi }\Phi \left(z\right)& =& {\left({\displaystyle \frac{1}{{\phi }^{\prime }\left(z\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{I}_{a^{+}}^{n-\varrho ;\phi }\Phi \left(z\right)\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (n-\varrho )}}{\left({\displaystyle \frac{1}{{\phi }^{\prime }\left(z\right)}}{\displaystyle \frac{d}{dz}}\right)}^n{\int }_a^z{\phi }^{\prime }\left(s\right){\left(\phi \left(z\right)-\phi \left(s\right)\right)}^{n-\varrho -1}\Phi \left(s\right)ds,\hfill \end{array}$$

where $n-1<\varrho \le n$.

Lemma 1

([31]). For $\Phi :[a,b]\to \mathbb{R}$ and $\varrho >\nu >0$, the next property of semi-group is verified as follows:

$$I_{a^{+}}^{\varrho +\nu ;\phi }\Phi \left(z\right)=I_{a^{+}}^{\varrho ;\phi }\Phi \left(z\right)I_{a^{+}}^{\nu ;\phi }\Phi \left(z\right).$$

Definition 2

([32]). Let $\phi \in C^m[a,b]$ with ${\phi }^{\prime }\left(z\right)\ne 0$ for all $z\in [a,b]$. Then, we define

$$A{C}^{m;\phi }[c,d]=\left\{\varphi :[c,d]\to \mathbb{R}and{\varphi }^{[m-1]}\in AC[c,d],{\varphi }^{[m-1]}={\left({\displaystyle \frac{1}{{\phi }^{\prime }\left(z\right)}}{\displaystyle \frac{d}{dz}}\right)}^m\varphi \right\},$$

where

$$AC[c,d]=\left\{f:[c,d]\to \mathbb{R},f\left(x\right)=f\left(c\right)+{\int }_c^{x}g\left(z\right)dz,g\in L^1\left([c,d]\right)\right\}.$$

Lemma 2

([33]). Let $\varrho >0$ and $\nu >0$. If $\omega \left(s\right)={\left[\phi \left(s\right)-\phi \left(a\right)\right]}^{\nu -1}$, then

$$\left({\mathcal{D}}_{a^{+}}^{\varrho ;\phi }\omega \left(s\right)\right)\left(z\right)={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu -\varrho )}}{\left[\phi \left(z\right)-\phi \left(a\right)\right]}^{\nu -\varrho -1},$$

and

$$\left(I_{a^{+}}^{\varrho ;\phi }\omega \left(s\right)\right)\left(z\right)={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu +\varrho )}}{\left[\phi \left(z\right)-\phi \left(a\right)\right]}^{\nu +\varrho -1}.$$

As a particular case, we have

$$\left({\mathcal{D}}_{a^{+}}^{\varrho ;z^{\gamma }}{s}^{\gamma (\nu -1)}\right)\left(z\right)={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu -\varrho )}}{z}^{\gamma (\nu -\varrho -1)},$$

and

$$\left(I_{a^{+}}^{\varrho ;z^{\gamma }}{s}^{\gamma (\nu -1)}\right)\left(z\right)={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu +\varrho )}}{z}^{\gamma (\nu +\varrho -1)}.$$

Lemma 3

([34]). Let $\varrho >n$ with $n\in \mathbb{N}$. Then,

$${\left({\displaystyle \frac{1}{{\phi }^{\prime }\left(z\right)}}{\displaystyle \frac{d}{dz}}\right)}^n{I}_{a^{+}}^{\varrho ;\phi }\Phi \left(z\right)=I_{a^{+}}^{\varrho -n;\phi }\Phi \left(z\right).$$

Lemma 4

([34]). Let $\varrho >\nu$, $n-1<\nu <n$, $n\in \mathbb{N}$. Then

$${\mathcal{D}}_{a^{+}}^{\nu ;\phi }{I}_{a^{+}}^{\varrho ;\phi }\Phi \left(z\right)=I_{a^{+}}^{\varrho -\nu ;\phi }\Phi \left(z\right).$$

In particular,

$${\mathcal{D}}_{a^{+}}^{\nu ;\phi }{I}_{a^{+}}^{\nu ;\phi }\Phi \left(z\right)=\Phi \left(z\right).$$

Lemma 5

([34]). Let $\varrho >0$ and ${\mathcal{D}}_{a^{+}}^{\varrho ;\phi }\Phi \left(z\right)\in A{C}^{n;\phi }[a,b]\cap L^1[a,b]$. Then,

$$I_{a^{+}}^{\varrho ;\phi }{\mathcal{D}}_{a^{+}}^{\varrho ;\phi }\Phi \left(z\right)=\Phi \left(z\right)+\sum _{i=1}^n{k}_i{(\phi \left(z\right)-\phi \left(a\right))}^{\varrho -i},$$

where $k_i\in \mathbb{R}$ for all $i\in \{1,\cdots ,n\}$ and $n-1<\varrho \le n$.

3. Existence and Uniqueness Results

Proposition 1.

Let $n-1<\kappa \le n$, with $n\ge 3$, and

$$0<{\eta }_1<{\eta }_2<\cdots <{\eta }_{m-2}<1,$$

$$0<{\kappa }_1<{\kappa }_2<\cdots <{\kappa }_{n-2}<n-2,$$

$$n-3<{\kappa }_{n-2}<n-2.$$

Let $p_j\in (0,\infty ),1\le j\le m-2$ with $m\ge 3$, $x_0\in \mathbb{R}$, $\kappa >\nu >{\kappa }_{n-2}$, and $1<\kappa -{\kappa }_{n-2}<2$.

If the following condition holds:

$${\kappa }_{n-2}-(n-i-2)<{\kappa }_i<{\kappa }_{n-2}-(n-i-2)+1,i\in \{1,2,\cdots ,n-3\},$$

(3)

then a function x is a solution for the multi-term FBVP (2) if and only if x verifies $y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)$, which satisfies the following integral equation:

$$\begin{array}{ccc}\hfill y\left(z\right)& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^t{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right),\hfill \end{array}$$

where $\gamma =\kappa -{\kappa }_{n-2}$, ${\gamma }_i={\kappa }_{n-2}-{\kappa }_i$ for every $i\in \{1,\cdots ,n-3\}$, ${\gamma }_0={\kappa }_{n-2}$, ${\nu }^{\ast }=\nu -{\kappa }_{n-2}$, and

$$\lambda =\sum _{j=1}^{m-2}{p}_j{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}-{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}\ne 0.$$

Proof. 

For $y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)$, then $y\left(0\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(0\right)=0$; from Lemma 5, it follows that

$$I_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)=x\left(z\right)+\sum _{i=1}^{n-2}{k}_i{(\phi \left(z\right)-\phi \left(0\right))}^{{\kappa }_{n-2}-i}.$$

(4)

Then, we have

$$I_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right){|}_{z=0}=x\left(0\right)+k_1{(\phi \left(z\right)-\phi \left(0\right))}^{{\kappa }_{n-2}-1}{|}_{z=0}+\cdots +k_{n-2}{(\phi \left(z\right)-\phi \left(0\right))}^{{\kappa }_{n-2}-n+2}{|}_{z=0}=0,$$

which implies that $k_{n-2}=0$. Consequently, in view of (3), by taking the operator ${\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }$ to both sides of (4), we find

$$I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_1;\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(z\right)+\sum _{i=1}^{n-3}{k}_i{\displaystyle \frac{\Gamma ({\kappa }_{n-2}-i+1)}{\Gamma ({\kappa }_{n-2}-i+1-{\kappa }_1)}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-i-{\kappa }_1}.$$

Then, we have

$$\begin{array}{ccc}\hfill I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_1;\phi }y\left(0\right)& =& {\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(0\right)+k_1{\displaystyle \frac{\Gamma \left({\kappa }_{n-2}\right)}{\Gamma ({\kappa }_{n-2}-{\kappa }_1)}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-{\kappa }_1-1}{|}_{z=0}\hfill \\ & & +k_2{\displaystyle \frac{\Gamma ({\kappa }_{n-2}-1)}{\Gamma ({\kappa }_{n-2}-{\kappa }_1-1)}}{\left(\phi \left(t\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-{\kappa }_1-2}{|}_{z=0}\hfill \\ & & +\cdots +k_{n-4}{\displaystyle \frac{\Gamma ({\kappa }_{n-2}-n+5)}{\Gamma ({\kappa }_{n-2}-{\kappa }_1-n+5)}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-{\kappa }_1-n+4}{|}_{z=0}\hfill \\ & & +k_{n-3}{\displaystyle \frac{\Gamma ({\kappa }_{n-2}-n+4)}{\Gamma ({\kappa }_{n-2}-{\kappa }_1-n+4)}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-{\kappa }_1-n+3}{|}_{z=0}.\hfill \end{array}$$

Hence, for ${\kappa }_{n-2}-n+3<{\kappa }_1<{\kappa }_{n-2}-n+4$, we obtain

$$I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_1;\phi }y\left(z\right){|}_{z=0}=k_{n-3}{\displaystyle \frac{\Gamma ({\kappa }_{n-2}-n+4)}{\Gamma ({\kappa }_{n-2}-{\kappa }_1-n+4)}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-{\kappa }_1-n+3}{|}_{z=0}=0,$$

which makes $k_{n-3}=0$.

In the same way as from condition (3), for $i\in \{1,\cdots ,n-4\}$, we find

$$k_{n-2}=k_{n-3}=\cdots =k_2=0,$$

and for $i=n-3$, we obtain

$$\begin{array}{ccc}\hfill I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_{n-3};\phi }y\left(0\right)& =& {\mathcal{D}}_{0^{+}}^{{\kappa }_{n-3};\phi }x\left(0\right)+k_1{\displaystyle \frac{\Gamma \left({\kappa }_{n-2}\right)}{\Gamma ({\kappa }_{n-2}-{\kappa }_{n-3})}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-{\kappa }_{n-3}-1}{|}_{z=0}\hfill \\ & =& k_1{\displaystyle \frac{\Gamma \left({\kappa }_{n-2}\right)}{\Gamma ({\kappa }_{n-2}-{\kappa }_{n-3})}}{\left(\phi \left(t\right)-\phi \left(0\right)\right)}^{{\kappa }_{n-2}-{\kappa }_{n-3}-1}{|}_{z=0}=0,\hfill \end{array}$$

which means that $k_1=0$. Finally, we obtain

$$I_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)=x\left(z\right),$$

and

$$\begin{array}{ccc}\hfill x\left(z\right)& =& I_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)=I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\hfill \\ \hfill {\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(t\right)& =& I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_1;\phi }y\left(t\right)=I_{0^{+}}^{{\gamma }_1;\phi }y\left(z\right),\hfill \\ \hfill ⋮& =& ⋮\hfill \\ \hfill {\mathcal{D}}_{0^{+}}^{{\kappa }_{n-3};\phi }x\left(z\right)& =& I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_{n-3};\phi }y\left(z\right)=I_{0^{+}}^{{\gamma }_{n-3};\phi }y\left(z\right),\hfill \\ \hfill {\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)& =& y\left(z\right),\hfill \end{array}$$

where ${\gamma }_i={\kappa }_{n-2}-{\kappa }_i$ for every $i\in \{1,\cdots ,n-3\}$ and ${\gamma }_0={\kappa }_{n-2}$; then, we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{\kappa ;\phi }x\left(z\right)={\mathcal{D}}_{0^{+}}^{\kappa ;\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)& =& {\mathcal{D}}_{0^{+}}^{n;\phi }{I}_{0^{+}}^{n-\kappa ;\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)\hfill \\ & =& {\mathcal{D}}_{0^{+}}^{n;\phi }{I}_{0^{+}}^{n-(\kappa -{\kappa }_{n-2});\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{\kappa -{\kappa }_{n-2};\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{\gamma ;\phi }y\left(z\right),\hfill \end{array}$$

where $\gamma =\kappa -{\kappa }_{n-2}$, ${\kappa }_{n-2}<\nu$ and $q-1<\nu <q$, with $q=\left[\nu \right]+1$, $q\in \mathbb{N}$. Thus, it follows that

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(z\right)={\mathcal{D}}_{0^{+}}^{\nu ;\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)& =& {\mathcal{D}}_{0^{+}}^{q;\phi }{I}_{0^{+}}^{q-\nu ;\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)\hfill \\ & =& {\mathcal{D}}_{0^{+}}^{q;\phi }{I}_{0^{+}}^{q-(\nu -{\kappa }_{n-2});\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{\nu -{\kappa }_{n-2};\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(z\right).\hfill \end{array}$$

Then,

$${\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)={\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(1\right)=\sum _{j=1}^{m-2}{p}_j{\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left({\eta }_j\right)+x_0,$$

where ${\nu }^{\ast }=\nu -{\kappa }_{n-2}$ and $x_0\in \mathbb{R}$. Consequently, our multi-point fractional boundary value problem (2) is equivalent to

$${\mathcal{D}}_{0^{+}}^{\gamma ;\phi }y\left(z\right)=f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),I_{0^{+}}^{{\gamma }_1;\phi }y\left(z\right),\cdots ,y\left(z\right)\right),z\in [0,1],$$

(5)

under the conditions

$$y\left(0\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(0\right)=0,$$

and

$${\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(1\right)=\sum _{j=1}^{m-2}{p}_j{\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left({\eta }_j\right)+x_0.$$

(6)

For $1<\gamma <2$, $f\in L^1\left([0,1]\times {\mathbb{R}}^{n-1}\right)$ and $y\in A{C}^{n;\phi }[0,1]$, we apply the $\phi$-RLFI $I_{0^{+}}^{\gamma ;\phi }$ to both sides of Equation (5), and thus we find that

$$I_{0^{+}}^{\gamma ;\phi }{\mathcal{D}}_{0^{+}}^{\gamma ;\phi }y\left(z\right)=I_{0^{+}}^{\gamma ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),I_{0^{+}}^{{\gamma }_1;\phi }y\left(z\right),\cdots ,y\left(z\right)\right).$$

Then,

$$y\left(z\right)=I_{0^{+}}^{\gamma ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),I_{0^{+}}^{{\gamma }_1;\phi }y\left(z\right),\cdots ,y\left(z\right)\right)+c_1{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}+c_2{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -2}.$$

Hence, from the first boundary condition, we obtain

$$\begin{array}{ccc}\hfill y\left(0\right)& =& I_{0^{+}}^{\gamma ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=0}+c_1{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}{|}_{z=0}+c_2{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -2}{|}_{z=0}\hfill \\ & =& c_2{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -2}{|}_{z=0}=0,\hfill \end{array}$$

which gives $c_2=0$; therefore,

$$y\left(z\right)=I_{0^{+}}^{\gamma ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right)+c_1{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1},$$

(7)

and from condition (6), we obtain

$${\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }{I}_{0^{+}}^{\gamma ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right)+c_1{\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}.$$

By $\kappa >\nu$, it follows that $\kappa -{\kappa }_{n-2}-(\nu -{\kappa }_{n-2})>0$, which means that $\gamma -{\nu }^{\ast }>0$. Hence, from Lemma 2, we obtain

$${\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(z\right)=I_{0^{+}}^{\gamma -{\nu }^{\ast };\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right)+c_1{\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\nu }^{\ast })}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}.$$

So, we find

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(1\right)& =& {\displaystyle \frac{1}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +c_1{\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\nu }^{\ast })}}{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1},\hfill \end{array}$$

and, on the other hand, we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left({\eta }_j\right)& =& {\displaystyle \frac{1}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +c_1{\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\nu }^{\ast })}}{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}.\hfill \end{array}$$

Then, by the last boundary condition, we find

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(1\right)& =& {\displaystyle \frac{1}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +c_1{\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\nu }^{\ast })}}{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}\hfill \\ & =& \sum _{j=1}^{m-2}{\displaystyle \frac{p_j}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +c_1\sum _{j=1}^{m-2}{\displaystyle \frac{p_j\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\nu }^{\ast })}}{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}+x_0,\hfill \end{array}$$

which means that

$$\begin{array}{ccc}& & c_1\left({\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\nu }^{\ast })}}\right)\left(\sum _{j=1}^{m-2}{p}_j{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}-{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}\right)\hfill \\ & =& {\displaystyle \frac{1}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & -\sum _{j=1}^{m-2}{\displaystyle \frac{p_j}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0.\hfill \end{array}$$

By putting

$$\lambda =\sum _{j=1}^{m-2}{p}_j{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}-{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}\ne 0,$$

we obtain

$$\begin{array}{ccc}\hfill c_1& =& {\displaystyle \frac{1}{\lambda \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -{\nu }^{\ast })}}}\left({\displaystyle \frac{1}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{\displaystyle \frac{p_j}{\Gamma (\gamma -{\nu }^{\ast })}}{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\right)\hfill \\ & =& {\displaystyle \frac{1}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right).\hfill \end{array}$$

Hence, Equation (7) can be expressed as

$$\begin{array}{ccc}\hfill y\left(z\right)& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right),\hfill \end{array}$$

(8)

where $y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\in A{C}^{n;\phi }[0,1]$; so, (8) is the integral equation, which is equivalent to Equation (5).

Conversely, let $y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\in A{C}^{n;\phi }[0,1]$ be a solution to Equation (5); then, we have

$$\begin{array}{ccc}\hfill I_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)& =& x\left(z\right),\hfill \\ \hfill I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_1;\phi }y\left(z\right)& =& {\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(z\right),\hfill \\ \hfill ⋮& =& ⋮\hfill \\ \hfill I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_{n-3};\phi }y\left(z\right)& =& {\mathcal{D}}_{0^{+}}^{{\kappa }_{n-3};\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-3};\phi }x\left(z\right),\hfill \end{array}$$

and by applying the operator $I_{0^{+}}^{{\kappa }_{n-2};\phi }$ to both sides of (8), we obtain

$$\begin{array}{ccc}& & I_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right)\hfill \\ & =& I_{0^{+}}^{{\kappa }_{n-2};\phi }{I}_{0^{+}}^{\gamma ;\phi }f\left(z,x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right)\hfill \\ & & +{\displaystyle \frac{1}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }_n-1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }_n-1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }_n)\right)\times \hfill \\ & & I_{0^{+}}^{{\kappa }_{n-2};\phi }{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill x\left(z\right)& =& I_{0^{+}}^{\kappa ;\phi }f\left(z,x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right)\hfill \\ & & +{\displaystyle \frac{1}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }_n)\right)\times \hfill \\ & & I_{0^{+}}^{{\kappa }_{n-2};\phi }{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\kappa -{\kappa }_{n-2}-1}.\hfill \end{array}$$

Taking the operator ${\mathcal{D}}_{0^{+}}^{\kappa ;\phi }$ to both sides, we find

$$\begin{array}{ccc}& & {\mathcal{D}}_{0^{+}}^{\kappa ;\phi }x\left(z\right)\hfill \\ & =& {\mathcal{D}}_{0^{+}}^{\kappa ;\phi }{I}_{0^{+}}^{\kappa ;\phi }f\left(z,x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right)\hfill \\ & & +{\displaystyle \frac{1}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }_n)\right)\times \hfill \\ & & {\mathcal{D}}_{0^{+}}^{\kappa ;\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\kappa -{\kappa }_{n-2}-1}\hfill \\ & =& f\left(z,x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right)\hfill \\ & & +{\displaystyle \frac{1}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},x\left(\mathfrak{r}\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }_n)\right)\times \hfill \\ & & {\mathcal{D}}_{0^{+}}^{\kappa -{\kappa }_{n-2};\phi }{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\kappa -{\kappa }_{n-2}-1}.\hfill \end{array}$$

Then, by Lemma 2, we obtain

$${\mathcal{D}}_{0^{+}}^{\kappa ;\phi }x\left(z\right)=f\left(z,x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right).$$

In checking the boundary conditions, we have

$$\begin{array}{ccc}\hfill y\left(0\right)& =& {\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(0\right)=I_{0^{+}}^{\kappa ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=0}\hfill \\ & & +{\displaystyle \frac{1}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }_n-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }_n-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }_n)\right)\hfill \\ & & {\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}{|}_{z=0}=0,\hfill \end{array}$$

and for any $i\in \{1,\cdots ,n-3\}$, we have

$${\mathcal{D}}_{0^{+}}^{{\kappa }_i;\phi }x\left(z\right){|}_{z=0}=I_{0^{+}}^{{\kappa }_{n-2}-{\kappa }_i;\phi }y\left(z\right){|}_{z=0}=0,$$

and

$$x\left(0\right)=I_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right){|}_{z=0}=0.$$

For $t=1$, we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)& =& {\mathcal{D}}_{0^{+}}^{\nu ;\phi }{I}_{0^{+}}^{{\kappa }_{n-2};\phi }y\left(z\right){|}_{z=1}={\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(z\right){|}_{z=1}\hfill \\ & =& I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\hfill \\ & & +{\displaystyle \frac{1}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right)\hfill \\ & & {\mathcal{D}}_{0^{+}}^{\nu -{\kappa }_{n-2};\phi }{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\kappa -{\kappa }_{n-2}-1}{|}_{z=1}\hfill \end{array}$$

and we have

$${\mathcal{D}}_{0^{+}}^{\nu -{\kappa }_{n-2};\phi }{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\kappa -{\kappa }_{n-2}-1}={\displaystyle \frac{\Gamma (\kappa -{\kappa }_{n-2})}{\Gamma (\kappa -\nu )}}{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}.$$

We obtain

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)& =& I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\hfill \\ & & +{\displaystyle \frac{1}{\lambda \Gamma (\kappa -{\kappa }_{n-2})}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\kappa -\nu -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }_n)\right)\times \hfill \\ & & {\displaystyle \frac{\Gamma (\kappa -{\kappa }_{n-2})}{\Gamma (\kappa -\nu )}}{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}\hfill \\ & =& I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}}{\lambda }}\left(I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}-x_0\right),\hfill \end{array}$$

and, on the other hand, we have

$$\begin{array}{ccc}\hfill {\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(z\right){|}_{z={\eta }_j}& =& {\mathcal{D}}_{0^{+}}^{{\nu }^{\ast };\phi }y\left(t\right){|}_{z={\eta }_j}\hfill \\ & =& I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}}{\lambda }}\left(I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}-x_0\right).\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & {\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)-\sum _{j=1}^{m-2}{p}_j{\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(z\right){|}_{z={\eta }_j}\hfill \\ & =& I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}}{\lambda }}\left(I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}-x_0\right)\hfill \\ & & -\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}\hfill \\ & & -\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}}{\lambda }}\left(I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{t={\eta }_j}-x_0\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & {\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)-\sum _{j=1}^{m-2}{p}_j{\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(z\right){|}_{z={\eta }_j}\hfill \\ & =& I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\hfill \\ & & -\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}\hfill \\ & & -{\displaystyle \frac{{\displaystyle \sum _{j=1}^{m-2}}{p}_j{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}-{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\kappa -\nu -1}}{\lambda }}\times \hfill \\ & & \left(I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}-x_0\right)\hfill \\ & =& I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\hfill \\ & & -\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}\hfill \\ & & -I_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z=1}\hfill \\ & & +\sum _{j=1}^{m-2}{p}_j{I}_{0^{+}}^{\kappa -\nu ;\phi }f\left(z,I_{0^{+}}^{{\gamma }_0;\phi }y\left(z\right),\cdots ,y\left(z\right)\right){|}_{z={\eta }_j}+x_0.\hfill \end{array}$$

Therefore,

$${\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)=\sum _{j=1}^{m-2}{p}_j{\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(z\right){|}_{z={\eta }_j}+x_0.$$

□

Now, we consider the Banach space $C=C\left(\right[0,1],\mathbb{R})$ of functions y with the norm $\parallel y\parallel ={max}_{z\in [0,1]}\left|y\left(z\right)\right|$.

The objective here is to discuss the existence of fixed points of the operator $K:C\to C$, defined by

$$\begin{array}{ccc}\hfill Ky\left(z\right)& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right).\hfill \end{array}$$

(9)

Our discussion depends on two-based theories: Krasnoselskii’s and Banach’s fixed-point theorems.

Theorem 1

(Krasnoselskii [11]). Let B be a closed, convex, bounded, and nonempty subset of a Banach space E, and let $k_1$ and $k_2$ be operators, satisfying the following

(i) 

$k_{1}x+k_{2}y\in B$, $\forall x,y\in B$;

(ii) 

$k_1$ is a compact and continuous operator;

(iii) 

$k_2$ is a contraction.

Then, there exists $x\in B$, such that $k_{1}x+k_{2}x=x$.

Theorem 2

(Banach [11]). Let X be a complete nonempty metric space and $k:X\to X$ be a contraction mapping; then, we can find a unique point $y\in X$ with $ky=y$.

At first, we shall show our first existence theorem by utilizing the following conditions:

• $\left[C1\right]$: Let f be a continuous function bounded with a continuous function $g:[0,1]\to {\mathbb{R}}^{+}$; then,
  $\forall (z,x_1,\cdots ,x_{n-1})\in [0,1]\times {\mathbb{R}}^{n-1}$

  $$\left|f(z,x_1,\cdots ,x_{n-1})\right|\le g\left(z\right),$$

  where $g\in C\left([0,1];{\mathbb{R}}^{+}\right)$ and $\parallel g\parallel ={sup}_{z\in [0,1]}\left|g\left(z\right)\right|$.
• $\left[C2\right]$: the function g is bounded and satisfies $\exists r>0$,

  $$\parallel g\parallel \le {\displaystyle \frac{1}{\Lambda }}\left(r-{\displaystyle \frac{|x_0|\Gamma (\gamma -{\nu }^{\ast })}{\left|\lambda \right|\Gamma \left(\gamma \right)}}{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}\right),$$

  where

  $${\displaystyle \frac{|x_0|\Gamma (\gamma -{\nu }^{\ast })}{\left|\lambda \right|\Gamma \left(\gamma \right)}}{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}<r,$$

  $$\lambda =\sum _{j=1}^{m-2}{p}_j{\left(\phi \left({\eta }_j\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}-{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -{\nu }^{\ast }-1}\ne 0,$$

  and

  $$\begin{array}{ccc}\hfill \Lambda & =& {\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right.\hfill \\ & & \left.+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right).\hfill \end{array}$$
• $\left[C3\right]$: The function $f:[0,1]\times {\mathbb{R}}^{n-1}\to \mathbb{R}$ satisfies ∃$M_i>0$, such that

  $$|f(z,x_0,\cdots ,x_{n-2})-f(z,y_0,\cdots ,y_{n-2})|\le \sum _{i=0}^{n-2}{M}_i|x_i-y_i|$$

  for any $i\in \{0,\cdots ,n-2\}$ and $(x_0,\cdots ,x_{n-2}),(y_0,\cdots ,y_{n-2})\in {\mathbb{R}}^{n-1}$.

Theorem 3.

Suppose that the conditions $\left[C1\right]$, $\left[C2\right]$, and $\left[C3\right]$ hold, and

$$\begin{array}{ccc}\hfill \Theta & =& {\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\times \hfill \\ & & \left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right)\hfill \\ & <& 1,\hfill \end{array}$$

where ${\gamma }_i={\kappa }_{n-2}-{\kappa }_i$; $i\in \{0,\cdots ,n-2\}$; ${\kappa }_0=0$; then, the multi-point FBVP (2) has at least one solution on $[0,1]$.

Proof. 

$\left(i\right)$ We set $B=\{y\in (C[0,1],\mathbb{R});\parallel y\parallel \le r\}$; $r>0$ is a closed, bounded, convex subset of $\left(C\right[0,1],\mathbb{R})$; and let us define the operators $k_1$ and $k_2$ from (9) in this way $K=k_1+k_2$ as follows:

$$\begin{array}{ccc}\hfill k_{1}y\left(z\right)& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill k_{2}y\left(z\right)& =& {\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right).\hfill \end{array}$$

For any $x,y\in B$, and $z\in [0,1]$, we have

$$\begin{array}{ccc}\hfill \parallel k_{1}x+k_{2}y\parallel & =& \underset{z\in [0,1]}{max}\left|{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.\left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right)\right|\hfill \\ & \le & \underset{z\in [0,1]}{max}\left\{{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}\left|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\right.\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\left|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\left|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}+\left|x_0\right|\Gamma (\gamma -{\nu }^{\ast })\right)\right\}.\hfill \end{array}$$

Therefore, from $\left[C1\right]$ and $\left[C2\right]$, we find

$$\begin{array}{ccc}\hfill \parallel k_{1}x+k_{2}y\parallel & \le & \underset{t\in [0,1]}{max}\left\{{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}\left|g\left(\mathfrak{r}\right)\right|d\mathfrak{r}\right.\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\left|g\left(\mathfrak{r}\right)\right|d\mathfrak{r}\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\left|g\left(\mathfrak{r}\right)\right|d\mathfrak{r}+\left|x_0\right|\Gamma (\gamma -{\nu }^{\ast })\right)\right\}.\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill \parallel k_{1}x+k_{2}y\parallel & \le & \parallel g\parallel \underset{z\in [0,1]}{max}\left\{{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}d\mathfrak{r}\right.\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}+{\displaystyle \frac{|x_0|\Gamma (\gamma -{\nu }^{\ast })}{\parallel g\parallel }}\right)\right\}\hfill \\ & \le & \parallel g\parallel \left({\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}d\mathfrak{r}\right.\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}+{\displaystyle \frac{|x_0|\Gamma (\gamma -{\nu }^{\ast })}{\parallel g\parallel }}\right)\right)\hfill \\ & \le & \parallel g\parallel \left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right.\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}+{\displaystyle \frac{|x_0|\Gamma (\gamma -{\nu }^{\ast })}{\parallel g\parallel }}\right)\right).\hfill \end{array}$$

Then,

$$\parallel k_{1}x+k_{2}y\parallel \le \parallel g\parallel \left(\Lambda +{\displaystyle \frac{|x_0|\Gamma (\gamma -{\nu }^{\ast })}{\left|\lambda \right|\parallel g\parallel \Gamma \left(\gamma \right)}}{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}\right)\le r.$$

This means that $k_{1}x+k_{2}y\in B$.

$\left(ii\right)$ We prove that $k_1$ is continuous and compact. We have

$$k_{1}y\left(z\right)={\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r},$$

since $f:[0,1]\times {\mathbb{R}}^{n-1}\to \mathbb{R}$ is continuous, meaning $k_1$ is also continuous.

We set $\Omega =\{k_{1}y\left(z\right);y\in B\}$; then, for all $y\in B$,

$$\begin{array}{ccc}\hfill \parallel k_{1}y\parallel & \le & \underset{z\in [0,1]}{max}\left\{{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}\left|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\right\}\hfill \\ & \le & {\displaystyle \frac{\parallel g\parallel }{\Gamma \left(\gamma \right)}}{\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}d\mathfrak{r}\le {\displaystyle \frac{\parallel g\parallel }{\Gamma (\gamma +1)}}{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma };\hfill \end{array}$$

then, $\Omega$ is bounded.

For all $z_1,z_2\in [0,1]$ such that $z_1<z_2$, we have

$$\begin{array}{ccc}& & |k_{1}y\left(z_2\right)-k_{1}y\left(z_1\right)|\hfill \\ & =& \left|{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^{z_2}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z_2\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^{z_1}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z_1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^{z_1}{\phi }^{\prime }\left(\mathfrak{r}\right)\left[{(\phi \left(z_2\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}-{(\phi \left(z_1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}\right]\left|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\hfill \\ & & +{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_{z_1}^{z_2}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z_2\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}\left|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\hfill \\ & \le & {\displaystyle \frac{\parallel g\parallel }{\Gamma \left(\gamma \right)}}\left({\int }_0^{z_1}{\phi }^{\prime }\left(\mathfrak{r}\right)\left[{(\phi \left(z_2\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}-{(\phi \left(z_1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}\right]d\mathfrak{r}\right.\hfill \\ & & \left.+{\int }_{z_1}^{z_2}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z_2\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}d\mathfrak{r}\right)\hfill \\ & \le & {\displaystyle \frac{\parallel g\parallel }{\Gamma \left(\gamma \right)}}\left({\left[-{\displaystyle \frac{{(\phi \left(z_2\right)-\phi \left(\mathfrak{r}\right))}^{\gamma }}{\gamma }}+{\displaystyle \frac{{(\phi \left(z_1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma }}{\gamma }}\right]}_0^{z_1}\right.\hfill \\ & & \left.-{\left[{\displaystyle \frac{{(\phi \left(z_2\right)-\phi \left(\mathfrak{r}\right))}^{\gamma }}{\gamma }}\right]}_{z_1}^{z_2}\right)\hfill \\ & \le & {\displaystyle \frac{\parallel g\parallel }{\Gamma (\gamma +1)}}\left({(\phi \left(z_2\right)-\phi \left(0\right))}^{\gamma }-{(\phi \left(z_1\right)-\phi \left(0\right))}^{\gamma }+2{(\phi \left(z_2\right)-\phi \left(z_1\right))}^{\gamma }\right).\hfill \end{array}$$

Since $\phi$ is continuous, then $|k_{1}y\left(z_2\right)-k_{1}y\left(z_1\right)|\to 0$ when $z_1\to z_2$, meaning $\Omega$ is equi-continuous; hence, by Arzelà–Ascoli’s theorem, the operator $k_1$ is compacted on B.

$\left(iii\right)$ We show that $k_2$ is a contraction. For any $x,y\in B$, we have

$$\begin{array}{ccc}& & \parallel k_{2}x-k_{2}y\parallel \hfill \\ & =& \underset{z\in [0,1]}{max}\left|{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right)\hfill \\ & & -{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.\left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right)\right|\hfill \\ & \le & \underset{z\in [0,1]}{max}\left\{{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\right.\right.\times \hfill \\ & & |f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)-f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)|d\mathfrak{r}\hfill \\ & & +\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}\times \hfill \\ & & \left.\left.|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }x\left(\mathfrak{r}\right),\cdots ,x\left(\mathfrak{r}\right)\right)-f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)|d\mathfrak{r}\right)\right\};\hfill \end{array}$$

hence, by exploiting $\left[C3\right]$, we obtain

$$\begin{array}{ccc}\hfill \parallel k_{2}x-k_{2}y\parallel & \le & \underset{z\in [0,1]}{max}\left\{{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\sum _{i=0}^{n-2}{M}_i\left|I_{0^{+}}^{{\gamma }_i;\phi }(x\left(\mathfrak{r}\right)-y\left(\mathfrak{r}\right))\right|d\mathfrak{r}\right.\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\sum _{i=0}^{n-2}{M}_i\left|I_{0^{+}}^{{\gamma }_i;\phi }(x\left(\mathfrak{r}\right)-y\left(\mathfrak{r}\right))\right|d\mathfrak{r}\right)\right\},\hfill \end{array}$$

where ${\gamma }_0={\kappa }_{n-2}$ and $I_{0^{+}}^{{\gamma }_{n-2};\phi }x\left(\mathfrak{r}\right)=x\left(\mathfrak{r}\right)$, and we have

$$\begin{array}{ccc}\hfill \left|I_{0^{+}}^{{\gamma }_i;\phi }(x\left(z\right)-y\left(z\right))\right|& =& \left|{\displaystyle \frac{1}{\Gamma \left({\gamma }_i\right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{{\gamma }_i-1}(x\left(\mathfrak{r}\right)-y\left(\mathfrak{r}\right))d\mathfrak{r}\right|\hfill \\ & \le & \underset{z\in [0,1]}{max}|x\left(z\right)-y\left(z\right)|{\displaystyle \frac{1}{\Gamma \left({\gamma }_i\right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{{\gamma }_i-1}d\mathfrak{r}\hfill \\ & \le & {\displaystyle \frac{{(\phi \left(z\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\parallel x-y\parallel .\hfill \end{array}$$

Then, we obtain

$$\begin{array}{ccc}& & \parallel k_{2}x-k_{2}y\parallel \hfill \\ & \le & {\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left(\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\parallel x-y\parallel {\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\right.\hfill \\ & & \left.+\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\parallel x-y\parallel \sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\right)\hfill \\ & \le & {\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\right.\hfill \\ & & \left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\right)\parallel x-y\parallel \hfill \\ & \le & {\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right.\hfill \\ & & \left.+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right)\parallel x-y\parallel .\hfill \end{array}$$

This means that

$$\parallel k_{2}x-k_{2}y\parallel \le \Theta \parallel x-y\parallel ,$$

where

$$\begin{array}{ccc}\hfill \Theta & =& {\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\times \hfill \\ & & \left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right)\hfill \\ & <& 1;\hfill \end{array}$$

thus, $k_2$ is contraction.

Therefore, using Krasnoselskii’s fixed-point theorem, we can conclude that the operator $k_1+k_2$ possesses at least one fixed point, which implies that the boundary value problem (2) has at least one solution on the interval $[0,1]$. □

Let us now examine the uniqueness of the solution to $\phi$-FBVP (2).

Theorem 4.

Assume that $\left[C3\right]$ and the following condition:

$$\begin{array}{ccc}\hfill \Delta & =& \sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\left[{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\right.\times \hfill \\ & & \left.\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right)\right]\hfill \\ & <& 1.\hfill \end{array}$$

Then, the FBVP (2) possesses a unique solution in $[0,1]$.

Proof. 

We can show that the operator $K:A\to A$ defined in (9) is a contraction map; thus, we have

$$\begin{array}{ccc}\hfill Ky\left(z\right)& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }y\left(\mathfrak{r}\right),\cdots ,y\left(\mathfrak{r}\right)\right)d\mathfrak{r}-x_0\Gamma (\gamma -{\nu }^{\ast })\right),\hfill \end{array}$$

where we have $A=A{C}^{n;\phi }[0,1]$ for any $y_1,y_2\in A$.

$$\begin{array}{ccc}& & |K{y}_1\left(z\right)-K{y}_2\left(z\right)|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }{y}_1\left(\mathfrak{r}\right),\cdots ,y_1\left(\mathfrak{r}\right)\right)-f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }{y}_2\left(\mathfrak{r}\right),\cdots ,y_2\left(\mathfrak{r}\right)\right)|d\mathfrak{r}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }{y}_1\left(\mathfrak{r}\right),\cdots ,y_1\left(\mathfrak{r}\right)\right)\right.\hfill \\ & & -f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }{y}_2\left(\mathfrak{r}\right),\cdots ,y_2\left(\mathfrak{r}\right)\right)|d\mathfrak{r}+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\times \hfill \\ & & \left.|f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }{y}_1\left(\mathfrak{r}\right),\cdots ,y_1\left(\mathfrak{r}\right)\right)-f\left(\mathfrak{r},I_{0^{+}}^{{\gamma }_0;\phi }{y}_2\left(\mathfrak{r}\right),\cdots ,y_2\left(\mathfrak{r}\right)\right)|d\mathfrak{r}\right).\hfill \end{array}$$

Hence, by exploiting $\left[C3\right]$, we obtain

$$\begin{array}{ccc}& & |K{y}_1\left(z\right)-K{y}_2\left(z\right)|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}\sum _{i=0}^{n-2}{M}_i\left|I_{0^{+}}^{{\gamma }_i;\phi }\left(y_1\left(\mathfrak{r}\right)-y_2\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\sum _{i=0}^{n-2}{M}_i\left|I_{0^{+}}^{{\gamma }_i;\phi }\left(y_1\left(\mathfrak{r}\right)-y_2\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\right.\hfill \\ & & \left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}\sum _{i=0}^{n-2}{M}_i\left|I_{0^{+}}^{{\gamma }_i;\phi }\left(y_1\left(\mathfrak{r}\right)-y_2\left(\mathfrak{r}\right)\right)\right|d\mathfrak{r}\right)\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(z\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -1}d\mathfrak{r}\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(t\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\parallel y_1-y_2\parallel \hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left(1\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(z\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\parallel y_1-y_2\parallel \right.\hfill \\ & & \left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(\mathfrak{r}\right){(\phi \left({\eta }_j\right)-\phi \left(\mathfrak{r}\right))}^{\gamma -{\nu }^{\ast }-1}d\mathfrak{r}\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(z\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\parallel y_1-y_2\parallel \right)\hfill \\ & \le & \left[{\displaystyle \frac{{(\phi \left(z\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right.\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right)\right]\sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(z\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\parallel y_1-y_2\parallel .\hfill \end{array}$$

Then,

$$\begin{array}{ccc}\hfill \parallel K{y}_1-K{y}_2\parallel & \le & \sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\left[{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\times \right.\hfill \\ & & \left.\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right)\right]\parallel y_1-y_2\parallel ,\hfill \end{array}$$

by setting

$$\begin{array}{ccc}\hfill \Delta & =& \sum _{i=0}^{n-2}{M}_i{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{{\gamma }_i}}{\Gamma ({\gamma }_i+1)}}\left[{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\times \right.\hfill \\ & & \left.\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right)\right]\hfill \\ & <& 1.\hfill \end{array}$$

Thus, we obtain

$$\parallel K{y}_1-K{y}_2\parallel \le \Delta \parallel y_1-y_2\parallel ;$$

(10)

then, the operator K is a contraction, meaning it has a unique fixed point y in A. Hence, FBVP (2) has a unique solution in $[0,1]$. □

Example 1.

Consider the following φ-FBVP:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{4.8;z^2+1}x\left(z\right)=2z+{\displaystyle \frac{1}{25}}cos\left(x\left(z\right)\right)+{\displaystyle \frac{1}{20}}sin\left({\mathcal{D}}_{0^{+}}^{1.5;z^2+1}x\left(z\right)\right)\hfill \\ +{\displaystyle \frac{1}{8}}sin\left({\mathcal{D}}_{0^{+}}^{2.1;z^2+1}x\left(z\right)\right)+{\displaystyle \frac{1}{16}}{tan}^{-1}\left({\mathcal{D}}_{0^{+}}^{2.9;z^2+1}x\left(z\right)\right),\hfill \\ x\left(0\right)={\mathcal{D}}_{0^{+}}^{1.5;z^2+1}x\left(0\right)={\mathcal{D}}_{0^{+}}^{2.1;z^2+1}x\left(0\right)={\mathcal{D}}_{0^{+}}^{2.9;z^2+1}x\left(0\right)=0,\hfill \\ {\mathcal{D}}_{0^{+}}^{3.7;z^2+1}x\left(1\right)=16{\mathcal{D}}_{0^{+}}^{3.7;z^2+1}x\left({\displaystyle \frac{1}{20}}\right)+2{\mathcal{D}}_{0^{+}}^{3.7;z^2+1}x\left({\displaystyle \frac{1}{10}}\right)+1.\hfill \end{array}\right.$$

(11)

In this example, we have $n=5$, $\phi \left(z\right)=z^2+1$, $\kappa =4.8$, ${\kappa }_1=1.5$, ${\kappa }_2=2.1$, ${\kappa }_3=2.9$, $m=4$, and $\nu =3.7$; then, $\gamma =\kappa -{\kappa }_3=1.9$, ${\nu }^{\ast }=\nu -{\kappa }_3=0.8$, $\gamma -{\nu }^{\ast }=\kappa -\nu =1.1$, ${\eta }_1={\displaystyle \frac{1}{20}}$, ${\eta }_2={\displaystyle \frac{1}{10}}$, $p_1=16$, $p_2=2$, and $x_0=1$.

For $z\in [0,1]$, we set

$$\begin{array}{ccc}& & f\left(z,x\left(z\right),{\mathcal{D}}_{0^{+}}^{1.5;z^2+1}x\left(z\right),{\mathcal{D}}_{0^{+}}^{2.1;z^2+1}x\left(z\right),{\mathcal{D}}_{0^{+}}^{2.9;z^2+1}x\left(z\right)\right)\hfill \\ & =& 2z+{\displaystyle \frac{1}{25}}cos\left(x\left(z\right)\right)+{\displaystyle \frac{1}{20}}sin\left({\mathcal{D}}_{0^{+}}^{1.5;z^2+1}x\left(z\right)\right)\hfill \\ & & +{\displaystyle \frac{1}{8}}sin\left({\mathcal{D}}_{0^{+}}^{2.1;z^2+1}x\left(z\right)\right)+{\displaystyle \frac{1}{16}}{tan}^{-1}\left({\mathcal{D}}_{0^{+}}^{2.9;z^2+1}x\left(z\right)\right);\hfill \end{array}$$

then, for every $(z,x_1,x_2,x_3,x_4)\in [0,1]\times {\mathbb{R}}^4$, we have

$$\begin{array}{ccc}\hfill |f(z,x_1,x_2,x_3,x_4)|\le g\left(z\right)& =& 2z+{\displaystyle \frac{1}{25}}+{\displaystyle \frac{1}{20}}+{\displaystyle \frac{1}{8}}+{\displaystyle \frac{\pi }{32}}\hfill \\ & =& 2z+{\displaystyle \frac{\pi }{32}}+{\displaystyle \frac{43}{200}},z\in [0,1],\hfill \end{array}$$

which gives

$$\parallel g\parallel ={\displaystyle \frac{\pi }{32}}+{\displaystyle \frac{443}{200}};$$

hence, by computing λ from condition $\left[C2\right]$, we obtain

$$\lambda =16{\left({\left({\displaystyle \frac{1}{20}}\right)}^2\right)}^{1.1-1}+2{\left({\left({\displaystyle \frac{1}{10}}\right)}^2\right)}^{1.1-1}-{\left(1\right)}^{1.1-1}\approx 9.050399035.$$

From Theorem 3, and with $M_0=0.04$, $M_1=0.05$, $M_2=0.125$, $M_3=0.03125\pi$, we find

$$\begin{array}{ccc}\hfill \Theta & =& {\displaystyle \frac{{\left(1\right)}^{0.9}}{\left|\lambda \right|\Gamma \left(1.9\right)}}\left(0.04{\displaystyle \frac{{\left(1\right)}^{2.9}}{\Gamma \left(3.9\right)}}+0.05{\displaystyle \frac{{\left(1\right)}^{1.4}}{\Gamma \left(2.4\right)}}+0.125{\displaystyle \frac{{\left(1\right)}^{0.8}}{\Gamma \left(1.8\right)}}+0.03125\pi {\displaystyle \frac{{\left(1\right)}^0}{\Gamma \left(1\right)}}\right)\times \hfill \\ & & \left[{\displaystyle \frac{{\left(1\right)}^{1.1}}{1.1}}+16{\displaystyle \frac{{\left({\left(\frac{1}{20}\right)}^2\right)}^{1.1}}{1.1}}+2{\displaystyle \frac{{\left({\left(\frac{1}{10}\right)}^2\right)}^{1.1}}{1.1}}\right]\hfill \\ & \approx & 0.03027<1;\hfill \end{array}$$

therefore, by applying Krasnoselskii’s fixed-point theorem, we can assert that the fractional boundary value problem (11) has at least one solution on the interval $[0,1]$.

Example 2.

Consider the following φ-FBVP:

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{3.7;z^2}x\left(z\right)={\displaystyle \frac{1}{3\sqrt{3}}}sin\left({\mathcal{D}}_{0^{+}}^{1.3;z^2}x\left(z\right)\right)-{\displaystyle \frac{cos(z^2-1)}{4e^{x{\left(z\right)}^2+1}}}+{\displaystyle \frac{1}{6\sqrt{e^z\left(cos\left({\mathcal{D}}_{0^{+}}^{1.9;z^2}x\left(z\right)\right)+\pi \right)}}}-{\displaystyle \frac{3}{5\Gamma \left(2.1\right)}}{z}^{2.2}\hfill \\ {\displaystyle x\left(0\right)={\mathcal{D}}_{0^{+}}^{1.3;z^2}x\left(0\right)={\mathcal{D}}_{0^{+}}^{1.9;z^2}x\left(0\right)=0,}\hfill \\ {\displaystyle {\mathcal{D}}_{0^{+}}^{2.6;z^2}x\left(1\right)=2{\left(\frac{1}{5}\right)}^{\frac{1}{5}}{\mathcal{D}}_{0^{+}}^{2.6;z^2}x\left(0.2\right)+6{\left(\frac{3}{10}\right)}^{\frac{6}{5}}{\mathcal{D}}_{0^{+}}^{2.6;z^2}x\left(0.3\right)+\frac{36}{100\Gamma \left(1.4\right)},}\hfill \end{array}\right.$$

(12)

In this example, we choose $n=4$, $\kappa =3.7$, ${\kappa }_1=1.3$, ${\kappa }_2=1.9$, and $\nu =2.6$; $m=4$, $p_1=2{\left({\displaystyle \frac{1}{5}}\right)}^{{\displaystyle \frac{1}{5}}}$, $p_2=6{\left({\displaystyle \frac{3}{10}}\right)}^{{\displaystyle \frac{6}{5}}}$, and ${\eta }_1=0.2$; and ${\eta }_2=0.3$, $x_0={\displaystyle \frac{36}{100\Gamma \left(1.4\right)}}$, and $\phi \left(z\right)=z^2$. Therefore, we obtian $\gamma =\kappa -{\kappa }_2=1.8$, ${\nu }^{\ast }=\nu -{\kappa }_2=0.7$, ${\kappa }_2-{\kappa }_1=0.6$, $\gamma -{\nu }^{\ast }=\kappa -\nu =1.1$. For $z\in [0,1]$, we have

$$\begin{array}{ccc}\hfill f\left(z,x\left(z\right),{\mathcal{D}}_{0^{+}}^{1.3;z^2}x\left(z\right),{\mathcal{D}}_{0^{+}}^{1.9;z^2}x\left(z\right)\right)& =& {\displaystyle \frac{1}{3\sqrt{3}}}sin\left({\mathcal{D}}_{0^{+}}^{1.3;z^2}x\left(z\right)\right)-{\displaystyle \frac{cos(z^2-1)}{4e^{x{\left(z\right)}^2+1}}}\hfill \\ & & +{\displaystyle \frac{1}{6\sqrt{e^z\left(cos\left({\mathcal{D}}_{0^{+}}^{1.9;z^2}x\left(z\right)\right)+\pi \right)}}}-{\displaystyle \frac{3}{5\Gamma \left(2.1\right)}}{z}^{2.2};\hfill \end{array}$$

then, for every $(z,x_1,x_2,x_3)\in [0,1]\times {\mathbb{R}}^3$, we have

$$\begin{array}{ccc}\hfill |f(z,x_1,x_2,x_3)|\le g\left(z\right)& =& {\displaystyle \frac{1}{3\sqrt{3}}}+{\displaystyle \frac{cos(z^2-1)}{4e}}+{\displaystyle \frac{1}{6\sqrt{e^z\left(\pi -1\right)}}}+{\displaystyle \frac{3}{5\Gamma \left(2.1\right)}}{z}^{2.2},z\in [0,1],\hfill \end{array}$$

and so

$$\parallel g\parallel ={\displaystyle \frac{1}{3\sqrt{3}}}+{\displaystyle \frac{1}{4e}}+{\displaystyle \frac{1}{6\sqrt{\pi -1}}}+{\displaystyle \frac{3}{5\Gamma \left(1.4\right)}}.$$

From condition $\left[C2\right]$, we have

$$\lambda =2{\left({\displaystyle \frac{1}{5}}\right)}^{{\displaystyle \frac{1}{5}}}{\left({\left(0.2\right)}^2\right)}^{0.1}+6{\left({\displaystyle \frac{3}{10}}\right)}^{{\displaystyle \frac{6}{5}}}{\left({\left(0.3\right)}^2\right)}^{0.1}-{\left(1\right)}^{0.1}=1.16265\cdots$$

$$\begin{array}{ccc}\hfill \Lambda & =& {\displaystyle \frac{{\left(1\right)}^{1.8}}{\Gamma \left(2.8\right)}}+{\displaystyle \frac{{\left(1\right)}^{0.8}}{\left|\lambda \right|\Gamma \left(1.8\right)}}\left[{\displaystyle \frac{{\left(1\right)}^{1.1}}{1.1}}+2{\left({\displaystyle \frac{1}{5}}\right)}^{{\displaystyle \frac{1}{5}}}{\displaystyle \frac{{\left({\left(0.2\right)}^2\right)}^{1.1}}{1.1}}+6{\left({\displaystyle \frac{3}{10}}\right)}^{{\displaystyle \frac{6}{5}}}{\displaystyle \frac{{\left({\left(0.3\right)}^2\right)}^{1.1}}{1.1}}\right]\hfill \\ & \approx & 1.5553014,\hfill \end{array}$$

and with $M_0={\displaystyle \frac{\sqrt{2}{e}^{-\frac{3}{2}}}{4}}$, $M_1={\displaystyle \frac{1}{3\sqrt{3}}}$, and $M_2={\displaystyle \frac{1}{12{(\pi -1)}^{\frac{3}{2}}}}$, from Theorem 4, we obtain

$$\begin{array}{ccc}\hfill \Delta & =& \left({\displaystyle \frac{\sqrt{2}{e}^{-\frac{3}{2}}}{4}}{\displaystyle \frac{{\left(1\right)}^{1.9}}{\Gamma \left(2.9\right)}}+{\displaystyle \frac{1}{3\sqrt{3}}}{\displaystyle \frac{{\left(1\right)}^{0.6}}{\Gamma \left(1.6\right)}}+{\displaystyle \frac{1}{12{(\pi -1)}^{\frac{3}{2}}}}{\displaystyle \frac{{\left(1\right)}^0}{\Gamma \left(1\right)}}\right)\times \Lambda \hfill \\ & \approx & 0.443487<1;\hfill \end{array}$$

thus, by applying Banach’s fixed-point theorem, we can deduce that the fractional boundary value problem (12) has a unique solution on the interval $[0,1]$.

4. The Criterion for (HU) Stability

Fractional differential equations are crucial in mathematical analysis and the modeling of physical phenomena, and they have been extensively examined from various perspectives. One key aspect that has received significant attention is the stability analysis in the Hyers–Ulam sense [35,36]. The original definition of Hyers–Ulam stability has been expanded to more general forms with time [37,38]. In this section, we establish certain sufficient conditions to derive Hyers–Ulam-type stability results for our primary problem.

Definition 3

([35]). Consider the Banach space $C=C\left(\right[0,1],\mathbb{R})$ and an operator $K:C\to C$. The operator equation

$$\begin{array}{c}\hfill Ky=y,\end{array}$$

(13)

is considered to be Hyers–Ulam-stable if

$$\left|y\right(z)-Ky(z\left)\right|\le ϵ$$

for all $z\in [0,1]$, which implies that there exists a constant $P_K>0$ such that for any $y\in C\left([0,1],\mathbb{R}\right)$ satisfying (13), one can find a unique solution $\widehat{y}\in \mathcal{C}\left([0,1],\mathbb{R}\right)$ of the operator Equation (13), provided that for any $z\in [0,1]$, we have

$$|y\left(z\right)-\widehat{y}\left(z\right)|\le P_K.ϵ.$$

Definition 4

([39]). Consider the operator $K:C\to C$. Inspired by Definition 3, we define the operator equation

$$\begin{array}{c}\hfill y\left(z\right)=Ky\left(z\right)\end{array}$$

(14)

as Hyers–Ulam-stable if, for the inequality

$$\begin{array}{c}\hfill \left|y\right(z)-Ky(z\left)\right|\le ϵ,z\in [0,1],\end{array}$$

(15)

we can find a constant $P_K$, such that for y satisfying (14), there exists a unique solution $\widehat{y}$ of the operator Equation (14), provided that for any $z\in [0,1]$

$$|y\left(z\right)-\widehat{y}\left(z\right)|\le P_K.ϵ.$$

Theorem 5.

Assume that $\varphi \in C\left(\right[0,1],\mathbb{R})$ is a solution of the inequality (15) satisfying the following conditions:

(i) 

$\left|\varphi \right(z\left)\right|\le ϵ$ for every $z\in [0,1]$,

(ii) 

${\mathcal{D}}_{0^{+}}^{\kappa ;\phi }x\left(z\right)-f\left(z,x\left(z\right),{\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right)-\varphi \left(z\right)=0,\mathit{for}\mathit{every}z\in [0,1]$.

Then,

$$\left|y\right(z)-Ky(z\left)\right|\le \Lambda ϵ,z\in [0,1],$$

where $y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)$ with

$$\begin{array}{ccc}\hfill \Lambda & :=& {\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\left|\lambda \right|\Gamma \left(\gamma \right)}}\left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right.\hfill \\ & & \left.+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right),\hfill \end{array}$$

and K denotes the operator defined in (9).

Proof. 

Based on condition $\left(ii\right)$ and for every $z\in [0,1]$, we have

$${\mathcal{D}}_{0^{+}}^{\kappa ;\phi }x\left(z\right)-f\left(z,x\left(z\right),{\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right)-\varphi \left(z\right)=0,$$

(16)

$$x\left(0\right)=0,{\mathcal{D}}_{0^{+}}^{{\kappa }_i;\phi }x\left(0\right)=0,1\le i\le n-2,{\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left(1\right)=\sum _{j=1}^{m-2}{p}_j{\mathcal{D}}_{0^{+}}^{\nu ;\phi }x\left({\eta }_j\right)+x_0.$$

From Proposition 1 and by assuming that $y\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)$, the solution of $\phi$-FBVP (16) can written as

$$\begin{array}{ccc}\hfill y\left(z\right)& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(s\right){(\phi \left(z\right)-\phi \left(s\right))}^{\gamma -1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }y\left(s\right),\cdots ,y\left(s\right)\right)ds\hfill \\ & & +{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(s\right){(\phi \left(z\right)-\phi \left(s\right))}^{\gamma -1}\varphi \left(s\right)ds\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(s\right){(\phi \left(1\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }y\left(s\right),\cdots ,y\left(s\right)\right)ds\right.\hfill \\ & & +{\int }_0^1{\phi }^{\prime }\left(s\right){(\phi \left(1\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}\varphi \left(s\right)ds\hfill \\ & & -\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }y\left(s\right),\cdots ,y\left(s\right)\right)ds\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}\varphi \left(s\right)ds-x_0\Gamma (\gamma -{\nu }^{\ast })\right);\hfill \end{array}$$

then, we have

$$\begin{array}{ccc}& & \left|y\right(z)-Ky(z\left)\right|\hfill \\ & =& \left|y\left(z\right)-{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(s\right){(\phi \left(z\right)-\phi \left(s\right))}^{\gamma -1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }y\left(s\right),\cdots ,y\left(s\right)\right)ds\right.\hfill \\ & & -{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(s\right){(\phi \left(1\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }y\left(s\right),\cdots ,y\left(s\right)\right)ds\right.\hfill \\ & & \left.\left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }y\left(s\right),\cdots ,y\left(s\right)\right)ds-x_0\Gamma (\gamma -{\nu }^{\ast })\right)\right|\hfill \\ & =& \left|{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(s\right){(\phi \left(z\right)-\phi \left(s\right))}^{\gamma -1}\varphi \left(s\right)ds\right.\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(s\right){(\phi \left(1\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}\varphi \left(s\right)ds\right.\hfill \\ & & \left.\left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}\varphi \left(s\right)ds\right)\right|\hfill \\ & \le & {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(s\right){(\phi \left(z\right)-\phi \left(s\right))}^{\gamma -1}\left|\varphi \left(s\right)\right|ds\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(s\right){(\phi \left(1\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}\left|\varphi \left(s\right)\right|ds\right.\hfill \\ & & \left.+\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}\left|\varphi \left(s\right)\right|ds\right);\hfill \end{array}$$

hence, by using condition $\left(i\right)$, we obtain

$$\begin{array}{ccc}\hfill \left|y\right(z)-Ky(z\left)\right|& \le & \left({\displaystyle \frac{{(\phi \left(z\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left[{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right.\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right]\right)ϵ\hfill \\ & \le & \left({\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma }}{\Gamma (\gamma +1)}}+{\displaystyle \frac{{\left(\phi \left(1\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left[{\displaystyle \frac{{(\phi \left(1\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right.\right.\hfill \\ & & \left.\left.+\sum _{j=1}^{m-2}{p}_j{\displaystyle \frac{{(\phi \left({\eta }_j\right)-\phi \left(0\right))}^{\gamma -{\nu }^{\ast }}}{\gamma -{\nu }^{\ast }}}\right]\right)ϵ,\hfill \end{array}$$

which leads to

$$\left|y\right(z)-Ky(z\left)\right|\le \Lambda ϵ,z\in [0,1].$$

□

Theorem 6.

Under the hypotheses of Theorems 4 and 5, the solution of φ-FBVP (2) is Hyers-Ulam-stable.

Proof. 

Let $x\in C\left(\right[0,1],\mathbb{R})$ be one of the solutions of the following inequality:

$$\left|{\mathcal{D}}_{0^{+}}^{\kappa ;\phi }x\left(z\right)-f\left(z,x\left(z\right),{\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }x\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }x\left(z\right)\right)\right|\le ϵ,z\in [0,1],$$

and let $\widehat{x}\in C([0,1],\mathbb{R})$ be the unique solution of the problem

$$\left\{\begin{array}{c}{\mathcal{D}}_{0^{+}}^{\kappa ;\phi }\widehat{x}\left(z\right)=f\left(t,\widehat{x}\left(z\right),{\mathcal{D}}_{0^{+}}^{{\kappa }_1;\phi }\widehat{x}\left(z\right),\cdots ,{\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }\widehat{x}\left(z\right)\right),t\in [0,1],\hfill \\ {\displaystyle \widehat{x}\left(0\right)=0,{\mathcal{D}}_{0^{+}}^{{\kappa }_i;\phi }\widehat{x}\left(0\right)=0,1\le i\le n-2,}\hfill \\ {\mathcal{D}}_{0^{+}}^{\nu ;\phi }\widehat{x}\left(1\right)={\displaystyle \sum _{j=1}^{m-2}}{p}_j{\mathcal{D}}_{0^{+}}^{\nu ;\phi }\widehat{x}\left({\eta }_j\right)+x_0;\hfill \end{array}\right.$$

then, by Proposition 1, the solution $\widehat{y}\left(z\right)={\mathcal{D}}_{0^{+}}^{{\kappa }_{n-2};\phi }\widehat{x}\left(z\right)$ can be expressed as

$$\begin{array}{ccc}\hfill \widehat{y}\left(z\right)& =& {\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}{\int }_0^z{\phi }^{\prime }\left(s\right){(\phi \left(z\right)-\phi \left(s\right))}^{\gamma -1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }\widehat{y}\left(s\right),\cdots ,\widehat{y}\left(s\right)\right)ds\hfill \\ & & +{\displaystyle \frac{{\left(\phi \left(z\right)-\phi \left(0\right)\right)}^{\gamma -1}}{\lambda \Gamma \left(\gamma \right)}}\left({\int }_0^1{\phi }^{\prime }\left(s\right){(\phi \left(1\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }\widehat{y}\left(s\right),\cdots ,\widehat{y}\left(s\right)\right)ds\right.\hfill \\ & & \left.-\sum _{j=1}^{m-2}{p}_j{\int }_0^{{\eta }_j}{\phi }^{\prime }\left(s\right){(\phi \left({\eta }_j\right)-\phi \left(s\right))}^{\gamma -{\nu }^{\ast }-1}f\left(s,I_{0^{+}}^{{\gamma }_0;\phi }\widehat{y}\left(s\right),\cdots ,\widehat{y}\left(s\right)\right)ds-x_0\Gamma (\gamma -{\nu }^{\ast })\right).\hfill \end{array}$$

Then, we obtain

$$\begin{array}{ccc}\hfill |y\left(z\right)-\widehat{y}\left(z\right)|& =& |y\left(z\right)-K\widehat{y}\left(z\right)|\hfill \\ & =& |y\left(z\right)-Ky\left(z\right)+Ky\left(z\right)-K\widehat{y}\left(z\right)|\hfill \\ & \le & |y\left(z\right)-Ky\left(z\right)|+|Ky\left(z\right)-K\widehat{y}\left(z\right)|;\hfill \end{array}$$

hence, Theorem 5 and the inequality (10) lead to

$$\parallel y-\widehat{y}\parallel \le \Lambda ϵ+\Delta \parallel y-\widehat{y}\parallel ,$$

which gives

$$\parallel y-\widehat{y}\parallel \le {\displaystyle \frac{\Lambda }{1-\Delta }}ϵ.$$

Therefore, the solution of $\phi$-FBVP (2) is Hyers-Ulam-stable. □

5. Conclusions and Perspectives

In this work, we have explored the existence, uniqueness, and Hyers–Ulam stability of solutions for a multi-point $\phi$-Riemann–Liouville fractional boundary value problem (FBVP). By leveraging fixed-point theorems such as Banach’s and Krasnoselskii’s, we established the conditions under which unique solutions to the FBVP can be guaranteed. Our analysis provided detailed proofs and propositions that validate the theoretical findings, ensuring a rigorous mathematical framework. Furthermore, we delved into the concept of Hyers–Ulam stability, demonstrating its significance in the context of differential equations. We identified sufficient conditions for the stability of solutions and illustrated these theoretical results with practical examples. This comprehensive approach ensures that the theoretical contributions are robust and applicable to real-world problems.

Future research can extend this work by generalizing the results to higher-dimensional problems, developing numerical methods to approximate solutions, exploring different boundary conditions, and applying the theoretical framework to real-world problems. These directions will enhance the applicability and robustness of the current theoretical contributions.