Stability of p(·)-Integrable Solutions for Fractional Boundary Value Problem via Piecewise Constant Functions

Abstract

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space ($L^{p(·)}$). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers are established. Our results are obtained using two fixed-point theorems, then illustrating the results with a comprehensive example.

1. Introduction

The variable exponent Lebesgue spaces $L^{p(·)}$ have been systemically studied with the foundational publication by KovLa.cik and Rakosn L.k [1] in 1991. Following this work, numerous researchers developed an interest in these spaces and their properties (see [2,3]).

Although this concept seems to be simple, it has remarkable outcomes that accurately describe some physical, energy, electric and velocity phenomena. In recent years, we see many works dealing with this topic (see, for example, [4,5,6,7,8,9]).

In recent years, many authors have studied the existence and uniqueness of solutions to problems for fractional differential equations in the $C(J,\mathbb{R})$ space of continuous functions; on the other hand, the number of works discussing the existence of solutions in the $L^p(J,\mathbb{R})$ space of integrable functions is quite limited. Burton et al. [10] studied the existence of solutions in the $L^1(J,\mathbb{R})$ space to Caputo derivative problems.

Arshad established $L^p$-solutions of fractional integral equations involving the Riemann–Liouville integral operator in Banach spaces.

Due to the difficulty of studying the existence of solutions to fractional differential equations in $L^{p(·)}$, there are very few studies in the literature dealing with this topic. In 2018, Dong et al. [11] established that there exists a unique solution to a Cauchy type issue for fractional differential equations in $L^{p(·)}$.

Recently, Refice et al. [12] investigated the following boundary value problem of fractional differential equations in $L^{p(·)}$:

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\alpha }y\left(t\right)=\mathsf{\Phi }(t,y\left(t\right)),t\in [0,\zeta ],0<\alpha <1\hfill \\ \gamma y\left(0\right)+\mu y\left(T\right)=c,\hfill \end{array}\right.$$

(1)

where $\zeta ,c,\gamma ,\mu ,$ with $\mu \ne 0$ are real constants, and $D_{0^{+}}^{\alpha }$ is the left Riemann–Liouville fractional derivative of order $\alpha$ for function $y\left(t\right)$.

In [13], Refice et al. established the following terminal value problem of fractional differential equations in $L^{p(·)}$.

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\alpha }y\left(t\right)=\mathsf{\Phi }(t,y\left(t\right),I_{0^{+}}^{\alpha }y\left(t\right)),0\le t\le \zeta \hfill \\ y\left(T\right)=c,c\in \mathbb{R}\hfill \end{array}\right.$$

(2)

where $D_{0^{+}}^{\alpha }$, $I_{0^{+}}^{\alpha }$ are the left Riemann–Liouville fractional derivative and integral of order $\alpha$ for the function y, respectively.

Several authors have studied the Ulam stability properties and their existence for fractional differential equations (see [14,15,16,17,18,19,20] and the references therein).

Benchohra et al. [21] investigated the existing solutions to the following BVP in $C\left(\right[0,\zeta ],\mathbb{R})$:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{D}_a^{\alpha }y\left(t\right)=\mathsf{\Phi }(t,y\left(t\right)),0\le t\le \zeta \hfill \\ \gamma y\left(0\right)+\mu y\left(\zeta \right)=\varpi ,\hfill \end{array}\right.\end{array}$$

where $0<\alpha <1$, and ${}^c{D}_a^{\alpha }$ is the Caputo fractional derivative of order $\alpha$ for function $y\left(t\right)$.

Abdulahad et al. [22] proved the existing solutions to the BVP in $L^p([{\zeta }_1,{\zeta }_2],\mathbb{R})$ as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{D}_a^{\alpha }y\left(t\right)=\mathsf{\Phi }(t,y\left(t\right)),{\zeta }_1\le t\le {\zeta }_2,0<\alpha <1,\hfill \\ \gamma y\left({\zeta }_1\right)+\mu y\left({\zeta }_2\right)=\varpi ,\hfill \end{array}\right.\end{array}$$

where ${}^c{D}_a^{\alpha }$ is the Caputo fractional derivative of order $\alpha$ for function $y\left(t\right)$.

Inspired by [12,13,21,22], we consider the existing solutions to the following BVP in $L^{p(·)}(J,\mathbb{R})$:

$$\left\{\begin{array}{c}{D}_{0^{+}}^{\alpha }y\left(t\right)=\mathsf{\Phi }(t,y\left(t\right),I_{0^{+}}^{\alpha }y\left(t\right)),t\in J:=[0,\zeta ]\hfill \\ \gamma y\left(0\right)+\mu y\left(\zeta \right)=c,\hfill \end{array}\right.$$

(3)

where $0<\alpha <1$, $\mathsf{\Phi }(.,y(·),I_{0^{+}}^{\alpha }y(·))\in L^{p(·)}(J\times \mathbb{R}\times \mathbb{R},\mathbb{R})$, and $y\in L^{p(·)}(J,\mathbb{R})$, $\zeta ,c,\gamma ,\mu$ are real constants with $\mu \ne 0$ and $D_{0^{+}}^{\alpha }$, $I_{0^{+}}^{\alpha }$ are the the left Riemann–Liouville fractional derivative and integral of order $\alpha$ for function $y\left(t\right)$, respectively.

This paper can be considered as a generalization and extension of several articles. For example, if we choose the function $\mathsf{\Phi }$ as a two-variable function t, $y\left(t\right)$ only and independently of the integration $I_{0^{+}}^{\alpha }y\left(t\right)$, our problem reduces to the problem studied in [12], and, if we choose the constants $\gamma =0$ and $\mu =1$, our problem reduces to the terminal value problem studied in [13].

2. Preliminaries

In this section, we list some definitions and lemmas that are used in the following sections.

Definition 1

([11]). Letting $0<\alpha <1$, the left Riemann–Liouville fractional derivative of order α for function $y\left(t\right)$ in $L^{p(·)}$ is defined by:

$$\left(D_{0^{+}}^{\alpha }y\right)\left(t\right)=\frac{1}{\mathsf{\Gamma }(1-\alpha )}\frac{d}{dt}{\int }_0^t{(t-s)}^{-\alpha }y\left(s\right)ds,$$

(4)

and the left Riemann–Liouville fractional integral for function $y\left(t\right)$ of order α in $L^{p(·)}$ is defined by:

$$I_{0^{+}}^{\alpha }y\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_0^t{(t-s)}^{\alpha -1}y\left(s\right)ds.$$

(5)

Definition 2

([23,24]). Letting $1\le p\le \infty$, we denote by $L^p([{\zeta }_1,{\zeta }_2],\mathbb{R})$ the Lebesgue space $L^p([{\zeta }_1,{\zeta }_2],\mathbb{R})$=$\{ \mathsf{\Phi }:[{\zeta }_1,{\zeta }_2]\to \mathbb{R}$ a measurable functions such that ${\parallel \mathsf{\Phi }\parallel }_p<\infty \}$, with the norm

$$\begin{array}{c}\hfill {\parallel \mathsf{\Phi }\parallel }_p={\left({\int }_{{\zeta }_1}^{{\zeta }_2}{\left|\mathsf{\Phi }\left(t\right)\right|}^{p}dt\right)}^{\frac{1}{p}}if1\le p<\infty \end{array}$$

and

$$\begin{array}{c}\hfill {\parallel \mathsf{\Phi }\parallel }_{\infty }=esssu{p}_{{\zeta }_1\le t\le {\zeta }_2}\left|\mathsf{\Phi }\left(t\right)\right|ifp=\infty .\end{array}$$

Next, we will present some important properties of $D_{{}_{{\zeta }_1^{+}}}^{\alpha }$, $I_{{}_{{\zeta }_1^{+}}}^{\alpha }$.

Lemma 1

([23,24]). Let $1\le p<\infty$, ${\mathsf{\Phi }}_1,{\mathsf{\Phi }}_2\in L^p([{\zeta }_1,{\zeta }_2],\mathbb{R})$ and $0<\alpha ,\beta <1$

Then,

(1) $I_{{}_{{\zeta }_1^{+}}}^{\alpha }{I}_{{}_{{\zeta }_1^{+}}}^{\beta }{\mathsf{\Phi }}_1\left(t\right)=I_{{}_{{\zeta }_1^{+}}}^{\alpha +\beta }{\mathsf{\Phi }}_1\left(t\right)$

(2) $I_{{}_{{\zeta }_1^{+}}}^{\alpha }[{\mathsf{\Phi }}_1\left(t\right)+{\mathsf{\Phi }}_2\left(t\right)]=I_{{}_{{\zeta }_1^{+}}}^{\alpha }{\mathsf{\Phi }}_1\left(t\right)+I_{{}_{{\zeta }_1^{+}}}^{\alpha }{\mathsf{\Phi }}_2\left(t\right)$

(3) $D_{{}_{{\zeta }_1^{+}}}^{\alpha }{I}_{{}_{{\zeta }_1^{+}}}^{\alpha }{\mathsf{\Phi }}_1\left(t\right)={\mathsf{\Phi }}_1\left(t\right)$

(4) $\parallel I_{{}_{{\zeta }_1^{+}}}^{\alpha }{\mathsf{\Phi }}_1\parallel {}_p\le \frac{{({\zeta }_2-{\zeta }_1)}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}{\parallel {\mathsf{\Phi }}_1\parallel }_p.$

Lemma 2

([25]). Let $1\le p<\infty$, $0<\alpha <1$, then $I_{{}_{a^{+}}}^{\alpha }\mathsf{\Phi }\in L^p([{\zeta }_1,{\zeta }_2],\mathbb{R})$ for any $\mathsf{\Phi }\in L^p([{\zeta }_1,{\zeta }_2],\mathbb{R})$.

Lemma 3

([23]). Let $0<\alpha <1$, then the differential equation

$$D_{{\zeta }_1^{+}}^{\alpha }h=0$$

has a unique solution

$$h\left(t\right)={\omega }_1{(t-{\zeta }_1)}^{\alpha -1}$$

${\omega }_1\in \mathbb{R}.$

Lemma 4

([23]). Let $0<\alpha <1$, ${\zeta }_1>0$, $h\in L^1(J,\mathbb{R})$, $D_{{\zeta }_1^{+}}^{\alpha }h\in L^1(J,\mathbb{R})$, then

$$I_{{\zeta }_1^{+}}^{\alpha }{D}_{{\zeta }_1^{+}}^{\alpha }h\left(t\right)=h\left(t\right)+{\omega }_1{(t-{\zeta }_1)}^{\alpha -1}$$

${\omega }_1\in \mathbb{R}$.

Lemma 5

([26] (H$\ddot{o}$lder’s inequality)). Let p and q satisfy $\frac{1}{p}+\frac{1}{q}=1$ such that, $1<p<\infty ,1<q<\infty$. If ${\mathsf{\Phi }}_1\in L^p(J,\mathbb{R})$ and ${\mathsf{\Phi }}_2\in L^q(J,\mathbb{R})$, then ${\mathsf{\Phi }}_1{\mathsf{\Phi }}_2$ belongs to $L^1(J,\mathbb{R})$ and satisfies

$$\begin{array}{c}\hfill {\int }_J\left|{\mathsf{\Phi }}_1{\mathsf{\Phi }}_2\right|dx\le {\left[{\int }_J{\left|{\mathsf{\Phi }}_1\right|}^{p}dx\right]}^{\frac{1}{p}}{\left[{\int }_J{\left|{\mathsf{\Phi }}_2\right|}^{q}dx\right]}^{\frac{1}{q}}.\end{array}$$

Definition 3

([27]). For $\mathsf{\Omega }\subseteq {\mathbb{R}}^n$ to be an open set in ${\mathbb{R}}^n$, we denote by $L^{p(·)}(\mathsf{\Omega },\mathbb{R})$ the set $L^{p(·)}(\mathsf{\Omega },\mathbb{R})=\{\mathsf{\Phi }:\mathsf{\Omega }\to \mathbb{R}$, which is a measurable function such that $I_{p(·)}\left(\mathsf{\Phi }\right)={\int }_{\mathsf{\Omega }}{\left|\mathsf{\Phi }\left(t\right)\right|}^{p\left(t\right)}dt<\infty ,$ where $p\left(t\right):\mathsf{\Omega }\to [1,\infty )$ is a measurable function $\}.$

Note that the set $L^{p(·)}(\mathsf{\Omega },\mathbb{R})$ is a Banach space with a norm defined as

$$\begin{array}{c}\hfill {\parallel \mathsf{\Phi }\parallel }_{p(·)}=inf\{\eta >0:I_{p(·)}(\mathsf{\Phi }/\eta )\le 1\}.\end{array}$$

We use the following notation:

$$\begin{array}{c}\hfill p_{-}=in{f}_{t\in \mathsf{\Omega }}p\left(t\right),p_{+}=su{p}_{t\in \mathsf{\Omega }}p\left(t\right),\end{array}$$

and $q(·)$ is denoted by the conjugate exponent of $p(·)$ with

$$\begin{array}{c}\hfill \frac{1}{p(·)}+\frac{1}{q(·)}=1.\end{array}$$

Consider the sets of functions

$$\mathcal{P}\left(\mathsf{\Omega }\right)=\{p:\mathsf{\Omega }\to [1,\infty ),pisaboundedmeasurablefunction\},$$

and

$${\mathcal{P}}^{log}\left(\mathsf{\Omega }\right)=\{p\in \mathcal{P}\left(\mathsf{\Omega }\right):|p\left(t\right)-p\left(s\right)|\le \frac{A_p}{-log|t-s|},|t-s|\le \frac{1}{2},t,s\in \mathsf{\Omega },$$

$$suchthat{A}_p>0isindependentoftands\}$$

Lemma 6

([28] (H$\ddot{o}$lder’s inequality in $L^{p(·)}(\mathsf{\Omega },\mathbb{R}))$). Let $\mathsf{\Omega }\subseteq {\mathbb{R}}^n$ be an open set and ${\mathsf{\Phi }}_1\in L^{p(·)}(J,\mathbb{R})$, ${\mathsf{\Phi }}_2\in L^{q(·)}(J,\mathbb{R})$ with $\frac{1}{p(·)}+\frac{1}{q(·)}=1$. Then, we have

$$\begin{array}{c}\hfill {\int }_{\mathsf{\Omega }}|{\mathsf{\Phi }}_1\left(t\right){\mathsf{\Phi }}_2\left(t\right)|dt\le r\parallel {\mathsf{\Phi }}_1{\parallel }_{p(·)}{\parallel {\mathsf{\Phi }}_2\parallel }_{q(·)},\end{array}$$

where $r=su{p}_{t\in \mathsf{\Omega }}\frac{1}{p\left(t\right)}+su{p}_{t\in \mathsf{\Omega }}\frac{1}{q\left(t\right)}$.

Theorem 1

([11]). $I_{{}_{0^{+}}}^{\alpha }$ is bounded in $L^{p(·)}([0,\zeta ],\mathbb{R})$ for any $p(·)\in \mathcal{P}[0,\zeta ]$ and $0<\frac{1}{p_{-}}<\alpha <1$.

Definition 4

([29]). Let $J\subset \mathbb{R}$, $n\in \mathbb{N}$ and the finite set $\mathcal{P}$ of generalized intervals $E_i,i=1,2,...,n$. We say that the $\mathcal{P}$ is a partition of J if $E_i$ are pairwise disjointed and J is the union of $E_i$ exactly. Additionally, the function $p:J\to \mathbb{R}$ is piecewise constant with respect to $\mathcal{P}$ if p adopts a value on $E_i$, for any $E_i$ of $\mathcal{P}$.

Theorem 2

((Riesz compactaness Criteria) [30]). $M\subset L^p([t_0,t_1],\mathbb{R})$ is relatively compact if and only if the following hold:

(1) M is bounded in $L^p$,

(2) ${\int }_{t_0}^{t_1}{|x(t+h)-x\left(t\right)|}^{p}dt\to 0ash\to 0.$

The following fixed point theorem will be used in the proof of our main results.

Theorem 3

([23]). Let Υ be a convex subset of a Banach space Θ and $\mathsf{\Lambda }:\mathsf{{\rm Y}}⟶\mathsf{{\rm Y}}$ is a continuous, compact map. Thus, Λ has a fixed point in Υ.

Theorem 4

([23]). Let $(\mathsf{{\rm Y}},d)$ be a complete metric space, $\mathsf{\Lambda }:\mathsf{{\rm Y}}⟶\mathsf{{\rm Y}}$ a contraction. Then, Λ has a unique fixed point in Υ.

Finally, we will provide a definition of the stability of the BVP (3) as follows:

Definition 5

([14]). The BVP (3) is Ulam–Hyers stable (UH) if there exists $c_{\mathsf{\Phi }}>0$, so that, for every $ϵ>0$, for any solution $z\in L^p(J,\mathbb{R})$ of the following inequality

$$|D_{0^{+}}^{\alpha }z\left(t\right)-\mathsf{\Phi }(t,z\left(t\right))|\le ϵ,t\in J$$

(6)

there exists a solution $y\in L^p(J,\mathbb{R})$ of BVP (3) with

$$|z\left(t\right)-y\left(t\right)|\le c_{\mathsf{\Phi }}ϵ,t\in J.$$

3. Existence of Solutions

Throughout this paper, we assume that:

(H1)

Let $n\in \mathbb{N}$, $\mathcal{P}=\{J_{\kappa }:=({\zeta }_{\kappa -1},{\zeta }_{\kappa }],\kappa =1,2,...,n\}$ be a partition of the interval J, such that, ${\zeta }_0=0,{\zeta }_n=\zeta$ and let $u\left(t\right):J\to [1,\infty )$ be a piecewise constant function with respect to $\mathcal{P}$, i.e., $p\left(t\right)={\sum }_{\kappa =1}^n{p}_{\kappa }{I}_{\kappa }\left(t\right)$, where

$$I_{\kappa }\left(t\right)=\left\{\begin{array}{c}1,fort\in J_{\kappa }\hfill \\ 0,forelsewhere,\hfill \end{array}\right.\kappa =1,2,...,n$$

and $1\le p_{\kappa }<\infty$ are constants.

For $\kappa =1,2,...,n$, $E_{\kappa }=L^{p_{\kappa }}(J_{\kappa }\times \mathbb{R}\times \mathbb{R},\mathbb{R})$ represents the Banach space of measurable functions from $J_{\kappa }$ into $\mathbb{R}$ with the norm:

$$\begin{array}{c}\hfill {\parallel y\parallel }_{E_{\kappa }}={\left({\int }_{J_{\kappa }}{\left|y\right|}^{p_{\kappa }}dx\right)}^{\frac{1}{p_{\kappa }}}<\infty .\end{array}$$

(H2)

Let $\mathsf{\Phi }\in C(J_{\kappa }\times \mathbb{R}\times \mathbb{R},\mathbb{R})$ and there exist constants, $M_1,M_2>0$, such that

$$\begin{array}{c}\hfill |\mathsf{\Phi }(t,y_1,z_1)-\mathsf{\Phi }(t,y_2,z_2)|\le M_1|y_1-y_2|+M_2|z_1-z_2|,\end{array}$$

for every $y_i,z_i\in L^{p_{\kappa }}(J_{\kappa }\times \mathbb{R}\times \mathbb{R},\mathbb{R})$ i = 1, 2 and $t\in J_{\kappa }.$

(H3)

Let $\mathsf{\Phi }:J_{\kappa }\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ such that $\mathsf{\Phi }$ is strongly measurable in t and continuous in y and z, and there exist constants, $b,d>0$, such that

$$\begin{array}{c}\hfill {\parallel \mathsf{\Phi }(t,y\left(\right),z\left(\right))\parallel }_{E_{\kappa }}\le a\left(t\right)+{b\parallel y\parallel }_{E_{\kappa }}+d{\parallel z\parallel }_{E_{\kappa }},\end{array}$$

for any $y,z\in L^{p_{\kappa }}(J_{\kappa },\mathbb{R})$ and $t\in J_{\kappa },$ and $a\in L^{p_{\kappa }}(J_{\kappa },\mathbb{R}).$

Thus, for $t\in J_{\kappa }$, $1\le \kappa \le n$, the left Riemann–Liouville fractional derivative of order $\alpha$ for function $y\left(t\right)$ defined by (4), is provided by

$$\begin{array}{c}\hfill \left(D_{0^{+}}^{\alpha }y\right)\left(t\right)=\frac{1}{\mathsf{\Gamma }(1-\alpha )}\frac{d}{dt}{\int }_0^t{(t-s)}^{-\alpha }y\left(s\right)ds\end{array}$$

$$\begin{array}{c}=\frac{1}{\mathsf{\Gamma }(1-\alpha )}\left(\sum _{ı=1}^{\kappa -1}\frac{d}{dt}{\int }_{{\zeta }_{ı-1}}^{{\zeta }_{ı}}{(t-s)}^{-\alpha }y\left(s\right)ds+\frac{d}{dt}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{-\alpha }y\left(s\right)ds\right).\hfill \end{array}$$

(7)

Thus, we can write the BVP (3) in the form:

$$\begin{array}{c}\frac{1}{\mathsf{\Gamma }(1-\alpha )}\left(\sum _{ı=1}^{\kappa -1}\frac{d}{dt}{\int }_{{\zeta }_{ı-1}}^{{\zeta }_{ı}}{(t-s)}^{-\alpha }y\left(s\right)ds+\frac{d}{dt}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{-\alpha }y\left(s\right)ds\right)=\mathsf{\Phi }(t,y\left(t\right),I_{0^{+}}^{\alpha }y\left(t\right)).\hfill \end{array}$$

(8)

For $t\in [0,{\zeta }_{\kappa -1}]$, we use $y\equiv 0$, then (8) is written as

$$\begin{array}{c}\hfill \left(D_{{\zeta }_{\kappa -1}}^{\alpha }y\right)\left(t\right)=\mathsf{\Phi }(t,y\left(t\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(t\right)),t\in J_{\kappa }.\end{array}$$

For any $1\le \kappa \le n$, we deal with the following BVP:

$$\left\{\begin{array}{c}{D}_{{\zeta }_{\kappa -1}}^{\alpha }y\left(t\right)=\mathsf{\Phi }(t,y\left(t\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(t\right)),t\in J_{\kappa }\hfill \\ \gamma y\left({\zeta }_{\kappa -1}\right)+\mu y\left({\zeta }_{\kappa }\right)=c.\hfill \end{array}\right.$$

(9)

Lemma 7.

Let $1\le \kappa \le n$, $0<\alpha <1$, $\mathsf{\Phi }\in L^{p_{\kappa }}(J_{\kappa }\times \mathbb{R}\times \mathbb{R},\mathbb{R})$.

Then, the solution y of (9) is written as the following integral equation:

$$\begin{array}{c}\hfill y\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds-\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\end{array}$$

$${\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds+\frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }.$$

(10)

Proof. 

Let y be a solution of the BVP (9), we find (see Lemma 4)

$$\begin{array}{c}\hfill y\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds-{\omega }_1{(t-{\zeta }_{\kappa -1})}^{\alpha -1}.\end{array}$$

By the value boundary of (9), we obtain:

$$\begin{array}{c}\hfill {\omega }_1=\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds-\frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }}{\mu }\end{array}$$

Hence, we obtain Equation (10).

Inversely, it is clear that y satisfies the BVP (9). □

Our first existence result is based on Theorem 2.

Theorem 5.

Assume that (H2), (H3), and inequality

$$L(\alpha ,p_{\kappa },{\zeta }_{\kappa -1},{\zeta }_{\kappa })<1,$$

(11)

holds, where:

$$L(\alpha ,p_{\kappa },{\zeta }_{\kappa -1},{\zeta }_{\kappa })$$

$$\begin{array}{ccc}& =& 2^{p_{\kappa }-1}[\frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{({\zeta }_{\kappa -1}-{\zeta }_{\kappa })}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}+\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}\right)\hfill \\ & +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right)\left(b+\frac{d{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)].\hfill \end{array}$$

Then, the BVP (9) has at least one solution on $E_{\kappa }$.

Proof. 

For any function $y\in E_{\kappa }$ we define the operator

$$\begin{array}{ccc}\hfill Sy\left(t\right)& =& \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & -& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & +& \frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }.\hfill \end{array}$$

By the properties of fractional integrals, the operator $S:E_{\kappa }\to E_{\kappa }$ is well defined.

Let

$$\begin{array}{ccc}\hfill R_{\kappa }& \ge & \frac{A}{B}\hfill \end{array}$$

where:

$$\begin{array}{ccc}\hfill A& =& 2^{p_{\kappa }-1}[\frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{-((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}({\zeta }_{\kappa }-1)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\hfill \\ & & \left({(-1)}^{-(\alpha -1)p_{\kappa }}\mathsf{\Gamma }\left((\alpha -1)p_{\kappa }\right)I_{{\zeta }_{\kappa -1}}^{(\alpha -1)p_{\kappa }}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)\right)\hfill \\ & +& \left(\frac{2^{2p_{\kappa }-2}{c}^{p_{\kappa }}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }}}{{\mu }^{p_{\kappa }}{\alpha }^{p_{\kappa }}}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right){]}^{\frac{1}{p_{\kappa }}}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill B& =& [1-2^{p_{\kappa }-1}[\frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}(\frac{{({\zeta }_{\kappa -1}-{\zeta }_{\kappa })}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}\hfill \\ & +& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)})\hfill \end{array}$$

$$\begin{array}{ccc}& +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\hfill \\ & & \left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right)\left(b+\frac{d{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)]{]}^{\frac{1}{p_{\kappa }}}\hfill \end{array}$$

We consider the set

$$\begin{array}{c}\hfill B_{R_{\kappa }}=\{y\in E_{\kappa }{,\parallel y\parallel }_{E_{\kappa }}\le R_{\kappa }\}.\end{array}$$

Obviously, $B_{R_{\kappa }}$ is nonempty, bounded, convex, and closed.

Now, it must be shown that S fulfills the premise of Theorem 2.

STEP 1:$S\left(B_{R_{\kappa }}\right)\subseteq \left(B_{R_{\kappa }}\right)$

For $y\in B_{R_{\kappa }}$, using Holder’s inequality and by (H3), we obtain

$$\begin{array}{ccc}\hfill \parallel Sy\left(t\right){\parallel }_{E_{\kappa }}^{p_{\kappa }}& =& \parallel \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & -& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & +& \frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }{\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & \le & \frac{2^{p_{\kappa }-1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{∥{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds∥}_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& \frac{2^{2p_{\kappa }-2}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{∥{(t-{\zeta }_{\kappa -1})}^{\alpha -1}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds∥}_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& \frac{2^{2p_{\kappa }-2}{c}^{p_{\kappa }}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mu }^{p_{\kappa }}}{∥{(t-{\zeta }_{\kappa -1})}^{\alpha -1}∥}_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & \le & \frac{2^{p_{\kappa }-1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{∥{\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-s)}^{(\alpha -1)q_{\kappa }}ds\right)}^{\frac{1}{q_{\kappa }}}{\parallel \mathsf{\Phi }(s,y,I_{{\zeta }_{\kappa -1}}^{\alpha }y)\parallel }_{E_{\kappa }}∥}_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& \frac{2^{2p_{\kappa }-2}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\parallel {(t-{\zeta }_{\kappa -1})}^{\alpha -1}{\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{(\alpha -1)q_{\kappa }}ds\right)}^{\frac{1}{q_{\kappa }}}{\parallel \mathsf{\Phi }(s,y,I_{{\zeta }_{\kappa -1}}^{\alpha }y)\parallel }_{E_{\kappa }}{\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& \frac{2^{2p_{\kappa }-2}{c}^{p_{\kappa }}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mu }^{p_{\kappa }}}{∥{(t-{\zeta }_{\kappa -1})}^{\alpha -1}∥}_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & \le & \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{-((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}({\zeta }_{\kappa }-1)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{({\zeta }_{\kappa -1}-{\zeta }_{\kappa })}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}+\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}\right)\hfill \\ & & {\left({b\parallel y\parallel }_{E_{\kappa }}+d{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }y\parallel }_{E_{\kappa }}\right)}^{p_{\kappa }}\hfill \\ & +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\hfill \\ & & \left({(-1)}^{-(\alpha -1)p_{\kappa }}\mathsf{\Gamma }\left((\alpha -1)p_{\kappa }\right)I_{{\zeta }_{\kappa -1}}^{(\alpha -1)p_{\kappa }}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)\right)\hfill \end{array}$$

$$\begin{array}{ccc}& +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\hfill \\ & & \left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right){\left({b\parallel y\parallel }_{E_{\kappa }}+d{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }y\parallel }_{E_{\kappa }}\right)}^{p_{\kappa }}\hfill \\ & +& \left(\frac{2^{2p_{\kappa }-2}{c}^{p_{\kappa }}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }}}{{\mu }^{p_{\kappa }}{\alpha }^{p_{\kappa }}}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right)\hfill \\ & \le & \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{-((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}({\zeta }_{\kappa }-1)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\hfill \\ & & \left({(-1)}^{-(\alpha -1)p_{\kappa }}\mathsf{\Gamma }\left((\alpha -1)p_{\kappa }\right)I_{{\zeta }_{\kappa -1}}^{(\alpha -1)p_{\kappa }}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)\right)\hfill \\ & +& \left(\frac{2^{2p_{\kappa }-2}{c}^{p_{\kappa }}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }}}{{\mu }^{p_{\kappa }}{\alpha }^{p_{\kappa }}}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right)\hfill \\ & +& [\frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{({\zeta }_{\kappa -1}-{\zeta }_{\kappa })}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}+\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}\right)\hfill \\ & +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right)\hfill \\ & & \left(b+\frac{d{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)]{\parallel y\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{(-1)}^{\frac{-((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\mathsf{\Gamma }(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)I_{{\zeta }_{\kappa -1}}^{(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1)}{a}^{p_{\kappa }}({\zeta }_{\kappa }-1)}{{((\alpha -1)q_{\kappa }+1)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\right)\hfill \\ & +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\hfill \\ & & \left({(-1)}^{-(\alpha -1)p_{\kappa }}\mathsf{\Gamma }\left((\alpha -1)p_{\kappa }\right)I_{{\zeta }_{\kappa -1}}^{(\alpha -1)p_{\kappa }}{a}^{p_{\kappa }}\left({\zeta }_{\kappa }\right)\right)\hfill \\ & +& \left(\frac{2^{2p_{\kappa }-2}{c}^{p_{\kappa }}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }}}{{\mu }^{p_{\kappa }}{\alpha }^{p_{\kappa }}}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right)\hfill \\ & +& [\frac{2^{2p_{\kappa }-2}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\left(\frac{{({\zeta }_{\kappa -1}-{\zeta }_{\kappa })}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}+\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1}}{\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}\right)\left(\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}+1\right)}\right)\hfill \\ & +& \left(\frac{2^{3p_{\kappa }-3}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\right)\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}}{\frac{((\alpha -1)q_{\kappa }+1)p_{\kappa }}{q_{\kappa }}}\right)\hfill \\ & & \left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }+1}}{(\alpha -1)p_{\kappa }+1}\right)\left(b+\frac{d{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\right)]R_{\kappa }^{p_{\kappa }}\hfill \end{array}$$

which means that $S\left(B_{R_{\kappa }}\right)\subseteq B_{R_{\kappa }}$.

STEP 2:S is continuous.

Assume the sequence $\left(y_n\right)$ is convergent to y in $E_{\kappa }$. Then, for $t\in J_{\kappa }$, by (H2) and Holder’s inequality we have

$$\begin{array}{ccc}& & \parallel S\left(y_n\left(t\right)\right)-S\left(y\left(t\right)\right){\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & =& \parallel \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_n\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right))-\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))\right)ds\hfill \\ & -& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_n\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right))-\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))\right){ds\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & =& {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_n\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right))-\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))\right)ds\hfill \\ & -& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_n\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right))-\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\right){|}^{p_{\kappa }}dt\hfill \\ & \le & \frac{2^{p_{\kappa }-1}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_n\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right))-\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))\right)ds{|}^{p_{\kappa }}dt\hfill \\ & +& \frac{2^{p_{\kappa }-1}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}\hfill \\ & & \left(\mathsf{\Phi }(s,y_n\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right))-\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))\right)ds{|}^{p_{\kappa }}dt\hfill \\ & \le & \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}|y_n\left(s\right)-y\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & +& \frac{{\left(2^{1-\frac{1}{p_{\kappa }}}{M}_2\right)}^{p_{\kappa }}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({(t-{\zeta }_{\kappa -1})}^{\alpha -1}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}|y_n\left(s\right)-y\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({(t-{\zeta }_{\kappa -1})}^{\alpha -1}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & \le & \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left[{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}{\left({\int }_{{\zeta }_{\kappa -1}}^t|y_n\left(s\right)-y\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}\right]}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}[{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^t|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}[{\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|y_n\left(s\right)-y\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}[{\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \\ & \le & \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\left({\int }_{{\zeta }_{\kappa -1}}^t|y_n\left(s\right)-y\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\left({\int }_{{\zeta }_{\kappa -1}}^t|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}{\parallel y_n-y\parallel }_{E_{\kappa }}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n-I_{{\zeta }_{\kappa -1}}^{\alpha }y\parallel }_{E_{\kappa }}^{p_{\kappa }}dt\hfill \end{array}$$

$$\begin{array}{ccc}& \le & {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}\left({\int }_{{\zeta }_{\kappa -1}}^t|y_n\left(s\right)-y\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}\hfill \\ & & \left({\int }_{{\zeta }_{\kappa -1}}^t|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}{\parallel y_n-y\parallel }_{E_{\kappa }}^{p_{\kappa }}dt.\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n-I_{{\zeta }_{\kappa -1}}^{\alpha }y\parallel }_{E_{\kappa }}^{p_{\kappa }}dt.\hfill \end{array}$$

We put:

$$\begin{array}{c}\hfill {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}\left({\int }_{{\zeta }_{\kappa -1}}^t|y_n\left(s\right)-y\left(s\right){|}^{p_{\kappa }}ds\right)dt={\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }}{r}_{\kappa }\left(t\right)dt=\mathsf{\Delta }\end{array}$$

where

$$\begin{array}{c}\hfill {\theta }_{\kappa }=\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }},r_{\kappa }\left(t\right)={\int }_{{\zeta }_{\kappa -1}}^t|y_n\left(s\right)-y\left(s\right){|}^{p_{\kappa }}ds.\end{array}$$

Integrating by parts, we have:

$$\begin{array}{ccc}& & \mathsf{\Delta }=\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{r}_{\kappa }\left({\zeta }_{\kappa }\right)-{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{r}_{\kappa }^{\prime }\left(t\right)dt\hfill \\ & =& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{\parallel y_n-y\parallel }_{E_{\kappa }}^{p_{\kappa }}-{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{r}_{\kappa }^{\prime }\left(t\right)dt.\hfill \end{array}$$

Since the integral is

$${\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{r}_{\kappa }^{\prime }\left(t\right)dt\ge 0,$$

then,

$$\begin{array}{ccc}& & \parallel S\left(y_n\left(t\right)\right)-S\left(y\left(t\right)\right){\parallel }_{E_{\kappa }}^{p_{\kappa }}\le {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{\parallel y_n-y\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n-I_{{\zeta }_{\kappa -1}}^{\alpha }y\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}{\parallel y_n-y\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n-I_{{\zeta }_{\kappa -1}}^{\alpha }y\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \end{array}$$

$$\begin{array}{ccc}& \le & ({\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}){\parallel y_n-y\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& ({\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}){\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_n-I_{{\zeta }_{\kappa -1}}^{\alpha }y\parallel }_{E_{\kappa }}^{p_{\kappa }}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & \parallel S\left(y_n\left(t\right)\right)-S\left(y\left(t\right)\right){\parallel }_{E_{\kappa }}\hfill \\ & \le & ({\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }\alpha }}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}\right){)}^{\frac{1}{p_{\kappa }}}\hfill \\ & & \parallel y_n{-y\parallel }_{E_{\kappa }}\hfill \end{array}$$

$$\parallel \left(S{y}_n\right)-\left(Sy\right){\parallel }_{E_{\kappa }}\to 0asn\to \infty .$$

Consequently, S is continuous.

STEP 3:S is compact.

We demonstrate that S is completely continuous. By Step 1, we have $S\left(B_{R_{\kappa }}\right)=\{S\left(y\right):y\in B_{R_{\kappa }}\}\subset B_{R_{\kappa }}$, which means that ${\parallel S\left(y\right)\parallel }_{E_{\kappa }}\le R_{\kappa }$ for any $y\in B_{R_{\kappa }}$; thus, $S\left(B_{R_{\kappa }}\right)$ is uniformly bounded. We need to prove that $S\left(B_{R_{\kappa }}\right)$ is relatively compact. We will use the Riesz compactness criteria Theorem.

For $t_1,t_2\in J_{\kappa },t_1<t_2$, and $y\in B_{R_{\kappa }}$, we have

$$\begin{array}{ccc}& & {\int }_{t_1}^{t_2}{\left(y(t+h)-y\left(t\right)\right)}^{p_{\kappa }}dt={\int }_{t_1}^{t_2}(\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{t+h}{(t+h-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & -& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t+h-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & +& \frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t+h-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }\hfill \\ & -& \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & +& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,y\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }y\left(s\right))ds\hfill \\ & -& \frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }{)}^{p_{\kappa }}dt.\hfill \end{array}$$

Now, using the fact that translations of $L^{p_{\kappa }}$ functions are continuous in norm, we see that

$$\begin{array}{c}\hfill {\int }_{t_1}^{t_2}{\left(y(t+h)-y\left(t\right)\right)}^{p_{\kappa }}dt\to 0as{t}_2\to t_1\end{array}$$

uniformly in $y\in B_{R_{\kappa }}$, and (2) of Theorem 2 is proved.

Therefore, using Theorem 2, we have that $S\left(B_{R_{\kappa }}\right)$ is relatively compact. According to Theorem 3, the BVP (8) has at least solution $\widetilde{y_{\kappa }}$ in $B_{R_{\kappa }}$. □

The following result is depended on the Banach contraction principle.

Theorem 6.

Assume that (H1), (H2), and inequality

$$W(\alpha ,M_1,M_2,p_{\kappa },{\zeta }_{\kappa -1},{\zeta }_{\kappa })<1,$$

(12)

holds, where:

$$\begin{array}{ccc}& & W(\alpha ,M_1,M_2,p_{\kappa },{\zeta }_{\kappa -1},{\zeta }_{\kappa })={\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }\alpha }}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}\right).\hfill \end{array}$$

Then, the BVP (9) possesses a unique solution on $E_{\kappa },\kappa =1,2,...,n$.

Proof. 

Let $y_1,y_2\in L^{p_{\kappa }}\left(J_{\kappa }\right)$; using Holder’s inequality and (H2) we have

$$\begin{array}{ccc}& & \parallel S\left(y_1\left(t\right)\right)-S\left(y_2\left(t\right)\right){\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & =& \parallel \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_1\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right))-\mathsf{\Phi }(s,y_2\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right))\right)ds\hfill \\ & -& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_1\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right))-\mathsf{\Phi }(s,y_2\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right))\right){ds\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & =& {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_1\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right))-\mathsf{\Phi }(s,y_2\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right))\right)ds\hfill \\ & -& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\left(\mathsf{\Phi }(s,y_1\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right))-\mathsf{\Phi }(s,y_2\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right))ds\right){|}^{p_{\kappa }}dt\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \frac{2^{p_{\kappa }-1}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\hfill \\ & & \left(\mathsf{\Phi }(s,y_1\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right))-\mathsf{\Phi }(s,y_2\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right))\right)ds{|}^{p_{\kappa }}dt\hfill \\ & +& \frac{2^{p_{\kappa }-1}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|{(t-{\zeta }_{\kappa -1})}^{\alpha -1}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\hfill \\ & & \left(\mathsf{\Phi }(s,y_1\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right))-\mathsf{\Phi }(s,y_2\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right))\right)ds{|}^{p_{\kappa }}dt\hfill \\ & \le & \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}|y_1\left(s\right)-y_2\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({(t-{\zeta }_{\kappa -1})}^{\alpha -1}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}|y_1\left(s\right)-y_2\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left({(t-{\zeta }_{\kappa -1})}^{\alpha -1}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right)|ds\right)}^{p_{\kappa }}dt\hfill \\ & \le & \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{\left[{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}{\left({\int }_{{\zeta }_{\kappa -1}}^t|y_1\left(s\right)-y_2\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}\right]}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}[{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^t|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}[{\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|y_1\left(s\right)-y_2\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}[{\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \end{array}$$

$$\begin{array}{ccc}& \le & \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right)}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\left({\int }_{{\zeta }_{\kappa -1}}^t|y_1\left(s\right)-y_2\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& \frac{{\left(2^{1-\frac{1}{p_{\kappa }}}{M}_2\right)}^{p_{\kappa }}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\left({\int }_{{\zeta }_{\kappa -1}}^t|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}{\parallel y_1-y_2\parallel }_{E_{\kappa }}^{p_{\kappa }}dt\hfill \\ & +& \frac{{\left(2^{1-\frac{1}{p_{\kappa }}}{M}_2\right)}^{p_{\kappa }}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )}}{{\left(\mathsf{\Gamma }\left(\alpha \right)\right)}^{p_{\kappa }}}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\parallel }_{E_{\kappa }}^{p_{\kappa }}dt\hfill \end{array}$$

$$\begin{array}{ccc}& \le & {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}\left({\int }_{{\zeta }_{\kappa -1}}^t|y_1\left(s\right)-y_2\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}\left({\int }_{{\zeta }_{\kappa -1}}^t|I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1\left(s\right)-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\left(s\right){|}^{p_{\kappa }}ds\right)dt\hfill \end{array}$$

$$\begin{array}{ccc}& +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}{\parallel y_1-y_2\parallel }_{E_{\kappa }}^{p_{\kappa }}dt.\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\parallel }_{E_{\kappa }}^{p_{\kappa }}dt.\hfill \end{array}$$

In a similar way to Step 2, we find:

$$\begin{array}{ccc}& & \parallel S\left(y_1\left(t\right)\right)-S\left(y_2\left(t\right)\right){\parallel }_{E_{\kappa }}^{p_{\kappa }}\le {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{\parallel y_1-y_2\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{{\theta }_{\kappa }+1}}{{\theta }_{\kappa }+1}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}{\parallel y_1-y_2\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}{\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & \le & ({\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}){\parallel y_1-y_2\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& ({\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}){\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_1-I_{{\zeta }_{\kappa -1}}^{\alpha }{y}_2\parallel }_{E_{\kappa }}^{p_{\kappa }},\hfill \end{array}$$

Then,

$$\begin{array}{ccc}& & \parallel S\left(y_1\left(t\right)\right)-S\left(y_2\left(t\right)\right){\parallel }_{E_{\kappa }}\le ({\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \end{array}$$

$$\begin{array}{ccc}& +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_1{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& {\left[\frac{2^{1-\frac{1}{p_{\kappa }}}{M}_2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )+\frac{q_{\kappa }(\alpha -1)+1}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{1}{q_{\kappa }}}\mathsf{\Gamma }\left(\alpha \right)}\right]}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}\left(\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }\alpha }}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}\right){)}^{\frac{1}{p_{\kappa }}}{\parallel y_1-y_2\parallel }_{E_{\kappa }}\hfill \end{array}$$

Hence, by (12) and according the Banach’s contraction principle, S has a unique fixed point $\widetilde{y_{\kappa }}\in L^{p_{\kappa }}(J_{\kappa },\mathbb{R})$, which is a unique solution of the BVP (9). □

Now, we will prove the existence result for BVP (3).

Theorem 7.

Assume that (H1), (H2), and (11) hold for any $1\le \kappa \le n$. Thus, the BVP (3) has a unique solution on $L^{p(·)}(J,\mathbb{R})$.

Proof. 

We have proved in Theorem 6 that the BVP for the Riemann–Liouville fractional differential Equation (9) possesses unique solution ${\tilde{y}}_{\kappa }\in E_{\kappa }$ for any $1\le \kappa \le n$.

For $1\le \kappa \le n$, we define the function:

$$\begin{array}{c}\hfill y_{\kappa }=\left\{\begin{array}{c}0,ift\in \left[0{\zeta }_{\kappa -1}\right]\hfill \\ {\tilde{y}}_{\kappa },ift\in J_{\kappa }.\hfill \end{array}\right.\end{array}$$

Thus, the function $y_{\kappa }\in L^p([0,{\zeta }_{\kappa -1}],\mathbb{R})$ is a solution to the integral Equation (8) for $t\in J_{\kappa }$, with

$\gamma y_{\kappa }\left(0\right)+\mu y_{\kappa }\left({\zeta }_{\kappa }\right)=\gamma y_{\kappa }\left(0\right)+\mu {\tilde{y}}_{\kappa }\left({\zeta }_{\kappa }\right)=c.$

Then, the function:

$$\begin{array}{c}\hfill y\left(t\right)=\left\{\begin{array}{c}{y}_1\left(t\right)\in L^{p_1}(J_1,\mathbb{R}),\hfill \\ y_2\left(t\right)\in L^{p_2}(J_2,\mathbb{R}),\hfill \\ .\hfill \\ .\hfill \\ .\hfill \\ y_n\left(t\right)\in L^{p_n}(J_n,\mathbb{R})\hfill \end{array}\right.\end{array}$$

forms a unique solution of the BVP (3) in $L^{p(·)}\left(J\right)$. □

4. Stability of Solutions

Theorem 8.

Let (H2), (H3), and (11) be satisfied for every $1\le \kappa \le n$. Thus, BVP (3) is (UH) stable.

Proof. 

Let $ϵ>0$ and the function $z\in L^{p(·)}(J,\mathbb{R})$ be a solution of the inequality (6).

We define the functions $z_1\left(t\right)\equiv z\left(t\right),t\in [0,{\zeta }_1]$ and for $\kappa =2,3,...,n$

$$z_{\kappa }\left(t\right)=\left\{\begin{array}{c}0,t\in [0,{\zeta }_{\kappa -1}]\hfill \\ z\left(t\right),t\in J_{\kappa }.\hfill \end{array}\right.$$

(13)

For $t\in J_{\kappa }$, $\kappa \in \{1,2,...,n\}$ and according to (7), we obtain

$$\begin{array}{c}\hfill \left(D_{0^{+}}^{\alpha }{z}_{\kappa }\right)\left(t\right)=\frac{1}{\mathsf{\Gamma }(1-\alpha )}\frac{d}{dt}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{-\alpha }z\left(s\right)ds.\end{array}$$

Using $I_{{\zeta }_{\kappa -1}^{+}}^{\alpha }$ of both parts of (6), we obtain

$$\begin{array}{ccc}\hfill |z_{\kappa }\left(t\right)& -& \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))ds+\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))ds-\frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }|\hfill \\ & \le & ϵ\frac{{(t-{\zeta }_{\kappa -1})}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}\hfill \\ & \le & ϵ\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha }}{\mathsf{\Gamma }(\alpha +1)}.\hfill \end{array}$$

By Theorem 7, BVP (3) possesses unique solution $y\in L^{p(·)}\left(J\right)$ defined by $y\left(t\right)=y_{\kappa }\left(t\right)$ for $t\in J_{\kappa },\kappa =1,2,...,n$, where

$$y_{\kappa }=\left\{\begin{array}{c}0,t\in [0,{\zeta }_{\kappa -1}]\hfill \\ {\tilde{y}}_{\kappa },t\in J_{\kappa }\hfill \end{array}\right.$$

(14)

and ${\tilde{y}}_{\kappa }\in E_{\kappa }$ is a unique solution of BVP (9) defined by

$$\begin{array}{c}\hfill {\tilde{y}}_{\kappa }\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right))ds-\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\end{array}$$

$${\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right))ds+\frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }.$$

(15)

For $t\in J_{\kappa },\kappa =1,2,...,n$, by (13), (14) we obtain,

$$\begin{array}{ccc}& & |z\left(t\right)-y\left(t\right)|=|z\left(t\right)-y_{\kappa }\left(t\right)|=|z_{\kappa }\left(t\right)-{\tilde{y}}_{\kappa }\left(t\right)|.\hfill \end{array}$$

Hence, by (15) we have

$$\begin{array}{ccc}& & {\parallel z-y\parallel }_{E_{\kappa }}^{p_{\kappa }}=\parallel z-y_{\kappa }{\parallel }_{E_{\kappa }}^{p_{\kappa }}={\parallel z_{\kappa }-{\tilde{y}}_{\kappa }\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & =& \parallel z_{\kappa }\left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right))ds\hfill \\ & +& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right))ds\hfill \\ & -& \frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }{\parallel }_{p_{\kappa }}^{p_{\kappa }}\hfill \\ & \le & 2^{p_{\kappa }-1}\parallel z_{\kappa }\left(t\right)-\frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))ds\hfill \\ & +& \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))ds\hfill \\ & -& \frac{c{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mu }{\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& 2^{p_{\kappa }-1}{\parallel \frac{1}{\mathsf{\Gamma }\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}(\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right)))ds\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & +& 2^{p_{\kappa }-1}\parallel \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{1-\alpha }{(t-{\zeta }_{\kappa -1})}^{\alpha -1}}{\mathsf{\Gamma }\left(\alpha \right)}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}(\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right))){ds\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & \le & 2^{p_{\kappa }-1}{ϵ}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }+1}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}+\frac{2^{p_{\kappa }-1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\hfill \\ & & {\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{\alpha -1}|\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right)){\left|ds\right)}^{p_{\kappa }}dt\hfill \\ & +& \frac{2^{p_{\kappa }-1}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{(\alpha -1)p_{\kappa }}\hfill \\ & & ({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{\alpha -1}|\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right)){\left|ds\right)}^{p_{\kappa }}dt\hfill \\ & \le & 2^{p_{\kappa }-1}{ϵ}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }+1}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}+\frac{2^{p_{\kappa }-1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}[{\left({\int }_{{\zeta }_{\kappa -1}}^t{(t-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^t|\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right)){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \end{array}$$

$$\begin{array}{ccc}& +& \frac{2^{p_{\kappa }-1}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}[{\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{({\zeta }_{\kappa }-s)}^{q_{\kappa }(\alpha -1)}ds\right)}^{\frac{1}{q_{\kappa }}}\hfill \\ & & {\left({\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}|\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right)){|}^{p_{\kappa }}ds\right)}^{\frac{1}{p_{\kappa }}}{]}^{p_{\kappa }}dt\hfill \\ & \le & 2^{p_{\kappa }-1}{ϵ}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }+1}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}+\frac{2^{p_{\kappa }-1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}\frac{{(t-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\hfill \\ & & \left({\int }_{{\zeta }_{\kappa -1}}^t|\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right)){|}^{p_{\kappa }}ds\right)dt\hfill \\ & +& \frac{2^{p_{\kappa }-1}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}{\int }_{{\zeta }_{\kappa -1}}^{{\zeta }_{\kappa }}{(t-{\zeta }_{\kappa -1})}^{p_{\kappa }(\alpha -1)}\hfill \\ & & \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}{|\mathsf{\Phi }(s,z_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }\left(s\right))-\mathsf{\Phi }(s,{\tilde{y}}_{\kappa }\left(s\right),I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }\left(s\right))|}_{p_{\kappa }}^{p_{\kappa }}dt\hfill \\ & \le & 2^{p_{\kappa }-1}{ϵ}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }+1}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}+\frac{2^{p_{\kappa }-1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right){\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & & (2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\parallel z_{\kappa }-{\tilde{y}}_{\kappa }{\parallel }_{E_{\kappa }}^{p_{\kappa }}+2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }-I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }{\parallel }_{E_{\kappa }}^{p_{\kappa }})\hfill \\ & +& \frac{2^{p_{\kappa }-1}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}\hfill \\ & & (2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}\parallel z_{\kappa }-{\tilde{y}}_{\kappa }{\parallel }_{E_{\kappa }}^{p_{\kappa }}+2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\parallel I_{{\zeta }_{\kappa -1}}^{\alpha }{z}_{\kappa }-I_{{\zeta }_{\kappa -1}}^{\alpha }{\tilde{y}}_{\kappa }{\parallel }_{E_{\kappa }}^{p_{\kappa }})\hfill \\ & \le & 2^{p_{\kappa }-1}{ϵ}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }+1}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}+[\frac{2^{p_{\kappa }-1}\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}+2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}\right)}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right){\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& \frac{2^{p_{\kappa }-1}\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}+2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}\right){({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right)}\hfill \\ & & \frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}}}{{\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{p_{\kappa }(1-\alpha )+1}}{p_{\kappa }(1-\alpha )+1}]{\parallel z_{\kappa }-{\tilde{y}}_{\kappa }\parallel }_{E_{\kappa }}^{p_{\kappa }}\hfill \\ & \le & 2^{p_{\kappa }-1}{ϵ}^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }+1}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}+\vartheta {\parallel z-y\parallel }_{E_{\kappa }}^{p_{\kappa }},\hfill \end{array}$$

where

$$\begin{array}{ccc}& & \vartheta =\underset{\kappa =1,2,...,n}{max}[\frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}+2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}\right)2^{p_{\kappa }-1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right){\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}}{\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+1}\hfill \\ & +& \frac{\left(2^{p_{\kappa }-1}{M}_1^{p_{\kappa }}+2^{p_{\kappa }-1}{M}_2^{p_{\kappa }}\frac{{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\alpha p_{\kappa }}}{{\mathsf{\Gamma }}^{p_{\kappa }}(\alpha +1)}\right)2^{p_{\kappa }-1}{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{(1-\alpha )p_{\kappa }+\frac{p_{\kappa }(q_{\kappa }(\alpha -1)+1)}{q_{\kappa }}+p_{\kappa }(1-\alpha )+1}}{{\mathsf{\Gamma }}^{p_{\kappa }}\left(\alpha \right){\left(q_{\kappa }(\alpha -1)+1\right)}^{\frac{p_{\kappa }}{q_{\kappa }}}{p}_{\kappa }(1-\alpha )+1}].\hfill \end{array}$$

Then,

$$\begin{array}{c}\hfill {\parallel z-y\parallel }_{E_{\kappa }}\le \frac{2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{\alpha p_{\kappa }+1}{p_{\kappa }}}}{{(1-\vartheta )}^{\frac{1}{p_{\kappa }}}\mathsf{\Gamma }(\alpha +1)}ϵ.\end{array}$$

Therefore,

$$\begin{array}{c}\hfill {\parallel z-y\parallel }_p\le \frac{1}{\mathsf{\Gamma }(\alpha +1)}\left(\sum _{\kappa =1}^{\kappa =n}\frac{2{({\zeta }_{\kappa }-{\zeta }_{\kappa -1})}^{\frac{\alpha p_{\kappa }+1}{p_{\kappa }}}}{{(1-\vartheta )}^{\frac{1}{p_{\kappa }}}}\right)ϵ:=c_{\mathsf{\Phi }}ϵ.\end{array}$$

As a consequence, the BVP (3) is (UH) stable. □

5. Example

Let the following BVP:

$$\left\{\begin{array}{c}{D}^{0.5}y\left(t\right)=\frac{1}{(2+e^t)(1+y\left(t\right)+I_0^{0.5}y\left(t\right))},t\in J:=[0,2]\hfill \\ y\left(0\right)+y\left(2\right)=0.\hfill \end{array}\right.$$

(16)

For $(t,y,z)\in \left(\right[0,2]\times [1,+\infty )\times [1,+\infty \left)\right)$, we define the function g as follows

$$\begin{array}{c}\hfill \mathsf{\Phi }(t,y,z)=\frac{1}{(2+e^t)(1+y\left(t\right)+z)}.\end{array}$$

Thus, we obtain

$$\begin{array}{ccc}\hfill |\mathsf{\Phi }(t,y_1,z_1)-\mathsf{\Phi }(t,y_2,z_2)|& =& \frac{1}{(2+e^t)}|\frac{1}{1+y_1+z_1}-\frac{1}{1+y_2+z_2}|\hfill \\ & \le & \frac{1}{2}|y_2-y_1|+\frac{1}{2}|z_2-z_1|.\hfill \end{array}$$

Hence, the condition (H2) holds with $M_1=M_2=\frac{1}{2}$.

Let

$$p\left(t\right)=\left\{\begin{array}{c}{p}_1=2.7,ift\in [0,1],\hfill \\ p_2=2.8,ift\in ]1,2].\hfill \end{array}\right.$$

(17)

We consider two auxiliary BVP:

$$\left\{\begin{array}{c}{D}^{0.5}y\left(t\right)=\frac{1}{(2+e^t)(1+y\left(t\right))},t\in J_1:=[0,1],\hfill \\ y\left(0\right)+y\left(1\right)=0,\hfill \end{array}\right.$$

(18)

and

$$\left\{\begin{array}{c}{D}^{0.5}y\left(t\right)=\frac{1}{(2+e^t)(1+y\left(t\right))},,t\in J_2:=[1,2],\hfill \\ y\left(1\right)+y\left(2\right)=0.\hfill \end{array}\right.$$

(19)

For $\kappa =1,p_1=2.7$, we have

$$\begin{array}{c}\hfill W_{\alpha ,M,p_1,{\zeta }_0,{\zeta }_1}=0,015523456<1.\end{array}$$

Ergo, the condition (12) is fulfilled.

Using Theorem (6), the BVP (18) possesses a unique solution ${\tilde{y}}_1\in L^{2.7}(J_1,\mathbb{R})$.

For $\kappa =2,p_2=2.8$, we have

$$\begin{array}{c}\hfill W_{\alpha ,M,p_2,{\zeta }_1,{\zeta }_2}=0,499862644<1.\end{array}$$

This means that the condition (12) is fulfilled.

According to Theorem 6, the BVP (19) possesses a unique solution ${\tilde{y}}_2\in L^{2.8}(J_2,\mathbb{R})$.

As a result, using Theorem (7), the BVP (19) has a unique solution.

$$y\left(t\right)=\left\{\begin{array}{c}{\tilde{y}}_1\left(t\right)\in L^{2.7}(J_1,\mathbb{R}),\hfill \\ y_2\left(t\right)\in L^{2.8}(J_2,\mathbb{R}),\hfill \end{array}\right.$$

where

$$y_2\left(t\right)=\left\{\begin{array}{c}0,t\in J_1,\hfill \\ {\tilde{y}}_2\left(t\right),t\in J_2.\hfill \end{array}\right.$$

As a result, according to Theorem 8, BVP (16) is (UH) stable.

6. Conclusions

The existence of a unique solution to the multiterm BVP (3) for fractional differential equations in Lebesgue spaces with variable exponent ($L^p(·)$) has been discussed in this paper. Based on the main difference between the classical Lebesgue spaces and the variable exponent Lebesgue spaces, we analyzed the BVP (3) and showed its existence and uniqueness. Further, the stability in the sense of Ulam–Hyers was examined. Lastly, an illustrative example was presented to demonstrate the correctness and applicability of the findings.

The obtained results in this paper generalize and extend the existing results on fractional differential equations in Lebesgue spaces with variable exponents ($L^p(·)$). In addition, they are important and have numerous applications in the qualitative study of such systems.