Existence and Uniqueness Results for Fractional (p, q)-Difference Equations with Separated Boundary Conditions

Abstract

In this paper, we study the existence of solutions to a fractional $(p, q)$-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented.

1. Introduction

Fractional calculus, dealing with the integrals and derivatives of arbitrary order, constitutes an important area of investigation in view of its extensive theoretical development and applications during the last few decades. For some interesting results on fractional differential equations ranging from the existence and uniqueness of solutions to the analytic and numerical methods for finding solutions, we refer the reader to the following articles: [1,2,3,4,5]. Concerning the applications of fractional differential equations in engineering, clinical disciplines, biology, physics, chemistry, economics, signal and image processing, and control theory, for example, see [6,7,8,9,10] for more details.

The study of q-calculus was introduced by Jackson in 1910, see [11,12] for more details. As one of the major driving forces behind the modern mathematical analysis, q-calculus has played important roles in both mathematical and physical problems. For instance, Fock [13] has studied the symmetry of hydrogen atoms using the q-difference equation. The concepts of q-calculus found numerous applications in a variety of fields, such as combinatorics, orthogonal polynomials, basic hypergeometric functions, number theory, quantum theory, quantum mechanics, and theory of relativity for details, see [14,15,16], and the references cited therein. One can find the basic concepts of q-calculus in the text by Kac and Cheung [17], while some details about fractional q-difference calculus can be found in [14,18,19,20,21].

The subject of $(p, q)$-calculus is known as the extension of q-calculus to its two-parameter $(p, q)$ variant and has efficient applications in many fields. One can find some useful information about the $(p, q)$-calculus in [22,23,24,25,26,27,28,29,30].

In 2021, Neang et al. [31] considered the nonlocal boundary value problem of nonlinear fractional $(p, q)$-difference equations with taking care of solutions of existence and uniqueness results obtained by

$$\begin{array}{cc}\hfill {}^c{D}_{p,q}^{\alpha }u\left(t\right)& =f\left(t,u\left(p^{\alpha }t\right)\right),t\in [0,T/p^{\alpha }],1<\alpha \le 2,\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill {\beta }_{1}u\left(0\right)+{\gamma }_1{D}_{p,q}u\left(0\right)& ={\zeta }_{1}u\left({\eta }_1\right),{\beta }_{2}u\left(T\right)+{\gamma }_2{D}_{p,q}u(T/p)={\zeta }_{2}u\left({\eta }_2\right),\hfill \end{array}$$

(2)

where $f\in \mathcal{C}\left([0,T/p^{\alpha }]\times \mathbb{R},\mathbb{R}\right)$, ${\beta }_i,{\gamma }_i,{\eta }_i(i=1,2)$ are constants, ${}^c{D}_{p,q}^{\alpha }$ denoted by Caputo fractional $(p,q)$ type, while $D_{p,q}$ denoted by first-order $(p,q)$-derivative.

Qin and Sun [32] studied on a nonlinear fractional $(p,q)$-difference Schrödinger equation in 2021, given by the following:

$$\begin{array}{cc}\hfill & D_{p,q}^{\alpha }u\left(x\right)+\alpha h\left(p^{\alpha }x\right)f\left(u\left(p^{\alpha }x\right)\right)=0,x\in (0,1),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & u\left(0\right)=D_{p,q}u\left(0\right)=D_{p,q}u\left(1\right)=0,\hfill \end{array}$$

(4)

where $0<q<p\le 1,2<\alpha \le 3,D_{p,q}^{\alpha }$ is a Riemann–Liouville-type fractional $(p,q)$-difference operator, and $f\in \mathcal{C}\left([0,1],(0,\infty )\right),h\in \mathcal{C}\left([0,1],(0,\infty )\right).$

Moreover, Qin and Sun [33] studied positive solutions for fractional $(p,q)$-difference boundary value problems given by the following:

$$\begin{array}{cc}\hfill & D_{p,q}^{\alpha }u\left(x\right)+f\left(p^{\alpha }x,u\left(p^{\alpha }x\right)\right)=0,x\in (0,1),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & u\left(0\right)=u\left(1\right)=0,\hfill \end{array}$$

(6)

where $0<q<p\le 1,1<\alpha \le 2,$$D_{p,q}^{\alpha }$ is a Riemann–Liouville-type fractional $(p,q)$-difference operator, and $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is a non-negative continuous function.

However, even though Neang et al. [31] investigated and proved the nonlocal boundary value problems by considering on existence results of a class of fractional $(p,q)$-difference equations, it still was a bit complicated with the domain of a function when the authors applied the fractional $(p,q)$-integral operators. In this paper, to make this paper more smooth and convenient, we have investigated the existence and uniqueness of solutions for the local boundary value problem of fractional $(p,q)$-difference equation with a new function obtained $g\in \mathcal{C}\left([0,b]\times \mathbb{R},\mathbb{R}\right)$, given by the following:

$$\begin{array}{cc}\hfill & {}^c{D}_{p,q}^{\alpha }x\left(t\right)=g\left(p^{\alpha }t,x\left(p^{\alpha }t\right)\right),t\in [0,b],1<\alpha \le 2,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & {\alpha }_{1}x\left(0\right)+{\beta }_1{D}_{p,q}x\left(0\right)={\gamma }_1,{\alpha }_{2}x\left(b\right)+{\beta }_2{D}_{p,q}x\left(pb\right)={\gamma }_2,\hfill \end{array}$$

(8)

where ${\alpha }_i,{\beta }_i,{\gamma }_i(i=1,2)$ are constants, ${}^c{D}_{p,q}^{\alpha }$ denoted by Caputo fractional $(p,q)$ type, while $D_{p,q}$ denoted by the first-order $(p,q)$-derivative.

2. Preliminaries

In this part, some fundamental results and definitions of the $(p,q)$-calculus, which can be found in [14,23,25] are given.

Let $[a,b]\subset \mathbb{R}$ be an interval with $a<b$ and $0<q<p\le 1$ be constants,

$$\begin{array}{cc}\hfill {\left[k\right]}_{p,q}& =\frac{p^k-q^k}{p-q},k\in \mathbb{N},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {\left[k\right]}_{p,q}!& =\left\{\begin{array}{c}{\displaystyle {\left[k\right]}_{p,q}{[k-1]}_{p,q}\cdots {\left[1\right]}_{p,q}=\prod _{i=1}^k\frac{p^i-q^i}{p-q},k\in \mathbb{N},}\hfill \\ 1,k=0.\hfill \end{array}\right.\hfill \end{array}$$

(10)

For $n\in {\mathbb{N}}_0:=\{0,1,2,\dots \}$, the q-analogue of the power function ${(a-b)}_q^{{}^{\left(n\right)}}$ is given by

$${\displaystyle {(a-b)}_q^{\left(0\right)}=1,{(a-b)}_q^{\left(n\right)}:=\prod _{k=0}^{n-1}(a-b{q}^k),a,b\in \mathbb{R}.}$$

For $n\in {\mathbb{N}}_0:=\{0,1,2,\dots \}$, the $(p,q)$-analogue of the power function ${(a-b)}_{p,q}^{{}^{\left(n\right)}}$ is given by

$${\displaystyle {(a-b)}_{p,q}^{\left(0\right)}=1,{(a-b)}_{p,q}^{\left(n\right)}:=\prod _{k=0}^{n-1}(a{p}^k-b{q}^k),a,b\in \mathbb{R}.}$$

The generalization of q-gamma function is called $(p,q)$-gamma is given by

$$\begin{array}{cc}\hfill {\mathrm{\Gamma }}_{p,q}\left(t\right)& =\frac{{\left(p-q\right)}_{p,q}^{(t-1)}}{{(p-q)}^{t-1}}\hfill \\ \hfill & =\frac{p^{\left(\genfrac{}{}{0pt}{}{t-1}{2}\right)}}{(1-{(q/p)}^{t-1}}\prod _{k=0}^{\infty }\frac{1-{(q/p)}^{k+1}}{1-{(q/p)}^{k+t+1}},\hfill \end{array}$$

(11)

For $t\in \mathbb{R}\backslash \{0,-1,-2,\cdots \}$, and an equivalent definition of (11) is given in [25] as

$${\mathrm{\Gamma }}_{p,q}\left(t\right)=p^{\frac{t(t-1)}{2}}{\int }_0^{\infty }{x}^{t-1}{E}_{p,q}^{-qx}{d}_{p,q}x$$

(12)

where

$$E_{p,q}^{-qx}=\sum _{n=0}^{\infty }\frac{q^{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}}{{\left[n\right]}_{p,q}!}{\left(-qx\right)}^n.$$

Remark 1.

${\mathrm{\Gamma }}_{p,q}(t+1)={\left[t\right]}_{p,q}{\mathrm{\Gamma }}_{p,q}\left(t\right)$ and ${\mathrm{\Gamma }}_{p,q}\left(t\right)\le p^{\left(\genfrac{}{}{0pt}{}{t-1}{2}\right)}\left(1-{(q/p)}_{p,q}^{1-t}\right).$

The definition of $(p,q)$-beta function for $s,t>0$ is defined by

$$B_{p,q}(s,t)={\int }_0^1{u}^{s-1}{\left(1-qu\right)}_{p,q}^{(t-1)}{d}_{p,q}u,$$

(13)

and (13) can also be written as

$$B_{p,q}(s,t)=p^{\left(t-1\right)\left(2s+t-2\right)/2}\frac{{\mathrm{\Gamma }}_{p,q}\left(s\right){\mathrm{\Gamma }}_{p,q}\left(t\right)}{{\mathrm{\Gamma }}_{p,q}(s+t)},$$

(14)

see [34,35] for more details.

Definition 1

([23]). Let $0<q<p\le 1.$ Then, the $(p,q)$-derivative of f is defined by

$$D_{p,q}g\left(t\right)=\frac{g\left(pt\right)-g\left(qt\right)}{(p-q)t},t\ne 0$$

(15)

and ${\displaystyle D_{p,q}g\left(0\right)=\underset{t\to 0}{lim}{D}_{p,q}g\left(t\right)}$, provided that g is differential at $0.$

Definition 2

([23]). Let $0<q<p\le 1,g$ be an arbitrary function, and t be a real number. The $(p,q)$-integral of g is defined as

$${\int }_0^{t}g\left(s\right)d_{p,q}s=(p-q)t\sum _{n=0}^{\infty }\frac{q^n}{p^{n+1}}g\left(\frac{q^n}{p^{n+1}}t\right)$$

(16)

provided that the series of the right-hand side in (16) converges.

Theorem 1

([23]). Let $g,h$ be differentiable on $[0,b]$ with a constant λ. Then,

(i) 

${\displaystyle D_{p,q}\left(g\left(t\right)+h\left(t\right)\right)=D_{p,q}g\left(t\right)+D_{p,q}h\left(t\right)}$;

(ii) 

${\displaystyle D_{p,q}\lambda g\left(s\right)=\lambda D_{p,q}g\left(s\right)}$;

(iii) 

${\displaystyle D_{p,q}\left(gh\right)\left(t\right)=g\left(pt\right)D_{p,q}h\left(t\right)+h\left(qt\right)D_{p,q}g\left(t\right)}$;

(iv) 

${\displaystyle D_{p,q}\left(g/h\right)\left(t\right)=\frac{h\left(qt\right)D_{p,q}g\left(t\right)-g\left(qt\right)D_{p,q}h\left(t\right)}{h\left(pt\right)h\left(qt\right)}}$,

where $h\left(t\right)\ne 0$ for $t\in [0,b]$.

Theorem 2

([30]). Let g be a continuous function on $[0,b]$. Then,

(i) 

${\displaystyle D_{p,q}{\int }_0^{t}g\left(s\right)d_{p,q}s=g\left(t\right)}$;

(ii) 

${\displaystyle {\int }_0^t{D}_{p,q}g\left(s\right)d_{p,q}s=g\left(t\right)-g\left(0\right)}$;

(iii) 

${\displaystyle {\int }_a^t{D}_{p,q}g\left(s\right)d_{p,q}s=g\left(t\right)-g\left(a\right)}$, for $a\in (0,t)$.

Definition 3

([34]). Let g be a continuous function defined on $[0,b]$. Then, the Riemann–Liouville fractional $(p,q)$-integral type is stated by for $\alpha >0$

$$\begin{array}{cc}\hfill \left(I_{p,q}^{\alpha }g\right)\left(t\right)& =\frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{\int }_0^t{\left(t-qs\right)}_{p,q}^{(\alpha -1)}g\left(\frac{s}{p^{\alpha -1}}\right)d_{p,q}s\hfill \\ \hfill & =\frac{(p-q)t}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\sum _{n=0}^{\infty }\frac{q^n}{p^{n+1}}{\left(t-\frac{q^{n+1}}{p^{n+1}}t\right)}_{p,q}^{(\alpha -1)}g\left(\frac{q^n}{p^{\alpha +n}}t\right)\hfill \end{array}$$

(17)

where $t\in [0,p^{\alpha }b]$. Notice that if $\alpha =0$, then $\left(I_{p,q}^{0}g\right)\left(t\right)=g\left(t\right).$

Definition 4

([34]). Let g be a continuous function defined on $[0,b]$. Then, the Riemann–Liouville fractional $(p,q)$-derivative type is stated by

$$\left(D_{p,q}^{\alpha }g\right)\left(t\right)=\left(D_{p,q}^{\left[\alpha \right]}{I}_{p,q}^{\left[\alpha \right]-\alpha }g\right)\left(t\right),for\alpha >0$$

(18)

where $⌈\alpha ⌉$ is the smallest integer greater than or equal to α. Notice that if $\alpha =0,$, then $\left(D_{p,q}^{0}h\right)\left(t\right)=g\left(t\right).$

Definition 5

([34]). Let g be a continuous function defined on $[0,b]$. If $\alpha >0$, then the Caputo fractional $(p,q)$-derivative is stated by

$${(}^c{D}_{p,q}^{\alpha }g)\left(t\right)=\left(I_{p,q}^{\left[\alpha \right]-\alpha }{D}_{p,q}^{\left[\alpha \right]}g\right)\left(t\right),$$

(19)

where $⌈\alpha ⌉$ is the smallest integer greater than or equal to α. Notice that if $\alpha =0$, then $\left({}^c{D}_{p,q}^{0}g\right)\left(t\right)=g\left(t\right)$.

To obtain the sufficient condition of existence and uniqueness of solutions of (7)–(8), employing the following Lemmas of fractional $(p,q)$-calculus play an important role in those main results.

Lemma 1

([34]). Let g be a continuous function on $[0,b]$. Then,

(i) 

${\displaystyle \left(I_{p,q}^{\beta }{I}_{p,q}^{\alpha }g\right)\left(t\right)=\left(I_{p,q}^{\alpha +\beta }g\right)\left(t\right)}$;

(ii) 

${\displaystyle \left(D_{p,q}^{\alpha }{I}_{p,q}^{\alpha }g\right)\left(t\right)=g\left(t\right).}$

Lemma 2

([34]). Let g be a continuous function on $[0,b]$. If $\alpha >0$, and $n\in \mathbb{N}$, then the following equality holds:

$$\left(I_{p,q}^{\alpha }{D}_{p,q}^{n}g\right)\left(t\right)=\left(D_{p,q}^n{I}_{p,q}^{\alpha }g\right)\left(t\right)-\sum _{k=0}^{\left[\alpha \right]-1}\frac{t^{\alpha -n+k}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -n+k+1)}\left(D_{p,q}^{k}g\right)\left(0\right).$$

Lemma 3

([34]). Let g be a continuous function on $[0,b]$. If $\alpha >0$ and $n\in \mathbb{N}$, then

$$\left(I_{p,q}^{\alpha }{}^c{D}_{p,q}^{\alpha }g\right)\left(t\right)=g\left(t\right)-\sum _{k=0}^{\left[\alpha \right]-1}\frac{t^k}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}(k+1)}\left(D_{p,q}^{k}g\right)\left(0\right).$$

Lemma 4.

In order to prove (7) and (8), we first give a useful Lemma, as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^c{D}_{p,q}^{\alpha }x\left(t\right)=g\left(p^{\alpha }t\right),t\in [0,b],\hfill \\ {\alpha }_{1}x\left(0\right)+{\beta }_1{D}_{p,q}x\left(0\right)={\gamma }_1,\hfill \\ {\alpha }_{2}x\left(b\right)+{\beta }_2{D}_{p,q}x\left(pb\right)={\gamma }_2,\hfill \end{array}\right.\end{array}$$

(20)

is defined by

$$\begin{array}{cc}\hfill x\left(t\right)& ={\eta }_1+{\eta }_{2}t+{\int }_0^t\frac{{\left(t-qs\right)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g\left(ps\right)d_{p,q}s\hfill \\ \hfill & +\frac{1}{\Delta }\left[{\alpha }_1{\alpha }_{2}t-{\beta }_1{\alpha }_2\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g\left(ps\right)d_{p,q}s\hfill \\ \hfill & +\frac{1}{\Delta }\left[{\alpha }_1{\beta }_{2}t-{\beta }_1{\beta }_2\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}g\left(p^{2}s\right)d_{p,q}s,\hfill \end{array}$$

(21)

where

$${\eta }_1=\frac{{\beta }_1{\gamma }_2-{\gamma }_1\left({\alpha }_{2}b+{\beta }_2\right)}{\Delta },{\eta }_2=\frac{\left({\alpha }_2{\gamma }_1-{\alpha }_1{\gamma }_2\right)}{\Delta },$$

and it is supposed that

$$\Delta ={\alpha }_2{\beta }_1-{\alpha }_1({\alpha }_{2}b+{\beta }_2)\ne 0.$$

Proof. 

Applying fractional $(p,q)$-integral on (20), we obtain the following:

$${\displaystyle x\left(t\right)={\int }_0^t\frac{{(t-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g\left(ps\right)d_{p,q}s+t{c}_0+c_1,}$$

(22)

where $c_0,c_1$ are constants and $t\in [0,b].$ Utilizing (20) again, we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\alpha }_1{c}_1+{\beta }_1{c}_0={\gamma }_1,\hfill \\ {\displaystyle c_0\left({\alpha }_{2}b+{\beta }_2\right)+{\alpha }_2{c}_1={\gamma }_2-{\alpha }_2{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g\left(ps\right)d_{p,q}s}\hfill \\ {\displaystyle -{\beta }_2{\int }_0^b\frac{{(b-qs)}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}g\left(p^{2}s\right)d_{p,q}s.}\hfill \end{array}\right.\end{array}$$

(23)

Solving the above system of equations to find the constants $c_0,c_1$, we have

$$\begin{array}{cc}\hfill c_0& =\frac{1}{\Delta }\left[{\alpha }_2{\gamma }_1+{\alpha }_1{\alpha }_2{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g\left(ps\right)d_{p,q}s\right.\hfill \\ \hfill & \left.+{\alpha }_1{\beta }_2{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g\left(p^{2}s\right)d_{p,q}s\right]\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill c_1& =\frac{1}{\Delta }\left[-{\gamma }_1\left({\alpha }_{2}b+{\beta }_1{\alpha }_2\right){\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g\left(ps\right)d_{p,q}s\right.\hfill \\ \hfill & \left.-{\beta }_1{\beta }_2{\int }_0^b\frac{{(b-qs)}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}g\left(p^{2}s\right)d_{p,q}s\right].\hfill \end{array}$$

Substituting the values of $c_0,c_1$ in (22), we derive (21). By direction computation, we obtain for the converse. Therefore, this completed the proof. □

3. Main Results

Let $\mathcal{C}:=\mathcal{C}\left([0,b],\mathbb{R}\right)$ denote the Banach space of all continuous functions from $[0,b]$ to $\mathbb{R}$, endowed with norm, defined by

$$\parallel x\parallel =\sup \left\{\right|x\left(t\right)|:t\in [0,b\left]\right\}.$$

In view of Lemma 4, we define an operator $\mathcal{F}:\mathcal{C}\to \mathcal{C}$ as

$$\begin{array}{cc}\hfill \left(\mathcal{F}x\right)\left(t\right)& ={\eta }_1+{\eta }_{2}t+{\int }_0^t\frac{{(t-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g(ps,x\left(ps\right))d_{p,q}s\hfill \\ \hfill & +\frac{1}{\Delta }\left[{\alpha }_1{\alpha }_{2}t-{\beta }_1{\alpha }_2\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g(ps,x\left(ps\right))d_{p,q}s\hfill \\ \hfill & +\frac{1}{\Delta }\left[{\alpha }_1{\beta }_{2}t-{\beta }_1{\beta }_2\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}g(p^{2}s,x\left(p^{2}s\right))d_{p,q}s,\hfill \end{array}$$

(24)

where $\Delta ={\alpha }_2{\beta }_1-{\alpha }_1({\alpha }_{2}b+{\beta }_2)\ne 0.$

Observe that x is a solution to (7) and (8) if—and only if—x is a fixed-point of $\mathcal{F}.$ For convenience, we denote

$$\begin{array}{cc}\hfill k& ={\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}L\left(ps\right)d_{p,q}s+{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}L\left(ps\right)d_{p,q}s\hfill \\ & +{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}L\left(p^{2}s\right)d_{p,q}s,\hfill \end{array}$$

(25)

where

$${\sigma }_1\left(b\right)=\frac{1}{\left|\Delta \right|}\left(\left|{\alpha }_1\right|\left|{\alpha }_2\right|b+\left|{\beta }_1\right|\left|{\alpha }_2\right|\right),{\sigma }_2\left(b\right)=\frac{1}{\left|\Delta \right|}\left(|{\alpha }_1\left|\right|{\beta }_2|b+\left|{\beta }_1\right|\left|{\beta }_2\right|\right).$$

Theorem 3.

Let g be a continuous function on $[0,b]\times \mathbb{R}$ and there exists a integrable function $L:[0,b]\to \mathbb{R}$, such that

$\left(A_1\right)$

${\displaystyle \left|g(t,x)-g(t,y)\right|\le L\left(t\right)\left|x-y\right|}$, for each $t\in [0,b]$ and $x,y\in \mathbb{R}.$

If $k<1$, then (7) and (8) has a unique solution.

Proof. 

We transform the problem (7) and (8) into a fixed-point problem $\mathcal{F}x=x,$ where the operator $\mathcal{F}$ is given by (24). Applying Banach’s contraction mapping principle, we will show that $\mathcal{F}$ has a unique fixed point. Define a ball, $B_r=\{x\in \mathcal{C}:\parallel x\parallel \le r\}$, with the radius, r, satisfying

$$r\ge \frac{|{\eta }_1|+|{\eta }_2|b+MA}{1-k},$$

where

$$\begin{array}{c}\hfill A=\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_1\left(b\right)\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_2\left(b\right)\frac{b^{\alpha -1}}{{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)},\end{array}$$

(26)

and ${\displaystyle M=\underset{t\in \left[0,b\right]}{sup}\left|f(t,0)\right|}$. We have

$$\begin{array}{cc}\hfill \left|g(t,x(t\left)\right)\right|& \le \left|g(t,x(t\left)\right)-g(t,0)\right|+\left|g(t,0)\right|\hfill \\ \hfill & \le L\left(t\right)r+M.\hfill \end{array}$$

Now, we shall show that $\mathcal{F}\subset B_r$. For any $x\in B_r$, consider

$$\begin{array}{cc}\hfill \left|\left(\mathcal{F}x\right)\left(t\right)\right|& \le \left|{\eta }_1\right|+\left|{\eta }_2\right|t+{\int }_0^t\frac{{(t-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left[\left|{\alpha }_1\right|\left|{\alpha }_2\right|t+\left|{\beta }_1\right|\left|{\alpha }_2\right|\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left[{\alpha }_1{\beta }_{2}t+\left|{\beta }_1\right|\left|{\beta }_2\right|\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left|g(p^{2}s,x\left(p^{2}s\right))\right|d_{p,q}s\hfill \\ \hfill & \le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +{\sigma }_2\left(T\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left|g(p^{2}s,x\left(p^{2}s\right))\right|d_{p,q}s,\hfill \\ \hfill & \le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left(L\left(ps\right)r+M\right)d_{p,q}s\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left(L\left(ps\right)r+M\right)d_{p,q}s\hfill \\ \hfill & +{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left(L\left(p^{2}s\right)r+M\right)d_{p,q}s\hfill \\ \hfill & \le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+M\left\{\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_1\left(b\right)\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_2\left(b\right)\frac{b^{\alpha -1}}{{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\right\}\hfill \\ \hfill & +r\left\{{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}L\left(ps\right)d_{p,q}s\right.\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}L\left(ps\right)d_{p,q}s\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.+{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}L\left(p^{2}s\right)d_{p,q}s\right\}\hfill \end{array}$$

from (25) and (26), we obtain

$$\parallel \mathcal{F}x\parallel \le |{\eta }_1|+|{\eta }_2|b+MA+rk\le r.$$

This shows that $\mathcal{F}{\mathcal{B}}_r\subset {\mathcal{B}}_r$.

Now, for $x,y\in \mathcal{C}$, we obtain

$$\begin{array}{cc}\hfill & ∥\mathcal{F}x-\mathcal{F}y∥\hfill \\ \hfill & {\displaystyle \le \underset{t\in [0,b]}{sup}\left\{{\int }_0^t\frac{{\left(t-qs\right)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)-g(ps,y(ps\left)\right)\right|d_{p,q}s\right.}\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{\left(b-qs\right)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)-g(ps,y(ps\left)\right)\right|d_{p,q}s\hfill \\ & \left.+{\sigma }_2\left(b\right){\int }_0^b\frac{{\left(b-qs\right)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(p^{2}s,x\left(p^{2}s\right))-g(p^{2}s,y\left(p^{2}s\right))\right|d_{p,q}s\right\}\hfill \\ \hfill & {\displaystyle \le ∥x-y∥\left\{{\int }_0^b\frac{{\left(b-qs\right)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}L\left(ps\right)d_{p,q}s\right.}\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{\left(b-qs\right)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}L\left(ps\right)d_{p,q}s\hfill \\ \hfill & \left.+{\sigma }_2\left(b\right){\int }_0^b\frac{{\left(b-qs\right)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}L\left(p^{2}s\right)d_{p,q}s\right\},\hfill \end{array}$$

which, in view of (25), we obtain

$$∥\mathcal{F}x-\mathcal{F}y∥\le k∥x-y∥.$$

This is because $k\in (0,1)$, $\mathcal{F}$ is a contraction. Therefore, (7) and (8) has a unique solution. The proof is completed. □

Remark 2.

If g is a continuous function on $[0,b]\times \mathbb{R}$, and there exists a constant $L>0$ with

$${\displaystyle \left|g(t,x)-g(t,y)\right|\le L\left|x-y\right|,}$$

then (7) and (8) has a unique solution, if $k<1$.

Lemma 5

(Kranoselskii’s fixed-point theorem [36]). Let M be a closed, bounded, convex, and non-empty subset of a Banach space X. Let $A,B$ be two operators, such that:

(i) 

$Ax+By\in M,$ whenever $x,y\in M$;

(ii) 

A is compact and continuous;

(iii) 

B is a contraction mapping.

Then, there exists $z\in M$, such that $z=Az+Bz.$

Theorem 4.

Let g be a continuous functions on $[0,b]\times \mathbb{R}$, satisfying $\left(A_1\right).$ Assume that

$\left(A_2\right)$

there exists a function, $\mu \in \mathcal{C}\left([0,b],{\mathbb{R}}^{+}\right)$, and a non-decreasing function, $\varphi \in \mathcal{C}\left([0,b],{\mathbb{R}}^{+}\right)$, with

$$\left|g(t,x)\right|\le \mu \left(t\right)\varphi \left(\left|x\right|\right),$$

where $\left(t,x\right)\in [0,b]\times [-b,b].$

If

$$\begin{array}{cc}\hfill {\sigma }_1\left(b\right)& {\int }_0^b\frac{{\left(b-qs\right)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}L\left(ps\right)d_{p,q}s\hfill \\ \hfill & +{\sigma }_2\left(b\right){\int }_0^b\frac{{\left(b-qs\right)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}L\left(p^{2}s\right)d_{p,q}s<1,\hfill \end{array}$$

(27)

then (7) and (8) has at least one solution on $[0,b].$

Proof. 

Define ${\mathcal{B}}_{\overline{r}}:=\left\{x\in \mathcal{C}:∥x∥\le \overline{r}\right\},$ where

$$\overline{r}\ge \varphi \left(\overline{r}\right)∥\mu ∥\left[\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_1\left(b\right)\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_2\left(b\right)\frac{b^{\alpha -1}}{{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\right],$$

where ${\displaystyle ∥\mu ∥=\underset{t\in [0,b]}{sup}\left|\mu \left(t\right)\right|,}$ and define the operators $\mathcal{P}$ and $\mathcal{Q}$ on ${\mathcal{B}}_{\overline{r}}$ as

$$\begin{array}{cc}\hfill \left(\mathcal{P}x\right)\left(t\right)& =\frac{1}{\Delta }\left[{\alpha }_1{\alpha }_{2}t-{\beta }_1{\alpha }_2\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g(ps,x\left(ps\right))d_{p,q}s\hfill \\ \hfill & +\frac{1}{\Delta }\left[{\alpha }_1{\beta }_{2}t-{\beta }_1{\beta }_2\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}g(p^{2}s,x\left(p^{2}s\right))d_{p,q}s\hfill \end{array}$$

and

$$\left(\mathcal{Q}y\right)\left(t\right)={\eta }_1+{\eta }_{2}t+{\int }_0^t\frac{{(t-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}g(ps,y\left(ps\right))d_{p,q}s.$$

Observe that $\mathcal{P}x+\mathcal{Q}x=\mathcal{F}x.$ For $x,y\in {\mathcal{B}}_{\overline{r}},$ we have

$$\begin{array}{cc}\hfill \left|\left(\mathcal{P}x+\mathcal{Q}y\right)\left(t\right)\right|& \le \left|{\eta }_1\right|+\left|{\eta }_2\right|t+{\int }_0^t\frac{{(t-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,y(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left(\left|{\alpha }_1\right|\left|{\alpha }_2\right|t+\left|{\beta }_1\right|\left|{\alpha }_2\right|\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left({\alpha }_1{\beta }_{2}t+\left|{\beta }_1\right|\left|{\beta }_2\right|\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left|g(p^{2}s,x\left(p^{2}s\right))\right|d_{p,q}s\hfill \\ \hfill & \le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\mu \left(t\right)\varphi \left(\left|x\right|\right)d_{p,q}s\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\mu \left(t\right)\varphi \left(\left|x\right|\right)d_{p,q}s\hfill \\ \hfill & +{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\mu \left(t\right)\varphi \left(\left|x\right|\right)d_{p,q}s,\hfill \\ \hfill & \le \varphi \left(\overline{r}\right)∥\mu ∥\left[\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_1\left(b\right)\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_2\left(b\right)\frac{b^{\alpha -1}}{{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\right]\hfill \\ \hfill & \le \overline{r}.\hfill \end{array}$$

Thus, $\mathcal{P}x+\mathcal{Q}y\in {\mathcal{B}}_{\overline{r}}.$ By $\left(A_1\right)$ and (27), $\mathcal{Q}$ is a contraction mapping. By continuity of f, we obtain that $\mathcal{P}$ is continuous. It is easy to see that

$$∥\mathcal{P}∥\le \varphi \left(\overline{r}\right)∥\mu ∥\left[{\sigma }_1\left(b\right)\frac{b^{\alpha }}{{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+{\sigma }_2\left(b\right)\frac{b^{\alpha -1}}{{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\right].$$

Thus, the set $\mathcal{P}\left({\mathcal{B}}_{\overline{r}}\right)$ is uniformly bounded. $\mathcal{P}$ is compact. First, Let

$${\displaystyle \overline{g}=\underset{(t,x)\in [0,b]\times {\mathcal{B}}_{\overline{r}}}{sup}\left|g(t,x)\right|<\infty }$$

and let $t_1,t_2\in [0,b]$ with $t_1<t_2.$ Then, we obtain

$$\begin{array}{cc}\hfill & \left|\left(\mathcal{P}x\right)\left(t_2\right)-\left(\mathcal{Q}x\right)\left(t_1\right)\right|\hfill \\ \hfill & \le \frac{1}{\left|\Delta \right|}\left|{\alpha }_1\right|\left|{\alpha }_2\right|(t_2-t_1){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}{\alpha }_1{\beta }_2(t_2-t_1){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left|g(p^{2}s,x\left(p^{2}s\right))\right|d_{p,q}s\hfill \\ \hfill & \le \overline{g}\left\{\frac{|{\alpha }_1\left|\right|{\alpha }_2|b^{\alpha }}{|\Delta |{\mathrm{\Gamma }}_{p,q}(\alpha +1)}+\frac{|{\alpha }_1\left|\right|{\beta }_2|b^{\alpha -1}}{|\Delta |{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\right\}(t_2-t_1),\hfill \end{array}$$

which is independent of x, and tends to zero as $t_1\to t_2.$ So, the set $\mathcal{P}\left(B_{\overline{r}}\right)$ is equicontinuous. By the Arzelá–Ascoli theorem, $\mathcal{P}$ is compact on $B_{\overline{r}}.$ Thus, (7) and (8) has at least one solution on $[0,b].$ □

Remark 3.

Let g be a continuous function on $[0,b]\times \mathbb{R}$, satisfying $\left(A_1\right).$ Assume that

$$\left|g(t,x)\right|\le \mu \left(t\right),\forall (t,x)\in [0,b]\times \mathbb{R}\mathrm{and}\mu \in \mathcal{C}\left([0,b],{\mathbb{R}}^{+}\right).$$

If (27) holds, then (7) and (8) has at least one solution on $[0,b].$

Lemma 6

(Nonlinear alternative for single value maps [37]). Let E be a Banach space, $\mathcal{C}$ a closed, convex subset of E and $\mathcal{U}$ an open subset of $\mathcal{C}$ with $u\in \mathcal{U}.$ Suppose that $\mathcal{F}:\overline{\mathcal{U}}\to \mathcal{C}$ is a continuous, compact function; that is, $\mathcal{F}\left(\overline{\mathcal{U}}\right)$ is a relatively compact subset of $\mathcal{C}$ map. Then, either

(i) 

$\mathcal{F}$ has a fixed point in $\overline{\mathcal{U}},$ or

(ii) 

there is a $u\in \partial \mathcal{U}$ (the boundary of $\mathcal{U}$ in $\mathcal{C}$) and $\lambda \in (0,1)$ with $u=\lambda \mathcal{F}u.$

Theorem 5.

Let g be a continuous function on $[0,b]\times \mathbb{R}$. Assume that

$\left(A_3\right)$

there exists functions $u_1,u_2\in \mathcal{C}\left([0,b],{\mathbb{R}}^{+}\right),$ and a non-decreasing function, $\mathrm{\Psi }:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$, such that

$$\left|g(t,x)\right|\le u_1\left(t\right)\mathrm{\Psi }\left(\right|x\left|\right)+u_2\left(t\right),\mathrm{t},\mathrm{x}\in [0,b]\times \mathbb{R};$$

$\left(A_4\right)$

there exists a number, $M>0$, such that

$$\frac{M}{|{\eta }_1|+|{\eta }_2|b+\mathrm{\Psi }\left(M\right){\omega }_1+{\omega }_2}>1,$$

(28)

where

$$\begin{array}{cc}\hfill {\omega }_i:& ={\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_i\left(ps\right)d_{p,q}s\hfill \\ & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_i\left(ps\right)d_{p,q}s\hfill \\ \hfill & +{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}{u}_i\left(p^{2}s\right)d_{p,q}s,i=1,2.\hfill \end{array}$$

(29)

Then, (7) and (8) has at least one solution on $[0,b].$

Proof. 

Notice that $\mathcal{F}:\mathcal{C}\to \mathcal{C}$ defined by (24). $\mathcal{F}$ is continuous. Let $\left\{x_n\right\}$ be a sequence of the function, such that $x_n\to x$ on $[0,b].$ Since

$$g\left(p^{\alpha }t,x_n\left(p^{\alpha }t\right)\right)\to g(p^{\alpha }t,x\left(p^{\alpha }t\right)).$$

Therefore, we obtain

$$\begin{array}{cc}\hfill & \left|\left(\mathcal{F}{x}_n\right)\left(t\right)-\left(\mathcal{F}x\right)\left(t\right)\right|\hfill \\ \hfill & \le {\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g\left(ps,x_n\left(ps\right)\right)-g(ps,x\left(ps\right))\right|d_{p,q}s\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g\left(ps,x_n\left(ps\right)\right)-g(ps,x\left(ps\right))\right|d_{p,q}s\hfill \\ \hfill & +{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left|g\left(p^{2}s,x_n\left(p^{2}s\right)\right)-g(p^{2}s,x\left(p^{2}s\right))\right|d_{p,q}s,\hfill \end{array}$$

which implies that

$$∥\mathcal{F}{x}_n-\mathcal{F}x∥\to 0\mathrm{as}n\to \infty .$$

Thus, the operator $\mathcal{F}$ is continuous.

Next, we show that $\mathcal{F}$ maps a bounded set into a bounded set in $\mathcal{C}\left([0,b],\mathbb{R}\right).$ For a positive number $r>0,$ let $B_r=\left\{x\in \mathcal{C}\left([0,b]\right):\parallel x\parallel \le r\right\}.$ Then, for any $x\in B_r,$ we have

$$\begin{array}{cc}\hfill \left|\left(\mathcal{F}x\right)\left(t\right)\right|& \le \left|{\eta }_1\right|+\left|{\eta }_2\right|t+{\int }_0^t\frac{{(t-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|f(s,x\left(p^{\alpha -1}s\right))\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left[\left|{\alpha }_1\right|\left|{\alpha }_2\right|t+\left|{\beta }_1\right|\left|{\alpha }_2\right|\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left[{\alpha }_1{\beta }_2+\left|{\beta }_1\right|\left|{\beta }_2\right|\right]{\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left|g(p^{2}s,x\left(p^{2}s\right))\right|d_{p,q}s\hfill \\ \hfill & \le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left[u_1\left(ps\right)\mathrm{\Psi }\left(\parallel x\parallel \right)+u_2\left(ps\right)\right]d_{p,q}s\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left[u_1\left(ps\right)\mathrm{\Psi }\left(\parallel x\parallel \right)+u_2\left(ps\right)\right]d_{p,q}s\hfill \\ \hfill & +{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left[u_1\left(p^2\right)\mathrm{\Psi }\left(\parallel x\parallel \right)+u_2\left(p^{2}s\right)\right]d_{p,q}s\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+\mathrm{\Psi }\left(\parallel r\parallel \right)\left\{{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_1\left(ps\right)d_{p,q}s\right.\hfill \\ & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_1\left(ps\right)d_{p,q}s\hfill \\ \hfill & \left.+{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}{u}_1\left(p^{2}s\right)d_{p,q}s\right\}\hfill \\ \hfill & +\left\{{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_1\left(ps\right)d_{p,q}s\right.\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{p}_1\left(ps\right)d_{p,q}s\hfill \\ \hfill & \left.+{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}{u}_1\left(p^{2}s\right)d_{p,q}s\right\}.\hfill \end{array}$$

We have

$$∥\mathcal{F}x∥\le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+\mathrm{\Psi }\left(r\right){\omega }_1+{\omega }_2\le M.$$

Next, $\mathcal{F}$ maps bounded sets into equicontinuous sets of $\mathcal{C}\left([0,b],\mathbb{R}\right).$ Let $t_1,t_2\in [0,b]$ with $t_1<t_2$ be two points and $B_r$ be a bounded ball in $\mathcal{F}.$ For $x\in B_r$, we obtain

$$\begin{array}{cc}\hfill & \left|\left(\mathcal{F}x\right)\left(t_2\right)-\left(\mathcal{F}x\right)\left(t_1\right)\right|\hfill \\ \hfill & \le |{\eta }_2|(t_2-t_1)+\left|{\int }_0^{t_2}\frac{{(t_2-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\right.\hfill \\ \hfill & \left.-{\int }_0^{t_1}\frac{{(t_1-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\right|\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left|{\alpha }_1\right|\left|{\alpha }_2\right|(t_2-t_1){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left|g(ps,x(ps\left)\right)\right|d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left|{\alpha }_1{\beta }_2\right|(t_2-t_1){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left|g(p^{2}s,x\left(p^{2}s\right))\right|d_{p,q}s\hfill \\ \hfill & \le |{\eta }_2|(t_2-t_1)+\left|{\int }_0^{t_1}\left[\frac{{(t_2-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}-\frac{{(t_1-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\right]\left[u_1\left(ps\right)\mathrm{\Psi }\left(r\right)+u_2\left(ps\right)\right]d_{p,q}s\right.\hfill \\ \hfill & \left.+{\int }_{t_1}^{t_2}\frac{{(t_1-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left[u_1\left(ps\right)\mathrm{\Psi }\left(r\right)+u_2\left(ps\right)\right]d_{p,q}s\right|\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left|{\alpha }_1\right|\left|{\alpha }_2\right|(t_2-t_1){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}\left[u_1\left(ps\right)\mathrm{\Psi }\left(r\right)+u_2\left(ps\right)\right]d_{p,q}s\hfill \\ \hfill & +\frac{1}{\left|\Delta \right|}\left|{\alpha }_1{\beta }_2\right|(t_2-t_1){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}\left[u_1\left(p^{2}s\right)\mathrm{\Psi }\left(r\right)+u_2\left(p^{2}s\right)\right]d_{p,q}s.\hfill \end{array}$$

Obviously, the right-hand side of the above inequality tends to zero independently of $x\in B_r$ as $t_2\to t_1.$ Thus, it follows by the Arzelá–Ascoli theorem that ${\displaystyle \mathcal{F}:\mathcal{C}\left([0,b],\mathbb{R}\right)\to \mathcal{C}\left([0,b],\mathbb{R}\right)}$ is completely continuous. Now, the operator $\mathcal{F}$ satisfies all the conditions of Lemma 6; therefore, by its conclusion, either condition $\left(i\right)$ or condition $\left(ii\right)$ holds.

Now, we show that the conclusion $\left(ii\right)$ is not possible. Let

$$\mathcal{U}=\left\{x\in \mathcal{C}\left([0,b],\mathbb{R}\right):\parallel x\parallel \le M\right\}$$

with $\left|{\eta }_1\right|+\left|{\eta }_2\right|b+\mathrm{\Psi }\left(M\right){\omega }_1+{\omega }_2<M.$ Then, it can be shown that

$$\begin{array}{cc}\hfill \left|\mathcal{F}x\right|\le & \left|{\eta }_1\right|+\left|{\eta }_2\right|b+\mathrm{\Psi }\left(\parallel x\parallel \right)\left\{{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_1\left(ps\right)d_{p,q}s\right.\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_1\left(ps\right)d_{p,q}s\hfill \\ \hfill & \left.+{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}{u}_2\left(p^{2}s\right)d_{p,q}s\right\}\hfill \\ \hfill & +\left\{{\int }_0^b\frac{{(b-qs)}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_2\left(ps\right)d_{p,q}s\right.\hfill \\ \hfill & +{\sigma }_1\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -1)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha }{2}\right)}{\mathrm{\Gamma }}_{p,q}\left(\alpha \right)}{u}_2\left(ps\right)d_{p,q}s\hfill \\ \hfill & \left.+{\sigma }_2\left(b\right){\int }_0^b\frac{{(b-qs)}_{p,q}^{(\alpha -2)}}{p^{\left(\genfrac{}{}{0pt}{}{\alpha -1}{2}\right)}{\mathrm{\Gamma }}_{p,q}(\alpha -1)}{u}_2\left(p^{2}s\right)d_{p,q}s\right\}\hfill \\ \hfill & \le \left|{\eta }_1\right|+\left|{\eta }_2\right|b+\mathrm{\Psi }\left(M\right){\omega }_1+{\omega }_2\le M.\hfill \end{array}$$

Suppose there exists $x\in \partial \mathcal{U}$ and $\lambda \in (0,1)$, such that $x=\lambda \mathcal{F}x.$ Then, for such choices of x and $\lambda$, we have

$$M=\parallel x\parallel =\lambda ∥\mathcal{F}x∥<\left|{\eta }_1\right|+\left|{\eta }_2\right|b+\mathrm{\Psi }\left(\parallel x\parallel \right){\omega }_1+{\omega }_2<M.$$

Thus, it leads to a contradiction. Accordingly, by Lemma 6, $x\in \overline{\mathcal{U}}$ is fixed point of $\mathcal{F}$. Therefore, x a solution of the problem (7) and (8). This completes the proof. □

Remark 4.

If $u_1,u_2$ in $\left(A_4\right)$ are continuous, then ${\omega }_i\le A\parallel u_i\parallel ,i=1,2,$ where A is defined by (26).

4. Examples

Example 1.

Let $p=1/4,q=1/5,\alpha =1.5,{\alpha }_1=1,{\alpha }_2=1/2,{\beta }_1=1/4,{\beta }_2=3/2,{\gamma }_1=1/3,{\gamma }_2=1,b=1.$ Given a non-negative function, $g(t,x)=|sin\left(t\right)|+\frac{{\mathrm{\Gamma }}_{p,q}\left(1.5\right)}{4}x.$ Consider

$$\begin{array}{cc}\hfill & {}^c{D}_{p,q}^{\alpha }x\left(t\right)=|sin\left(p^{\alpha }t\right)|+\frac{{\mathrm{\Gamma }}_{p,q}\left(1.5\right)}{4}x\left(p^{\alpha }t\right),t\in [0,1],\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & x\left(0\right)+\frac{1}{4}{D}_{p,q}x\left(0\right)=\frac{1}{3},\frac{1}{2}x\left(1\right)+\frac{3}{2}{D}_{p,q}x\left(4\right)=1.\hfill \end{array}$$

(31)

Since

$$\left|g(t,x)-g(t,y)\right|=\left|\frac{{\mathrm{\Gamma }}_{p,q}\left(1.5\right)}{4}x-\frac{{\mathrm{\Gamma }}_{p,q}\left(1.5\right)}{4}y\right|=\frac{{\mathrm{\Gamma }}_{p,q}\left(1.5\right)}{4}|x-y|,$$

it follows that the condition $\left(A_1\right)$ holds. Let $L=\frac{{\mathrm{\Gamma }}_{p,q}\left(1.5\right)}{4}.$ From (25), by direct computation, we obtain $\Delta =-15/8,{\eta }_1=2/9,{\eta }_2=4/9$ and ${\sigma }_1\left(1\right)=1/3,{\sigma }_2\left(1\right)=1$. It easy to see that

$$k=(1+{\sigma }_1\left(1\right))\frac{1}{4{\left[\alpha \right]}_{p,q}}+\frac{{\sigma }_2\left(1\right)}{4}\approx 0.7187272601<1.$$

This satisfies Theorem 3. Accordingly, by Theorem 3, (30) and (31) has a unique solution.

Example 2.

Let $p=1/4,q=1/5,\alpha =3/2,{\alpha }_1=1,{\alpha }_2=1/2,{\beta }_1=1/4,{\beta }_2=3/2,{\gamma }_1=1/3,{\gamma }_2=1,$ and $b=1$. Given a non-negative function

$$g(t,x)=\frac{1}{10}\frac{e^{-x^2}|sin\left(\left|x\right|\right)|}{x^2+1}+\frac{e^{-x}\left(t^2+1\right)}{2+t^2}+\frac{1}{3}.$$

Consider

$$\begin{array}{cc}\hfill & {}^c{D}_{p,q}^{\alpha }x\left(t\right)=\frac{1}{10}\frac{e^{-{\left(x\left(p^{\alpha }t\right)\right)}^2}|sin\left(\left|x\left(p^{\alpha }t\right)\right|\right)|}{{\left(x\left(p^{\alpha }t\right)\right)}^2+1}+\frac{e^{-x\left(p^{\alpha }t\right)}\left({\left(p^{\alpha }t\right)}^2+1\right)}{2+{\left(p^{\alpha }t\right)}^2}+\frac{1}{3},t\in [0,1],\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & x\left(0\right)+\frac{1}{4}{D}_{p,q}x\left(0\right)=\frac{1}{3},\frac{1}{2}x\left(1\right)+\frac{3}{2}{D}_{p,q}x\left(4\right)=1.\hfill \end{array}$$

(33)

By applying Theorem 5, through simple calculation, we have $\Delta =-15/8,{\eta }_1=2/9,{\eta }_2=4/9$ and ${\sigma }_1\left(1\right)=1/3,{\sigma }_2\left(1\right)=1$.

Since

$$\begin{array}{cc}\hfill \left|g(t,x)\right|& =\left|\frac{1}{10}\frac{e^{-x^2}|sin\left(\left|x\right|\right)|}{x^2+1}+\frac{e^{-x}\left(t^2+1\right)}{2+t^2}+\frac{1}{3}\right|\hfill \\ \hfill & \le \frac{1}{10}\left|x\right|+1,\hfill \end{array}$$

$\left(A_3\right)$ holds. In fact, $u_1=1/10,u_2=1,\mathrm{\Psi }\left(M\right)=M.$ By computation, we obtain ${\omega }_1\approx 0.1380279163,{\omega }_2\approx 1.38027916.$ Furthermore, from the condition (28), it follows that $M>2.374724041.$ Thus, it satisfies Theorem 5. So, (32) and (33) has at least one solution.

5. Conclusions

In this paper, we investigated the local separated boundary value problem of a class of fractional $(p,q)$-difference equations involving the Caputo fractional derivative. By applying some well-known tools in fixed-point theory, such as Banach’s contraction mapping principle, Krasnoselskii’s fixed-point theorem, and the Leary–Schauder nonlinear alternative, we derive the existence and uniqueness of solutions for the problem. Moreover, some illustrating examples were also presented.