# Positive solution for *q*-fractional four-point boundary value problems with *p*-Laplacian operator

- Qiaozhen Yuan
^{1}and - Wengui Yang
^{1}Email author

**2014**:481

https://doi.org/10.1186/1029-242X-2014-481

© Yuan and Yang; licensee Springer. 2014

**Received: **16 October 2014

**Accepted: **20 November 2014

**Published: **3 December 2014

## Abstract

This paper investigates a class of four-point boundary value problems of fractional *q*-difference equations with *p*-Laplacian operator ${D}_{q}^{\beta}({\phi}_{p}({D}_{q}^{\alpha}u(t)))=f(t,u(t))$, $t\in (0,1)$, $u(0)=0$, $u(1)=au(\xi )$, ${D}_{q}^{\alpha}u(0)=0$, and ${D}_{q}^{\alpha}u(1)=b{D}_{q}^{\alpha}u(\eta )$, where ${D}_{q}^{\alpha}$ and ${D}_{q}^{\beta}$ are the fractional *q*-derivative of the Riemann-Liouville type, *p*-Laplacian operator is defined as ${\phi}_{p}(s)={|s|}^{p-2}s$, $p>1$, and $f(t,u)$ may be singular at $t=0,1$ or $u=0$. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, some sufficient conditions for the existence of at least one positive solution are established. Furthermore, two examples are presented to illustrate the main results.

**MSC:**39A13, 34B18, 34A08.

## Keywords

*q*-difference equationsfour-point boundary conditions

*p*-Laplacian operatorpositive solutionupper and lower solutions method

## 1 Introduction

Recently, fractional differential equations with *p*-Laplacian operator have gained its popularity and importance due to its distinguished applications in numerous diverse fields of science and engineering, such as viscoelasticity mechanics, non-Newtonian mechanics, electrochemistry, fluid mechanics, combustion theory, and material science. There have appeared some results for the existence of solutions or positive solutions of boundary value problems for fractional differential equations with *p*-Laplacian operator; see [1–7] and the references therein. For example, under different conditions, Wang *et al.* [8] and Ren and Chen [9] established the existence of positive solutions to four-point boundary value problems for nonlinear fractional differential equations with *p*-Laplacian operator by using the upper and lower solutions method and fixed point theorems, respectively.

Since Al-Salam [10] and Agarwal [11] proposed the fractional *q*-difference calculus, new developments in this theory of fractional *q*-difference calculus have been made due to the explosion in research within the fractional differential calculus setting. For example, some researcher obtained *q*-analogs of the integral and differential fractional operators properties such as the *q*-Laplace transform, the *q*-Taylor formula, the Mittag-Leffler function [12–15], and so on.

Recently, the theory of boundary value problems for nonlinear fractional *q*-difference equations has been addressed extensively by several researchers. There have been some papers dealing with the existence and multiplicity of solutions or positive solutions for boundary value problems involving nonlinear fractional *q*-difference equations by the use of some well-known fixed point theorems and the upper and lower solutions method. For some recent developments on the subject, see [16–25] and the references therein. El-Shahed and Al-Askar [26] studied the existence of multiple positive solutions to the nonlinear *q*-fractional boundary value problems by using the Guo-Krasnoselskii fixed point theorem in a cone. Under different conditions, Graef and Kong [27, 28] investigated the existence of positive solutions for the boundary value problem with fractional *q*-derivatives in terms of different ranges of *λ*, respectively. Zhao *et al.* [29] showed some existence results of positive solutions to nonlocal *q*-integral boundary value problem of nonlinear fractional *q*-derivatives equation using the generalized Banach contraction principle, the monotone iterative method, and the Krasnoselskii fixed point theorem. Ahmad *et al.* [30] considered the existence of solutions for the nonlinear fractional *q*-difference equation with nonlocal boundary conditions by applying some well-known tools of fixed point theory such as the Banach contraction principle, the Krasnoselskii fixed point theorem, and the Leray-Schauder nonlinear alternative. By applying the nonlinear alternative of Leray-Schauder type and Krasnoselskii fixed point theorems, the author [31] established sufficient conditions for the existence of positive solutions for nonlinear semipositone fractional *q*-difference system with coupled integral boundary conditions. Relying on the standard tools of fixed point theory, Agarwal *et al.* [32] and Ahmad *et al.* [33] discussed the existence and uniqueness of solutions for a new class of sequential *q*-fractional integrodifferential equations with *q*-antiperiodic boundary conditions and nonlocal four-point boundary conditions, respectively.

*q*-fractional boundary value problem involving the

*p*-Laplacian operator:

where ${a}_{0},{a}_{1}\ne 0$, $1<\alpha \in \mathbb{R}$, and $f\in C([0,1]\times \mathbb{R},\mathbb{R})$. Under some conditions, the authors obtained the existence and uniqueness of the solution for the above boundary value problem by using the Banach contraction mapping principle.

*p*-Laplacian operator:

where $0<\beta {\eta}^{\alpha -2}<1$, $0<q<1$. The authors proved the existence and uniqueness of a positive and nondecreasing solution for the boundary value problems by using a fixed point theorem in partially ordered sets.

*q*-difference boundary value problem with

*p*-Laplacian operator:

where $1<\alpha ,\beta \le 2$. The existence results for the above boundary value problem were obtained by using the upper and lower solutions method associated with the Schauder fixed point theorem.

*q*-difference boundary value problem with

*p*-Laplacian operator:

where ${D}_{q}^{\alpha}$, ${D}_{q}^{\beta}$ are the fractional *q*-derivative of the Riemann-Liouville type with $1<\alpha ,\beta \le 2$, $0\le a,b\le 1$, $0<\xi ,\eta <1$, ${\phi}_{p}(s)={|s|}^{p-2}s$, $p>1$, ${({\varphi}_{p})}^{-1}={\varphi}_{r}$, $(1/p)+(1/r)=1$, and $f(t,u):(0,1)\times (0,+\mathrm{\infty})\to [0,\mathrm{\infty})$ is continuous and may be singular at $t=0,1$ or $u=0$. By applying the upper and lower solutions method associated with the Schauder fixed point theorem, the existence results of at least one positive solution for the above fractional *q*-difference boundary value problem with *p*-Laplacian operator are established. This work improves essentially the results of [36]. At the end of this paper, we will give two examples to show the effectiveness of the main results.

## 2 Preliminaries

For the convenience of the reader, we present some necessary definitions and lemmas of fractional *q*-calculus theory to facilitate the analysis of problem (1.1). These details can be found in the recent literature; see [37] and references therein.

*q*-analog of the Pochhammer symbol (the

*q*-shifted factorial) is defined by

*q*-analog of the power ${(a-b)}^{n}$ with $n\in {\mathbb{N}}_{0}=\{0,1,2,\dots \}$ is

*q*-gamma function is defined by

and satisfies ${\mathrm{\Gamma}}_{q}(x+1)={[x]}_{q}{\mathrm{\Gamma}}_{q}(x)$.

*q*-derivative of a function

*f*is here defined by

*q*-derivatives of higher order by

*q*-integral of a function

*f*defined in the interval $[0,b]$ is given by

*f*is defined in the interval $[0,b]$, its integral from

*a*to

*b*is defined by

*i.e.*,

*f*is continuous at $x=0$, then

*i*):

Denote that if $\alpha >0$ and $a\le b\le t$, then ${(t-a)}^{(\alpha )}\ge {(t-b)}^{(\alpha )}$ [18].

**Definition 2.1** ([11])

*f*be function defined on $[0,1]$. The fractional

*q*-integral of the Riemann-Liouville type is ${I}_{q}^{0}f(x)=f(x)$ and

**Definition 2.2** ([13])

*q*-derivative of the Riemann-Liouville type of order $\alpha \ge 0$ is defined by ${D}_{q}^{0}f(x)=f(x)$ and

where *m* is the smallest integer greater than or equal to *α*.

**Lemma 2.3** ([14])

*Let*$\alpha ,\beta \ge 0$

*and*

*f*

*be a function defined on*$[0,1]$.

*Then the next formulas hold*:

- (1)
$({I}_{q}^{\beta}{I}_{q}^{\alpha}f)(x)={I}_{q}^{\alpha +\beta}f(x)$,

- (2)
$({D}_{q}^{\alpha}{I}_{q}^{\alpha}f)(x)=f(x)$.

**Lemma 2.4** ([18])

*Let*$\alpha >0$

*and*

*p*

*be a positive integer*.

*Then the following equality holds*:

**Lemma 2.5**

*Let*$y\in C[0,1]$, $1<\alpha \le 2$, $0<\xi <1$,

*and*$0\le a\le 1$.

*Then the unique solution of the following linear fractional*

*q*-

*difference boundary value problem*:

*is given by*

*where*

*Proof*By applying Lemma 2.4, we may reduce (2.1) to an equivalent integral equation

where $G(t,s)$ is defined in (2.3). The proof is completed. □

**Lemma 2.6**

*Let*$y\in C[0,1]$, $1<\alpha ,\beta \le 2$, $0<\xi ,\eta <1$,

*and*$0\le a,b\le 1$.

*Then the following fractional*

*q*-

*difference boundary value problem with*

*p*-

*Laplacian operator*:

*has unique solution given by*

*where*${b}_{1}={b}^{p-1}$, $G(t,s)$

*is defined by*(2.3)

*and*

*Proof*By applying Lemma 2.4, we may reduce (2.7) to an equivalent integral equation,

The proof is completed. □

**Lemma 2.7**

*Let*$1<\alpha ,\beta \le 2$, $0<\xi ,\eta <1$,

*and*$0\le a,b\le 1$.

*Then functions*$G(t,s)$

*and*$H(t,s)$

*defined by*(2.3)

*and*(2.9),

*respectively*,

*are continuous on*$[0,1]\times [0,1]$

*satisfying*

- (a)
$G(t,qs)\ge 0$, $H(t,qs)\ge 0$,

*for all*$t,s\in [0,1]$; - (b)
*for all*$t,s\in [0,1]$, ${\sigma}_{1}(qs){t}^{\alpha -1}\le G(t,qs)\le {\sigma}_{2}(qs){t}^{\alpha -1}$,*where*${\sigma}_{1}(s)=\frac{ag(\xi ,s)}{1-a{\xi}^{\alpha -1}},\phantom{\rule{2em}{0ex}}{\sigma}_{2}(s)=\frac{{(1-s)}^{(\alpha -1)}}{{\mathrm{\Gamma}}_{q}(\alpha )}+\frac{ag(\xi ,s)}{1-a{\xi}^{\alpha -1}}.$

*Proof* The proof is obvious, so we omit the proof. □

From Lemmas 2.5 and 2.7, it is easy to obtain the following lemma.

**Lemma 2.8** *Let* $u(t)\in C([0,1],\mathbb{R})$ *satisfies* $u(0)=0$, $u(1)={\phi}_{p}(b)u(\eta )$, *and* ${D}_{q}^{\beta}u(t)\ge 0$ *for any* $t\in (0,1)$, *then* $u(t)\le 0$, *for* $t\in [0,1]$.

Let $E=\{u:u,{\phi}_{p}({D}_{q}^{\alpha}u)\in {C}^{2}[0,1]\}$. Now we introduce the following definitions about the upper and lower solutions of the fractional *q*-difference boundary value problem (1.1).

**Definition 2.9**A function $\varphi (t)$ is called a lower solution of the fractional

*q*-difference boundary value problem (1.1), if $\varphi (t)\in E$ and $\varphi (t)$ satisfies

**Definition 2.10**A function $\psi (t)$ is called an upper solution of the fractional

*q*-difference boundary value problem (1.1), if $\psi (t)\in E$ and $\psi (t)$ satisfies

## 3 Main results

For the sake of simplicity, we make the following assumptions throughout this paper.

(H_{1}) $f(t,u)\in C[(0,1)\times (0,+\mathrm{\infty}),[0,+\mathrm{\infty})]$ and $f(t,u)$ is decreasing in *u*.

_{2}) Set $e(t)={t}^{\alpha -1}$. For any constant $\rho >0$, $f(t,\rho )\not\equiv 0$, and

*P*={$u\in C[0,1]$: there exist two positive constants $0<{l}_{u}<{L}_{u}$ such that ${l}_{u}e(t)\le u(t)\le {L}_{u}e(t)$, $t\in [0,1]$}. Obviously, $e(t)\in P$. Therefore,

*P*is not empty. For any $u\in P$, define an operator

*T*by

**Theorem 3.1** *Suppose that conditions* (H_{1})-(H_{2}) *are satisfied*, *then the boundary value problem* (1.1) *has at least one positive solution* *u*, *and there exist two positive constants* $0<{\lambda}_{1}<1<{\lambda}_{2}$ *such that* ${\lambda}_{1}e(t)\le u(t)\le {\lambda}_{2}e(t)$, $t\in [0,1]$.

*Proof* We will divide our proof into four steps.

Step 1. We show that *T* is well defined on *P* and $T(P)\subset P$, and *T* is decreasing in *u*.

*P*, there exist two positive constants $0<{l}_{u}<1<{L}_{u}$ such that ${l}_{u}e(t)\le u(t)\le {L}_{u}e(t)$ for any $t\in [0,1]$. It follows from Lemma 2.7 and conditions (H

_{1})-(H

_{2}) that

*T*is well defined and $T(P)\subset P$. It follows from (H

_{1}) that the operator

*T*is decreasing in

*u*. By direct computations, we can state that

*q*-difference boundary value problem (1.1). Let

We will prove that the functions $\varphi (t)=Tn(t)$, $\psi (t)=Tm(t)$ are a couple of lower and upper solutions of the fractional *q*-difference boundary value problem (1.1), respectively.

_{1}), we know that

*T*is nonincreasing relative to

*u*. Thus it follows from (3.4) and (3.5) that

that is, $\varphi (t)$ and $\psi (t)$ are a couple of lower and upper solutions of the fractional *q*-difference boundary value problem (1.1), respectively.

*q*-difference boundary value problem

_{1}) and (3.9) that $g(t,u):[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous. To see this, we consider the operator $A:C[0,1]\to C[0,1]$ defined as follows:

*A*is a solution of the boundary value problem (3.8). Noting that $\varphi (t)\in P$, there exists a positive constant $0<{l}_{\varphi}<1$ such that $\varphi (t)\ge {l}_{\varphi}e(t)$, $t\in [0,1]$. It follows from Lemma 2.7, (3.9), and (H

_{2}) that

which implies that the operator *A* is uniformly bounded.

that is to say, *A* is equicontinuous. Thus, from the Arzela-Ascoli theorem, we know that *A* is a compact operator, by using the Schauder fixed point theorem, the operator *A* has a fixed point *u* such that $u=Au$; *i.e.*, the fractional *q*-difference boundary value problem (3.8) has a positive solution.

*u*is fixed point of

*A*and (3.7), we get

*g*, one verifies that

*ψ*is an upper solution to (1.1), we obviously have

Since ${\phi}_{p}$ is monotone increasing, we obtain ${D}_{q}^{\alpha}\psi (t)\le {D}_{q}^{\alpha}u(t)$, *i.e.*, ${D}_{q}^{\alpha}(\psi -u)(t)\le 0$. Combining Lemma 2.8, we have $(\psi -u)(t)\ge 0$. Therefore, $\psi (t)\ge u(t)$, $t\in [0,1]$, a contradiction to the assumption that $u(t)>\psi (t)$. Hence, $u(t)>\psi (t)$ is impossible.

*g*, one verifies that

*ϕ*is an upper solution to (1.1), we obviously have

Since ${\phi}_{p}$ is monotone increasing, we obtain ${D}_{q}^{\alpha}u(t)\le {D}_{q}^{\alpha}\varphi (t)$, *i.e.*, ${D}_{q}^{\alpha}(u-\varphi )(t)\le 0$. Combining Lemma 2.5, we have $(u-\varphi )(t)\ge 0$. Therefore, $u(t)\ge \varphi (t)$, $t\in [0,1]$, a contradiction to the assumption that $u(t)<\varphi (t)$. Hence, $u(t)<\varphi (t)$ is impossible.

Consequently, we have $\varphi (t)\le u(t)\le \psi (t)$, $t\in [0,1]$, that is, $u(t)$ is a positive solution of the boundary value problem (1.1). Furthermore, $\varphi (t),\psi (t)\in P$ implies that there exist two positive constants $0<{\lambda}_{1}<1<{\lambda}_{2}$ such that ${\lambda}_{1}e(t)\le u(t)\le {\lambda}_{2}e(t)$, $t\in [0,1]$. Thus, we have finished the proof of Theorem 3.1. □

**Theorem 3.2** *If* $f(t,u)\in C([0,1]\times [0,+\mathrm{\infty}),[0,+\mathrm{\infty}))$ *is decreasing in* *u* *and* $f(t,\rho )\not\equiv 0$ *for any* $\rho >0$, *then the boundary value problem* (1.1) *has at least one positive solution* *u*, *and there exist two positive constants* $0<{\lambda}_{1}<1<{\lambda}_{2}$ *such that* ${\lambda}_{1}e(t)\le u(t)\le {\lambda}_{2}e(t)$, $t\in [0,1]$.

*Proof* The proof is similar to Theorem 3.1, we omit it here. □

## 4 Two examples

**Example 4.1**Consider the

*p*-Laplacian fractional

*q*-difference boundary value problem

_{1}) holds. For any $\rho >0$, $f(t,\rho )\not\equiv 0$, we have

which implies that (H_{2}) holds. Theorem 3.1 implies that the boundary value problem (4.1) has at least one positive solution.

**Example 4.2**Consider the

*p*-Laplacian fractional

*q*-difference boundary value problem

It is not difficult to check that $f(t,u):[0,1]\times [0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is continuous and decreasing in *u* and $f(t,\rho )\not\equiv 0$ for any $\rho >0$. Theorem 3.2 implies that the boundary value problem (4.2) has at least one positive solution.

## Declarations

### Acknowledgements

The authors sincerely thank the editor and reviewers for their valuable suggestions and useful comments, which have led to the present improved version of the original manuscript.

## Authors’ Affiliations

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