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# Existence and multiplicity of positive solutions for nonhomogeneous boundary value problems with fractional q-derivatives

Yulin Zhao1*, Haibo Chen2 and Qiming Zhang1

Author Affiliations

1 School of Science, Hunan University of Technology, Zhuzhou, 412007, China

2 Department of Mathematics, Central South University, Changsha, 410075, China

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Boundary Value Problems 2013, 2013:103  doi:10.1186/1687-2770-2013-103

 Received: 17 January 2013 Accepted: 12 April 2013 Published: 25 April 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we study a class of fractional q-difference equations with nonhomogeneous boundary conditions. By applying the classical tools from functional analysis, sufficient conditions for the existence of single and multiple positive solutions to the boundary value problem are obtained in term of the explicit intervals for the nonhomogeneous term. In addition, some examples to illustrate our results are given.

MSC: 34A08, 34B18, 39A13.

##### Keywords:
fractional q-difference equation; nonhomogeneous boundary value problem; positive solution; multiplicity

### 1 Introduction

Fractional differential equations have attracted considerable interest because of its demonstrated applications in various fields of science and engineering including fluid flow, rheology, diffusive transport akin to diffusion, electrical networks, probability [1,2]. Many researchers have studied the existence of solutions (or positive solutions) to fractional boundary value problems; for example, see [3-10] and the references therein.

The early work on q-difference calculus or quantum calculus dates back to Jackson’s papers [11], basic definitions and properties of quantum calculus can be found in the book [12]. For some recent existence results on q-difference equations, we refer to [13-15] and the references therein.

The fractional q-difference calculus had its origin in the works by Al-Salam [16] and Agarwal [17]. More recently, there seems to be new interest in the study of this subject and many new developments were made in this theory of fractional q-difference calculus [18-22]. Specifically, fractional q-difference equations have attracted the attentions of several researchers. Some recent work on the existence theory of fractional q-difference equations can be found in [20,23-31]. However, the study of boundary value problems for nonlinear fractional q-difference equations is still in the initial stage and many aspects of this topic need to be explored.

By using a fixed-point theorem in a cone, M. El-Shahed and F. Al-Askar [25] were concerned with the existence of positive solutions to nonlinear q-difference equation:

where and is the fractional q-derivatives of the Caputo type.

In [27], Graef and Kong investigated the boundary value problem with fractional q-derivatives

where is a parameter, and the uniqueness, existence and nonexistence of positive solutions are considered in terms of different ranges of λ.

By applying the Banach contraction principle, Krasnoselskii’s fixed-point theorem, and the Leray-Schauder nonlinear alternative, Ahmad, Ntouyas and Purnaras [29] studied the existence of solution for the following nonlinear fractional q-difference equation with nonlocal boundary conditions:

where is the fractional q-derivative of the Caputo type, and .

Recently, in [32], the authors investigate the following singular semipositone integral boundary value problem for fractional q-derivatives equation:

where , is the q-derivative of Riemann-Liouville type of order α, is continuous and semipositone, and may be singular at .

Since finding positive solutions of boundary value problems is interest in various fields of sciences, fractional q-calculus equations has tremendous potential for applications. In this paper, we will deal with the following nonhomogeneous boundary value problem with fractional q-derivatives:

(1.1)

where , , , , and λ is a parameter, is the q-derivative of Riemann-Liouville type of order α, is continuous. In the present work, we gave the corresponding Green’s function of the boundary value problem (1.1) and its properties. By using the generalized Banach contraction principle and Krasnoselskii’s fixed-point theorem, the uniqueness, existence, and multiplicity of positive solution to the BVP (1.1) are obtained in term of the explicit intervals for the nonhomogeneous term. Our results are different from those of [25,27].

### 2 Preliminaries on q-calculus and lemmas

For the convenience of the reader, below we cite some definitions and fundamental results on q-calculus as well as the fractional q-calculus. The presentation here can be found in, for example, [12,18,20,22].

Let and define

The q-analogue of the power function with is

More generally, if , then

(2.1)

Clearly, if , then . The q-gamma function is defined by

and satisfies .

The q-derivative of a function f is defined by

and q-derivatives of higher order by

The q-integral of a function f defined in the interval is given by

If and f is defined in the interval , then its integral from a to b is defined by

Similar to that for derivatives, an operator is given by

The fundamental theorem of calculus applies to these operators and , i.e.,

and if f is continuous at , then

(2.2)

The following formulas will be used later, namely, the integration by parts formula:

and

(2.3)

(2.4)

(2.5)

(2.6)

where denotes the derivative with respect to the variable t.

Definition 2.1 Let and f be a function defined on . The fractional q-integral of Riemann-Liouville type is and

Definition 2.2 The fractional q-derivative of the Riemann-Liouville type of order is defined by and

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

Lemma 2.3 ([20])

Assume thatand, then.

Lemma 2.4Letandfbe a function defined on. Then the following formulas hold:

(1) ,

(2) .

Lemma 2.5 ([20])

Letandnbe a positive integer. Then the following equality holds:

Lemma 2.6 ([22])

Let, , the following is valid:

Particularly, for , , using q-integration by parts, we have

Obviously, we have , and

In order to define the solution for the problem (1.1), we need the following lemmas.

Lemma 2.7For given, the unique solution of the boundary value problem

(2.7)

subject to the boundary conditions

(2.8)

is given by

(2.9)

where

(2.10)

Proof Since , we put . In view of Definition 2.1 and Lemma 2.4, we see that

Then it follows from Lemma 2.5 that the solution of (2.7) and (2.8) is given by

(2.11)

for some constants . From , we have .

Differentiating both sides of (2.11) and with the help of (2.4) and (2.6), we obtain,

and

Then by the boundary condition , we get . Using the boundary condition , we get

Hence, we have

This completes the proof of the lemma. □

Lemma 2.8The functiondefined by (2.10) satisfies the following conditions:

(i) , andfor all.

(ii) for all.

Proof We start by defining the following two functions:

Obviously, . Now , and for

Therefore, .

Moreover, for , it follows from (2.4) and Lemma 2.3 that

which implies that is an increasing function with respect to t. It is clear that is increasing in t. Therefore, is an increasing function of t for all , and so .

When , then

Finally, we prove part (ii). When , we have

If , then we have

which implies that part (ii) holds. This completes the proof of the lemma. □

Remark 2.9 If we let , then

According to [20], we may take , .

### 3 The main results

Let be a Banach space endowed with the norm . Define the cone by .

Define the operator as follows:

(3.1)

Theorem 3.1Assume thatis continuous and there exists a nonnegative functionsuch that

(3.2)

Then the BVP (1.1) has a unique positive solution for any, provided

(3.3)

If, in addition, on, then the conclusion is true for.

Proof We will show that under the assumptions (3.2) and (3.3), is a contraction operator for m sufficiently large.

By the definition of , for , we have

where .

Consequently,

where .

By introduction, we get

From the condition (3.3), we have

for m sufficiently large. So, we get

Hence, it follows from the generalized Banach contraction principle that the BVP (1.1) has a unique positive solution for any . If , then the condition on and Lemma 2.8 imply that in . This completes the proof of the theorem. □

Remark 3.2 When is a constant, the condition (3.2) reduces to a Lipschitz condition.

Our next existence results is based on Krasnoselskii’s fixed-point theorem [33].

Lemma 3.3 (Krasnoselskii’s)

LetEbe a Banach space, and letbe a cone. Assume, are open subsets ofEwith, and letbe a completely continuous operator such that, either

(1) , and, , or

(2) , and, .

ThenThas at least one fixed point in.

Define a cone by

Obviously, K is a cone of nonnegative functions in X.

Lemma 3.4The operatoris completely continuous.

Proof Firstly, we prove that . By (2.9) and Lemma 2.8, we have

On the other hand,

Hence, we have .

Next, we show that T is uniformly bounded. For fixed , consider a bounded subset of K defined by , and let . Then for , we get

which implies that is bounded.

Finally, we show that T is equicontinuous. For all , setting

where

For any , we can prove that if and , then

In fact, we have

If , then

If , , then

By means of Arzela-Ascoli theorem, is completely continuous.

For the sake of convenience, we introduce the following weight functions:

and set

□

Theorem 3.5Suppose that there exists two positive numberssuch that one of the following conditions is satisfied

() , ;

() , .

Then the BVP (1.1) has at least one positive solution, such thatfor. If, in addition, on, then the conclusion is true for.

Proof Because the proofs are similar, we prove only the case (). Denote . Then for any , we get , , , and , . By assumption (), we have

In view of (2.9) and Lemma 2.8, we have

On the other hand, define . For any and , we have and , . Thus,

It follows

By Lemma 3.3, the operator T has at least one fixed point , and . Since , , then, the solution is positive for . As in the proof of Theorem 3.1, is a positive solution for . This completes the proof of the theorem. □

Theorem 3.6Suppose that there exists three positive numberssuch that one of the following conditions is satisfied

() , , ;

() , , .

Then the BVP (1.1) has at least two positive solutionssuch thatfor. If, in addition, on, then the conclusion is true for.

Proof We prove only the case (). Since is continuous and , there exist two positive numbers , such that and , . Thus, it follows from the assumption () that

From Theorem 3.5, the operator T has two fixed point , with . Therefore, the BVP (1.1) has at least two positive solutions for . As in the proof of Theorem 3.1, , are two positive solutions for . This completes the proof of the theorem. □

Denote the integer part of m by . Generally, we have the following theorem.

Theorem 3.7Suppose that there existspositive numberssuch that one of the following conditions is satisfied:

() , , ;

() , , .

Then the BVP (1.1) has at leastmpositive solutions, , such thatfor. If, in addition, on, then the conclusion is true for.

### 4 Examples

Example 4.1 The fractional q-difference boundary value problem

(4.1)

has a unique positive solution for any .

Proof In this case, , , , , . Let

and . It is easy to prove that

A simple computation showed

and

which implies that

Obviously, for any , we have

Thus, Theorem 3.1 implies that the boundary value problem (4.1) has a unique positive solution for any . □

Example 4.2 Consider the following fractional boundary value problem:

(4.2)

where , , , . Choosing , then .

By calculation, we get . By Lemma 2.6, Lemma 2.8 and with the aid of a computer, we obtain that

and

Let . Take , , then , and satisfies

(i) ;

(ii) .

So, by Theorem 3.5, the problem (4.2) has one positive solution such that for .

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

### Acknowledgements

Dedicated to Professor Hari M Srivastava.

The authors are highly grateful for the referees’ careful reading and comments on this paper. The research is supported by the National Natural Science Foundation of China (Grant No. 11271372, 11201138); it is also supported by the Hunan Provincial Natural Science Foundation of China (Grant No. 13JJ3106, 12JJ2004), and the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12B034).

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