Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2013 / Article
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Nonlinear Functional Difference Equations with Applications

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Research Article | Open Access

Volume 2013 |Article ID 376938 | 7 pages | https://doi.org/10.1155/2013/376938

Positive Solutions for Three-Point Boundary Value Problem of Fractional Differential Equation with -Laplacian Operator

Academic Editor: Hua Su
Received29 Oct 2012
Accepted05 Nov 2012
Published26 Feb 2013

Abstract

We investigate the existence of multiple positive solutions for three-point boundary value problem of fractional differential equation with -Laplacian operator , where are the standard Riemann-Liouville derivatives with , and the constant is a positive number satisfying ; -Laplacian operator is defined as . By applying monotone iterative technique, some sufficient conditions for the existence of multiple positive solutions are established; moreover iterative schemes for approximating these solutions are also obtained, which start off a known simple linear function. In the end, an example is worked out to illustrate our main results.

1. Introduction

In this paper, we study the existence of multiple positive solutions for the following three-point boundary value problem of fractional differential equation with -Laplacian operator where are the standard Riemann-Liouville derivatives with and the constant is a positive number satisfying ; -Laplacian operator is defined as .

Fractional differential equations have gained considerable importance due to their application in various sciences, such as physics, mechanics, chemistry, and engineering. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], and the papers [414] and the references therein.

In [15], Li et al. were concerned with the nonlinear differential equation of fractional order subject to the boundary conditions By using some fixed point theorems, the existence and multiplicity results of positive solutions were established.

On the other hand, the differential equations with -Laplacian have also been widely studied owing to the fact that -Laplacian boundary value problems have important application in theory and application of mathematics and physics. For example, in [16], by using the fixed point index, Yang and Yan investigated the existence of positive solution for the third-order Sturm-Liouville boundary value problems with -Laplacian operator: However, there are few articles dealing with the existence of solutions to boundary value problems for fractional differential equation with -Laplacian operator. In [17], the authors investigated the nonlinear nonlocal problem where . By using Krasnoselskii’s fixed point theorem and Leggett-Williams theorem, some sufficient conditions for the existence of positive solutions to the above BVP are obtained. In [18], by using upper and lower solutions method, under suitable monotone conditions, Wang et al. investigated the existence of positive solutions to the following nonlocal problem: where . Recently, Chai [19] investigated the two-point boundary value problem of fractional differential equation with -Laplacian operator: By means of the fixed point theorem on cones, some existence and multiplicity results of positive solutions are obtained.

Motivated by the above mentioned works, in this paper, we consider the multiplicity results of positive solutions for the three point boundary value problem of fractional differential equation with -Laplacian operator. Difference to [1519], by using monotone iterative technique, we not only establish the existence of multiple positive solutions but also obtain the iterative sequences of these positive solutions.

2. Preliminaries and Lemmas

In this section, we introduce some preliminary facts which are used throughout this paper.

Definition 1 (see [1–3]). The Riemann-Liouville fractional integral of order of a function is given by provided that the right-hand side is pointwise defined on .

Definition 2 (see [1–3]). The Riemann-Liouville fractional derivative of order of a function is given by where , denotes the integer part of number , provided that the right-hand side is pointwise defined on .

Proposition 3 (see [1–3]). (1) If , then
(2) If , then

Proposition 4 (see [1–3]). Let , and is integrable, then where , is the smallest integer greater than or equal to .

Definition 5. A function is called a nonnegative solution of BVP (1), if on and satisfies (1). Moreover, if , then is said to be a positive solution of BVP (1).
For forthcoming analysis, we first consider the following fractional differential equation:

Lemma 6 (see [15]). If and , then the boundary value problem (13) has the unique solution where where .

Lemma 7 (see [15]). The Green function in Lemma 6 has the following properties:(i) is continuous on ,(ii) for any .And if , the Green function also satisfies(iii) for any ,(iv)there exists a positive function such that where

Let be the set of positive integers, let be the set of real numbers, and let be the set of nonnegative real numbers. Let . Denote by the Banach space of all continuous functions from into with the norm

Define the cone in as Let satisfy the relation , where is given by (1).

To study BVP (1), we first consider the associated linear BVP: for and .

Let . By Proposition 4, the solution of initial value problem is given by . From the relations , it follows that , and so Noting that , from (22), we know that the solution of (20) satisfies By Lemma 6, the solution of (23) can be written as Since , , we have , , and so which implies that the solution of (23) is given by For the convenience, we make the following assumptions.(H1) is continuous and nondecreasing, and there exists a constant such that, for any , (H2) is nonnegative on (0, 1), and

Remark 8. By (27), for any , clearly, Now, for any , define one operator as follows: Then by (20) and (23), the BVP (1) is equivalent to the fixed point problems of the operators .

Lemma 9. Assume that (H1) and (H2) hold. Then are continuous, compact, and nondecreasing.

Proof. In fact, for any , On the other hand, by Lemma 7, So .
Next, supposing is a bounded set, then for any , there exists a constant such that . Thus for any , we have which implies is bounded. On the other hand, according to the Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we easily see is completely continuous. In the end, noticing the monotonicity of on and the definition of , we also have that the operator is nondecreasing.

3. Main Results

Define two constants

Theorem 10. Suppose conditions (H1) and (H2) hold. If there exists a positive constant such that where and are defined by (34), then the BVP (1) has the maximal and minimal solutions and , which are positive, and there exist two positive constants such that Moreover for initial values , define the iterative sequences by Then for uniformly, respectively.

Proof. Let ; we firstly prove . In fact, for any , we have By the assumption , we have It follows from Lemma 9 that is completely continuous operator; thus by (35) and (40), we have which implies that .
Let ; then . Leting , we have . Denote It follows from that . Since is compact, we obtain that is a sequentially compact set.
Since , we have By the induction, we get Consequently, there exists such that . Letting , from the continuity of and , we obtain , which implies that is a nonnegative solution of boundary value problem (1). Since , we know the zero function is not the solution of boundary value problem (1), thus ; by , we have that is, is a positive solution of boundary value problem (1).
On the other hand, let ; then . Leting , from the previous expressions, we have . Thus let us denote It follows from that Since is compact by Lemma 9, we can assert that is a sequentially compact set.
Now, since , we have It follows from Lemma 9 that is nondecreasing, so By the induction, we have Consequently, there exists such that . Letting , from the continuity of and , we obtain , which implies that is a nonnegative solution of boundary value problem (1).
Next, noting , thus it follows from monotonicity of that ; by the induction, we have , which implies that . Thus by (45) we have This means that is also a positive solution of boundary value problem (1).
In the end, let be any fixed point of in , then and then By induction, we have Taking the limit, we have This implies that and are maximal and minimal solutions of the BVP (1).Let , then we have The proof is completed.

Remark 11. If , then holds naturally, and in this case we take Thus we have the following Corollary 12.

Corollary 12. Suppose condition (H1) holds and . If there exists a positive constant such that where is defined by (34), then the BVP (1) has the maximal and minimal solutions and , which are positive, and there exist two positive constants such that Moreover for initial values , define the iterative sequences by Then for uniformly, respectively.

Corollary 13. Suppose conditions hold. If Then there exists a constant such that the BVP (1) has the maximal and minimal solutions and , which are positive, and there exist two positive constants such that Moreover for initial values , define the iterative sequences by Then for uniformly, respectively.

Proof . It follows from that which implies that there exists large enough such that Notice that ; (67) is equivalent to By Theorem 10, the conclusion of Corollary 13 holds.

Remark 14. In Corollary 13, we obtain that the BVP (1) has the maximal and minimal solutions and only by comparing to . But note that and are irrelative, so (62) is easy to be satisfied; this implies that Corollary 13 is very interesting.

Example 15. Consider the following boundary value problem:
Let , and , then For any and , we have Taking , then which implies that (62) holds. By Corollary 13, we know the BVP (69) has at least two positive solutions.

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Copyright © 2013 Shang-lin Yao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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