Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 376938, 7 pages

http://dx.doi.org/10.1155/2013/376938

Research Article

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

Shang-lin Yao,1,2 Guo-hui Wang,² Zhi-ping Li,¹ and Li-jun Yu²

¹School of Energy Resources, China University of Geosciences (Beijing), Beijing 100083, China

²Research Institute of Petroleum Exploration and Development, PetroChina, Beijing 100083, China

Received 29 October 2012; Accepted 5 November 2012

Academic Editor: Hua Su

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.

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 [4–14] 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 [15–19], 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.

References

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, NY, USA, 1993.

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, AcademicPress, New York, NY, USA, 1999.

A. Babakhani and V. Daftardar-Gejji, “Existence of positive solutions of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 434–442, 2003.

D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.

A. M. A. El-Sayed, “Nonlinear functional-differential equations of arbitrary orders,” Nonlinear Analysis. Theory, Methods & Applications, vol. 33, no. 2, pp. 181–186, 1998.

V. Lakshmikantham, “Theory of fractional functional differential equations,” Nonlinear Analysis, vol. 69, pp. 3337–3343, 2008.

S. Q. Zhang, “Existence of positive solution for some class of nonlinear fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 278, no. 1, pp. 136–148, 2003.

Y. Zhou, “Existence and uniqueness of solutions for a system of fractional differential equations,” Fractional Calculus & Applied Analysis, vol. 12, no. 2, pp. 195–204, 2009.

Y. Zhou, “Existence and uniqueness of fractional functional differential equations with unbounded delay,” International Journal of Dynamical Systems and Differential Equations, vol. 1, no. 4, pp. 239–244, 2008.

Z. B. Bai and H. S. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

N. Kosmatov, “A singular boundary value problem for nonlinear differential equations of fractionalorder,” Journal of Applied Mathematics and Computing, vol. 29, no. 1-2, pp. 125–135, 2008.

E. R. Kaufmann and E. Mboumi, “Positive solutions of a boundary value problem for a nonlinearfractional differential equation,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1–11, 2008.

C. Z. Bai, “Triple positive solutions for a boundary value problem of nonlinear fractional differentialequation,” Electronic Journal of Qualitative Theory of Differential Equations, no. 24, pp. 1–10, 2008.

C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010.

C. Yang and J. Yan, “Positive solutions for third-order Sturm-Liouville boundary value problems with -Laplacian,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2059–2066, 2010.

J. Wang, H. Xiang, and Z. Liu, “Positive solutions for three-point boundary value problems of nonlinear fractional differential equations with -Laplacian,” Far East Journal of Applied Mathematics, vol. 37, no. 1, pp. 33–47, 2009.

J. H. Wang and Z. Xiang, “Upper and lower solutions method for a class of singular fractionalboundary value problems with p-Laplacian operator,” Abstract and Applied Analysis, vol. 2010, Article ID 971824, 12 pages, 2010.

G. Chai, “Positive solutions for boundary value problem of fractional differential equation with p-Laplacian operator,” Boundary Value Problems, vol. 2012, article 18, 2012.