International Journal of Differential Equations

Volume 2014 (2014), Article ID 485647, 10 pages

http://dx.doi.org/10.1155/2014/485647

Research Article

Multiple Positive Solutions for a Coupled System of -Laplacian Fractional Order Two-Point Boundary Value Problems

K. R. Prasad¹ and B. M. B. Krushna²

¹Department of Applied Mathematics, Andhra University, Visakhapatnam 530 003, India

²Department of Mathematics, MVGR College of Engineering, Vizianagaram 535 005, India

Received 24 February 2014; Accepted 21 April 2014; Published 7 May 2014

Academic Editor: Bashir Ahmad

Copyright © 2014 K. R. Prasad and B. M. B. Krushna. 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.

This paper establishes the existence of at least three positive solutions for a coupled system of -Laplacian fractional order two-point boundary value problems, , , , , , , , , , , by applying five functionals fixed point theorem.

1. Introduction

The theory of differential equations offers a broad mathematical basis to understand the problems of modern society which are complex and interdisciplinary by nature. Fractional order differential equations have gained importance due to their applications to almost all areas of science, engineering, and technology. Among all the theories, the most applicable operator is the classical -Laplacian, given by , . These types of problems have a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design.

The positive solutions of boundary value problems associated with ordinary differential equations were studied by many authors [1–3] and extended to -Laplacian boundary value problems [4–6]. Later, these results are further extended to fractional order boundary value problems [7–15] by applying various fixed point theorems on cones. Recently, researchers are concentrating on the theory of fractional order boundary value problems associated with -Laplacian operator.

In 2012, Chai [16] investigated the existence and multiplicity of positive solutions for a class of boundary value problem of fractional differential equation with -Laplacian operator,

by means of the fixed point theorem on cones.

This paper is concerned with the existence of positive solutions for a coupled system of -Laplacian fractional order boundary value problems:

where ,  ,  ,  , are positive real numbers, ,  ,   are continuous functions, and ,  ,  ,  for are the standard Riemann-Liouville fractional order derivatives.

The rest of the paper is organized as follows. In Section 2, the Green functions for the homogeneous BVPs corresponding to (2), (4) are constructed and the bounds for the Green functions are estimated. In Section 3, sufficient conditions for the existence of at least three positive solutions for a coupled system of -Laplacian fractional order BVP (2)–(5) are established, by using five functionals fixed point theorem. In Section 4, as an application, the results are demonstrated with an example.

2. Green Functions and Bounds

In this section, the Green functions for the homogeneous BVPs are constructed and the bounds for the Green functions are estimated, which are essential to establish the main results.

Let be Green’s function for the homogeneous BVP:

Lemma 1. Let . If , then the fractional order BVP

with (7) has a unique solution

where

Proof. Let be the solution of fractional order BVP (8), (7). Then

and hence

Using the boundary conditions (7), , , and are determined as

Hence, the unique solution of (8), (7) is

Lemma 2. Let and , . Then the fractional order BVP

with (4) has a unique solution

where

Proof. An equivalent integral equation for (16) is given by

Using the conditions ,  ,  , and are determined as and . Then,

Therefore,

Consequently,

Hence, is the solution of fractional order BVP (16) and (4).

Lemma 3. Assume that . Then Green’s function satisfies the following inequalities:

(i), for all ,(ii), for all ,(iii), for all ,

where .

Proof. Green’s function is given in (10). For ,

For ,

Hence, the inequality (i) is proved. For ,

Therefore, is increasing with respect to , which implies that . Now, for ,

Therefore, is increasing with respect to , which implies that . Hence, the inequality (ii) is proved. Now, the inequality (iii) can be established.

Let and . Then

Let and . Then

Hence the inequality (iii) is proved.

Lemma 4. For , Green’s function satisfies the following inequalities:

(i), (ii).

Proof. Green’s function is given in (18). Clearly, it is observed that, for , .

For ,

Hence, the inequality (i) is proved. Now we establish the inequality (ii), for ,

Therefore, is increasing with respect to , for , which implies that . Similarly, it can be proved that for . Hence the inequality (ii) is proved.

Lemma 5. Green’s function satisfies the following inequality: there exists a positive function such that

Proof. Since is monotonic function, for all , we have

From (i) of Lemma 4, , for . For ,   is increasing with respect to for and decreasing with respect to for . For ,   is decreasing with respect to for and increasing with respect to for . If we define

Then,

where

and satisfy the equation . In particular, if ; as ; and as . Hence the inequality in (31) holds.

In a similar manner, the results of the Green functions and for the homogeneous BVPs corresponding to the fractional order BVP (3) and (5) are obtained.

Remark 6. Consider the following.

and , for all , where .

Remark 7. Consider the following.

and , for all , where .

3. Existence of Multiple Positive Solutions

In this section, the existence of at least three positive solutions for a coupled system of -Laplacian fractional order BVP (2)–(5) is established by using five functionals fixed point theorem.

Let , , be nonnegative continuous convex functionals on and let , be nonnegative continuous concave functionals on ; then for nonnegative numbers , , , , and , convex sets are defined:

In obtaining multiple positive solutions of the -Laplacian fractional order BVP (2)-(5), the following so-called five functionals fixed point theorem is fundamental.

Theorem 8 (see [17]). Let be a cone in the real Banach space . Suppose that and are nonnegative continuous concave functionals on and , , are nonnegative continuous convex functionals on , such that, for some positive numbers and , and , for all . Suppose further that is completely continuous and there exist constants , , , and with such that each of the following is satisfied:

(B1) and for ,(B2) and for ,(B3) provided that with ,(B4) provided that with .

Then, has at least three fixed points such that and with .

Consider the Banach space , where equipped with the norm , for and the norm, is defined as

Define a cone by

Define the nonnegative continuous concave functionals and the nonnegative continuous convex functionals , , on by

where . For any ,

Let

Theorem 9. Suppose that there exist such that , for satisfies the following conditions:

(A1),   and ,(A2) ,   and ,(A3) ,   and .

Then, the fractional order BVP (2)–(5) has at least three positive solutions, ,  , and such that ,   and with .

Proof. Let and be the operators defined by

It is obvious that a fixed point of is the solution of the fractional order BVP (2)–(5). Three fixed points of are sought. First, it is shown that . Let . Clearly, and , for . Also, for ,

Similarly, . Therefore,

Hence, and so . Moreover, is completely continuous operator. From (40), for each , , and . It is shown that . Let . Then . Condition (A3) is used to obtain

Therefore . Now conditions (B1) and (B2) of Theorem 8 are to be verified. It is obvious that

Next, let or . Then, and . Now, condition (A2) is applied to get

Clearly, condition (A1) leads to

To see that (B3) is satisfied, let with . Then

Finally, it is shown that (B4) holds. Let with . Then, we have

It has been proved that all the conditions of Theorem 8 are satisfied. Therefore, the fractional order BVP (2)–(5) has at least three positive solutions, ,  , and such that , , and with . This completes the proof of the theorem.

4. Example

In this section, as an application, the result is demonstrated with an example.

Consider a coupled system of -Laplacian fractional order BVP:

where

Then the Green functions and , for , are given by

Clearly, the Green functions and , for , are positive and ,   are continuous and increasing on . By direct calculations, ,  ,  , and . Choosing ,   and and then and , for satisfies

(i),   and ,(ii),   and ,(iii),   and .

Then, all the conditions of Theorem 9 are satisfied. Therefore, it follows from Theorem 8 that the -Laplacian fractional order BVP (51) has at least three positive solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the referees for their valuable comments and suggestions.

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