International Journal of Differential Equations

Volume 2014, Article ID 485647, 10 pages

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

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

^{1}Department of Applied Mathematics, Andhra University, Visakhapatnam 530 003, India^{2}Department 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*

*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*

*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*

*Conflict of Interests*

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

*Acknowledgment*

*Acknowledgment*

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

*References*

*References*

- R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,”
*Indiana University Mathematics Journal*, vol. 28, no. 4, pp. 673–688, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal, D. O'Regan, and P. J. Y. Wong,
*Positive Solutions of Differential, Difference and Integral Equations*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - D. R. Anderson and J. M. Davis, “Multiple solutions and eigenvalues for third-order right focal boundary value problems,”
*Journal of Mathematical Analysis and Applications*, vol. 267, no. 1, pp. 135–157, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Avery and J. Henderson, “Existence of three positive pseudo-symmetric solutions for a one-dimensional $p$-Laplacian,”
*Journal of Mathematical Analysis and Applications*, vol. 277, no. 2, pp. 395–404, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - C. Yang and J. Yan, “Positive solutions for third-order Sturm-Liouville boundary value problems with $p$-Laplacian,”
*Computers & Mathematics with Applications*, vol. 59, no. 6, pp. 2059–2066, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - R. P. Kapula, P. Murali, and K. Rajendrakumar, “Existence of positive solutions for higher order $(p,q)$-Laplacian two-point boundary value problems,”
*International Journal of Differential Equations*, vol. 2013, Article ID 743943, 9 pages, 2013. View at Google Scholar · View at MathSciNet - Z. Bai and H. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Dehghani and K. Ghanbari, “Triple positive solutions for boundary value problem of a nonlinear fractional differential equation,”
*Bulletin of the Iranian Mathematical Society*, vol. 33, no. 2, pp. 1–14, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Henderson and S. K. Ntouyas, “Positive solutions for systems of nonlinear boundary value problems,”
*Nonlinear Studies*, vol. 15, no. 1, pp. 51–60, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions,”
*Computers & Mathematics with Applications*, vol. 58, no. 9, pp. 1838–1843, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 64–69, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Bai, “Existence of positive solutions for boundary value problems of fractional functional differential equations,”
*Electronic Journal of Qualitative Theory of Differential Equations*, vol. 2010, no. 30, pp. 1–14, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1251–1268, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Nyamoradi and M. Javidi, “Existence of multiple positive solutions for fractional differential inclusions with $m$-point boundary conditions and two fractional orders,”
*Electronic Journal of Differential Equations*, vol. 2012, no. 187, pp. 1–26, 2012. View at Google Scholar · View at MathSciNet - K. R. Prasad and B. M. B. Krushna, “Multiple positive solutions for a coupled system of Riemann-Liouville fractional order two-point boundary value problems,”
*Nonlinear Studies*, vol. 20, no. 4, pp. 501–511, 2013. View at Google Scholar · View at MathSciNet - G. Chai, “Positive solutions for boundary value problem of fractional differential equation with $p$-Laplacian operator,”
*Boundary Value Problems*, vol. 2012, article 18, 20 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - R. I. Avery, “A generalization of the Leggett-Williams fixed point theorem,”
*Mathematical Sciences Research Hot-Line*, vol. 3, no. 7, pp. 9–14, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet

*
*