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International Journal of Differential Equations
Volume 2013 (2013), Article ID 743943, 9 pages
http://dx.doi.org/10.1155/2013/743943
Research Article

Existence of Positive Solutions for Higher Order -Laplacian Two-Point Boundary Value Problems

1Department of Applied Mathematics, Andhra University, Visakhapatnam 530003, India
2Department of Mathematics, VITAM College of Engineering, Visakhapatnam 531173, India

Received 17 April 2013; Revised 17 July 2013; Accepted 17 July 2013

Academic Editor: Kanishka Perera

Copyright © 2013 Rajendra Prasad Kapula 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 derive sufficient conditions for the existence of positive solutions to higher order -Laplacian two-point boundary value problem, , , , , , , , ; , , , , , and , where are continuous functions from to , and . We establish the existence of at least three positive solutions for the two-point coupled system by utilizing five-functional fixed point theorem. And also, we demonstrate our result with an example.

1. Introduction

The goal of differential equations is to understand the physical phenomena of nature by developing mathematical models. Among all, a class of differential equations governed by nonlinear differential operators, which have wide applications and interest, has been developed to study such type of equations. In this theory, the most investigated operator is the classical -Laplacian, given by with . These problems have a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design. Due to the importance in both theory and applications, -Laplacian boundary value problems have created a great deal of interest in recent years; we mention a few [111].

Recently, Prasad and Murali [12] established the existence of positive solutions of -Laplacian singular boundary value problem on time scale, by assuming suitable conditions on . Till now in the literature of boundary value problems, the theory was not developed to the system of higher order boundary value problems with -Laplacian. Mainly, this type of problems arises in radar invention models and microatom invention models.

Due to our interest in the literature, in this paper, we consider two-point higher order )-Laplacian boundary value problem (BVP) where are continuous functions from to ,   and .   If we take in the above problem then it reduces to -Laplacian problem.

To obtain a solution of the BVP (2), we construct Green's functions for the corresponding homogeneous BVPs. For , let be Green's function of the BVP, and it is given by Let be Green's function of the BVP and it can be recursively defined as where is Green's function of and is given by Then can be expressed in the form Since , we have For , For , let be Green’s function of the BVP and it is given by Let be Green's function of the BVP and it can be recursively defined as where is Green’s function of and is given by Then can be expressed in the form Since , we have For , Further, it is easily seen that , , and , all are nonnegative on .

A solution of the BVP (2) is a function such that , and satisfies the BVP (2).

A positive solution of the BVP (2) is a solution of the BVP (2) such that and are nonnegative on .

The rest of the paper is organized as follows. In Section 2, we estimate the bounds of Green's functions. In Section 3, we establish the existence of at least three positive solutions for two-point BVP (2) by using Avery's generalization of Leggett-Williams fixed point theorem. And also, we demonstrate our result with an example.

2. Bounds of Green's Functions

In this section, we state some lemmas to estimate bounds on Green’s functions which are needed in later discussions. The following lemma is included in the paper to prove the remaining lemmas.

Lemma 1. Let and . Then, for any where .

Proof. The proof is by induction. First, for inequality (21) is obvious. Next, for fixed , assume that (21) is true; from recursive formula we have, for each , Hence, by induction the proof is complete.

In the previous lemma by choosing , , and , , and , we get , , and . We know that .

Lemma 2. For , one has For , one has For , one has where .

Lemma 3. For , one has where .

For details refer to [9].

In Lemma 1, replacing by , we have the following.

Lemma 4. For , one has For , one has For , one has where .

Lemma 5. For , one has where .

For details refer to [9].

Lemma 6. For , one has For , one has For , one has

For details refer to [11].

Lemma 7. For , one has

For details refer to [11].

Lemma 8. For , one has For , one has For , one has

For details refer to [11].

Lemma 9. For , one has

For details refer to [11].

Denote

3. Existence of Multiple Positive Solutions

In this section, we establish the existence of at least three positive solutions for the system of BVP (2), by using Avery's generalization of the Leggett-Williams fixed point theorem.

Let be a real Banach space with cone . We define the nonnegative continuous convex functionals , , and and nonnegative continuous concave functionals , on , for nonnegative numbers , , , , and ; we define the following sets:

In obtaining multiple positive solutions of the BVP (2), the following so-called five-functionals fixed point theorem will be fundamental.

Theorem 10 (see [4]). Let be a cone in a real Banach space . Suppose and are nonnegative continuous concave functionals on and ; and are nonnegative continuous convex functionals on such that, for some positive numbers and , Suppose further that is completely continuous, and there exist constants with such that each of the following is satisfied (B1)   and for ,(B2) and for ,(B3) provided with , (B4) provided with .
Then has at least three fixed points such that

Let , and denote by the norm defined as , where Then the set with the norm is a complete normed linear space.

Let , , and . Define the cone by Now, define the nonnegative continuous concave functionals and the nonnegative continuous convex functionals , and on by We observe that, for any ,

We derive growth conditions on so that the BVP (2) has at least three positive solutions. We are now ready to present the main result of this section.

We denote

Theorem 11. Suppose there exists such that and satisfy the following conditions: (A1) and , for all and ,(A2) for all and or for all and , (A3) and , for all and .
Then the BVP (2) has at least three positive solutions.

Proof. Define the completely continuous operator by And also we denote
It is obvious that a fixed point of is a solution of the BVP (2). We seek three fixed points , , and of . First, we show that . Let . Clearly, ,   for . Consider Thus, . Next, for all , by (46), (47), respectively, we have and . To show that , let . This implies . We may now use condition (A3) to obtain Therefore, .
We first verify that condition (B1) of Theorem 10 is satisfied. The constant function is Next, let . It follows that from (A2) Similarly, and, hence, we have .
Next, we show that (B2) is fulfilled. The constant function is Let . And from (A1), To see that (B3) is satisfied, let with Using (23), (27), (31), and (35), we get
Finally, we show that (B4) holds. Let with . Using (23), (27), (31), and (35), we have
We have proved that all the conditions of Theorem 10 are satisfied, and so there exist at least three positive solutions. Therefore the BVP (2) has at least three positive solutions. This completed the proof of the theorem.

4. Example

Considering the higher order -Laplacian two-point boundary value problem, where are continuous functions from to . A simple calculation shows that , , , and . If we choose , , and , then conditions (A1)–(A3) are satisfied. Therefore, it follows from Theorem 11 that the BVP (61) has at least three positive solutions.

Acknowledgment

One of the authors (Dr. Penugurthi Murali) is thankful to CSIR, India, for awarding him a Research Associate.

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