Abstract and Applied Analysis

Volume 2012 (2012), Article ID 506716, 15 pages

doi:10.1155/2012/506716

Research Article

Existence of Positive Solutions for Third-Order -Point Boundary Value Problems with Sign-Changing Nonlinearity on Time Scales

Fatma Tokmak and Ilkay Yaslan Karaca

Department of Mathematics, Ege University, Izmir, 35100 Bornova, Turkey

Received 2 August 2012; Accepted 8 November 2012

Academic Editor: Yongfu Su

Copyright © 2012 Fatma Tokmak and Ilkay Yaslan Karaca. 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.

A four-functional fixed point theorem and a generalization of Leggett-Williams fixed point theorem are used, respectively, to investigate the existence of at least one positive solution and at least three positive solutions for third-order -point boundary value problem on time scales with an increasing homeomorphism and homomorphism, which generalizes the usual -Laplacian operator. In particular, the nonlinear term is allowed to change sign. As an application, we also give some examples to demonstrate our results.

1. Introduction

The theory of time scales was introduced by Hilger in his Ph.D. thesis in 1990 [1]. Theoretically, this new theory not only unifies continuous and discrete equations, but has also exhibited much more complicated dynamics on time scales. Moreover, the study of dynamic equations on time scales has led to several important applications, for example, insect population models, biology, neural networks, heat transfer, and epidemic models (see [2, 3] and references therein). Some preliminary definitions and theorems on time scales also can be found in books [2, 3], which are excellent references for calculus on time scales.

Multipoint boundary value problem (BVP) arises in a variety of different areas of applied mathematics and physics [4]. The study of multipoint BVPs for linear second-order ordinary differential equations was initiated by Ilin and Moiseev [5], since then many authors studied more general nonlinear multipoint boundary value problems ([6, 7] and references therein). Recently, when is -Laplacian operator, that is, and the nonlinear term does not depend on the first-order derivative, the existence problems of positive solutions of boundary value problems have attracted much attention. One can notice that the oddness of the -Laplacian operator is key to the proof. However, in the operator is not necessary odd, so it improves and generalizes the -Laplacian operator.

There are few works that investigate the existence of positive solutions by using four functionals fixed point theorem [8] or a generalization of Leggett-Williams fixed point theorem [9]. Avery et al. [8] present a four-functional fixed point theorem, which is a major generalization of the original Leggett-Williams fixed point theorem [10]. To our knowledge, this fixed point theorem so far has only been used by Sun et al. [11]. By using the four-functional fixed point theorem and five-functional fixed point theorem, they obtain the existence criteria of at least one positive solution and three positive solutions for the nonlocal BVP with -Laplacian. Liu and Sun [12] discussed the existence of at least three and arbitrary odd number positive solutions for -point boundary value problems on time scale. They used a generalization of Leggett-Williams fixed point theorem [9].

There have been extensive studies on BVP with sign-changing nonlinearity [13–17]. Most of these results are obtained by using the fixed point theorem on cone [18, 19]. To the best of our knowledge, positive solutions of third-order -point BVP for an increasing homeomorphism and homomorphism with sign-changing nonlinearity on time scales by using four functionals fixed point theorem [8] or a generalization of Leggett-Williams fixed point theorem [9] have not been considered till now. Our aim in this paper is to fill the gap.

In this paper, motivated by above results, we consider the existence of at least one or three positive solutions of the following third-order -point boundary value problem (BVP) on time scales

where is an increasing homeomorphism and homomorphism with . A projection is called an increasing homeomorphism and homomorphism if the following conditions are satisfied.

(i)If , then , for all ;(ii) is continuous bijection and its inverse mapping is also continuous;(iii), for all .

We will assume that the following conditions are satisfied throughout this paper:

(H1); (H2) is continuous, , and there exist such that ; (H3) exist.

By using a four-functional fixed point theorem [8], we establish the existence of at least one positive solution, and by using a generalization of Leggett-Williams fixed point theorem [9], we establish the existence of at least three positive solutions for the above boundary value problem. In particular, the nonlinear term is allowed to change sign. The remainder of this paper is organized as follows. Section 2 is devoted to some preliminary lemmas. We give and prove our main results in Section 3. Finally, in Section 4, we give two examples to demonstrate our results.

2. Preliminary Lemmas

In this section, we present some definitions and lemmas, which will be needed in the proof of the main results.

Lemma 2.1 (see [13]). If condition (H1) holds, then for , the boundary value problem (BVP),

has a unique solution

Lemma 2.2. If condition (H1) holds, then for , the boundary value problem (BVP),

has a unique solution

where .

Proof. Integrating both sides of equation in (2.3) on , we have

So,

By boundary value condition , we have

By (2.5) and (2.7), we know

This together with Lemma 2.1 implies that

where . The proof is complete.

Lemma 2.3. Let condition (H1) hold. If and , then the unique solution of (2.3) satisfies

(i), (ii),

where .

Proof. (i) Let . Consider the following equation:

So, we have . Hence .

(ii) By , we can know that the graph of is concave down on and is nonincreasing on . This together with the assumption that the boundary condition implies that for . This implies that

For all , we have from the concavity of that

that is,

This together with the boundary condition implies that

This completes the proof.

Let be equipped with the norm . Clearly, it follows that is a Banach space. For the convenience, let

We define two cones by

where is as in Lemma 2.3. Define the operators and by setting

where ,

where and . Obviously, is a solution of the BVP (1.1) if and only if is a fixed point of operator .

Lemma 2.4. is completely continuous.

Proof. It is easy to see that by and Lemma 2.3. By Arzela-Ascoli theorem and Lebesgue dominated convergence theorem, we can easily prove that operator is completely continuous.

3. Main Results

In this section, we prove the existence of at least one positive solution to the BVP (1.1) by applying a four-functional fixed point theorem [8]. Also, by using the generalization of Leggett-Williams fixed point theorem [9], we prove the existence of at least three positive solutions for the BVP (1.1).

Let and be nonnegative continuous concave functionals on , and let and be nonnegative continuous convex functionals on , then for positive numbers , , , and , we define the following sets:

Theorem 3.1 (see [8]). If is a cone in a real Banach space , and are nonnegative continuous concave functionals on , , and are nonnegative continuous convex functionals on and there exist positive numbers , , , and , such that

is a completely continuous operator, and is a bounded set. If

(i), (ii), for all , with and ,(iii), for all , with ,(iv), for all , with and ,(v), for all , with ,

then has a fixed point in .

Suppose with . For convenience, we take the notations

Theorem 3.2. Assume that (H1), (H2) hold if there exist constants , , , with , , and suppose that satisfies the following conditions:

(B1) for all ;(B2) for all ;(B3) for all .

Then the BVP (1.1) has a fixed point such that

Define maps

Let , , and be defined by (3.1).

Proof. We first show that is bounded and is completely continuous. For all , we have , which means that is a bounded set. According to Lemma 2.4, it is clear that is completely continuous.

Let

Clearly, . By direct calculation,

So, , which means that (i) in Theorem 3.1 is satisfied.

For all , with and , we have

and for all , with and , we have

Hence (ii) and (iv) in Theorem 3.1 are fulfilled.

Lastly, we consider Theorem 3.1 (iii) and (v). For all , with ,

and for all , with ,

Thus, all conditions of Theorem 3.1 are satisfied. has a fixed point in . Clearly, . By condition (B3), we have , that is, . Hence, . This means that is a fixed point of operator . Therefore, the BVP (1.1) has at least one positive solution.

Suppose and are two nonnegative continuous convex functionals satisfying

where is a positive constant, and

Let be given, nonnegative continuous convex functionals on satisfying the relation (3.12) and (3.13), and a nonnegative continuous concave functional on . We define the following convex sets:

Theorem 3.3 (see [9]). Let be a Banach space, a cone, and . Assume that and are nonnegative continuous convex functionals satisfying (3.12) and (3.13), is a nonnegative continuous concave functional on such that for all , and is a completely continuous operator. Suppose

(i) for ;(ii) for ;(iii) for with .

Then has at least three fixed points with

It is easy to see that for . So we get

where is as in (H3). Let , , and be defined on the cone by

respectively. Then it is easy to see that and (3.12), (3.13) hold. Now, for convenience, let (3.3) and .

Theorem 3.4. Assume (H1)–(H3) hold, is not equivalent to for . If there are positive numbers , with , such that the following conditions are satisfied.

(C1) for ,(C2) for ,(C3) for .

Then the BVP (1.1) has at least three positive solutions satisfying

Proof. By the definition of , it suffices to show that the conditions of Theorem 3.3 hold with respect to the operator . Firstly, we show that if the condition (C1) is satisfied, then

If , then assumption (C1) implies

Therefore, (3.19) holds. In the same way, if , then condition (C3) implies

As the argument above, we have . Thus, condition (ii) of Theorem 3.3 holds. To check the condition (i) of Theorem 3.3, we choose

It is easy to see that . Therefore, for , there are . Hence, condition (C2) implies . So, by the definition of ,

Hence, condition (i) of Theorem 3.3 holds. We finally prove that (iii) in Theorem 3.3 holds. In fact, for with , we have . Thus from Theorem 3.3, the BVP (1.1) has at least three fixed points . Clearly, , , , . By condition (C1), we have , , that is, . Hence, . This means that are fixed points of operator . Therefore, the BVP (1.1) has at least three positive solutions.

4. Example

Example 4.1. Let . If we choose , ,, , in the boundary value problem (1.1), then we have the following BVP on time scale :

where

Set , , by calculation,

and let , , and with . Clearly, we can verify that the conditions in Theorem 3.2 are fulfilled. Thus, by Theorem 3.2, the BVP (4.1) has a fixed point such that

Example 4.2. Let . If we choose , , , , in the boundary value problem (1.1), then we have the following BVP on time scale

where

Obviously the hypotheses (H1), (H2) hold and is not equivalent to on . By simple calculations, we have

If we choose , , , and , , then satisfies

(1) for ;(2) for ,(3).

So, all conditions of Theorem 3.4 hold. Thus by Theorem 3.4, the BVP (4.5) has at least three positive solutions , , such that

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