Nodal Solutions for Some Second-Order Semipositone Integral Boundary Value Problems

Abstract

Using bifurcation techniques, we first prove a global bifurcation theorem for nonlinear second-order semipositone integral boundary value problems. Then the existence and multiplicity of nodal solutions of the above problems are obtained. Finally, an example is worked out to illustrate our main results.

1. Introduction

In this paper, we consider the existence and multiplicity of nodal solutions for the following nonlinear second-order semipositone integral boundary value problems (BVP for short): where is a parameter, , and is nonnegative with .

Boundary value problems with integral boundary conditions for ordinary differential equations arise in different areas of applied mathematics and physics. Moreover, they include two, three, multipoint, and nonlocal boundary value problems as special cases. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to [1–12] and the references therein.

In [1], utilizing the fixed point index and Leray-Schauder degree theory, Zhang and Sun obtained some existence results for multiple solutions including sign-changing solutions under some technical hypotheses for the following integral boundary value problem: where , , is nonnegative with .

The purpose of this paper is to investigate the existence and multiplicity of sign-changing solutions of BVP (1), having a given number of zeros (so-called “nodal solution”). The existence of such solutions has been investigated for many types of nonlinear Sturm-Liouville problems with separated boundary conditions and multipoint boundary conditions in many recent papers; see [13–18]. Recently, Sun et al. [16] studied for the following -point boundary value problems: where is a parameter, , , , , , . By using Rabinowitz’s global bifurcation theorem, they obtained the existence and multiplicity of nodal solutions when , where To the authors’ knowledge, there are few papers that have considered the existence of nodal solutions for integral boundary value problems. In [1], Zhang and Sun have obtained sign-changing solutions of BVP (2), but no information is obtained regarding the number of zeros of the solution.

Motivated by [1, 15, 16], in this paper we investigate the existence and multiplicity of nodal solutions for BVP (1). The main features of this paper are as follows. First, the nonlinear term is semipositone, and , where is defined as in (4). Next, the methods used here are Rabinowitz’s global bifurcation theorem and some of the techniques used in [16], which are entirely different from [1, 7, 8]. Finally, the results we obtained are the existence of at least any given even number of nodal solutions.

Now we give some notations and a global bifurcation theorem which will be used in Section 3. Let be a real Banach space; Rabinowitz studied a nonlinear eigenvalue problem of the form where is a parameter, is a compact linear map, is completely continuous, and for near uniformly on bounded intervals. A solution of (5) is a pair which satisfies (5). The closure of the set of nontrivial solutions of (5) is denoted by . If there exist and such that , is said to be a positive eigenvalue of and is said to be an eigenfunction corresponding to . The set of positive eigenvalues of will be denoted by . The algebraic multiplicity of is , where denotes the null space of . The following was shown in Theorem 1.3 and Theorem 1.25 of Rabinowitz [19] and Theorem 2 of Dancer [20].

Theorem A. If is simple, then possess a maximal subcontinuum which can be decomposed into two subcontinua and such that, for some neighborhood of , , and imply , where and , for near 0. Moreover, either and are both unbounded or .

This paper is arranged as the follows: some preliminaries and some lemmas are given including the study of the eigenvalues and eigenfunctions of the linearization of BVP (1) in Section 2. The main results are proved by using Theorem A in Section 3. A concrete example is given to illustrate the application of the main results in Section 4.

2. Some Preliminaries and Lemmas

Let with the norm , , , with the norm , , , with the norm . Then , , are Banach spaces.

For any function , if , then is said to be a simple zero of if . For any integer and any , as in [15], we define sets consisting of the set of functions satisfying the following conditions:(i), , and ;(ii) has only simple zeros in and has exactly such zeros;(iii) has a zero strictly between each two consecutive zeros of .

Note that and let . It is easy to see that the sets and are disjoint and open in . Let under the product topology, , , and .

In the following, we give some information on the spectrum structure of the linear integral boundary value problem corresponding to BVP (1):

Define the operators on by where

It is easy to prove the following lemma.

Lemma 1. The linear operator is completely continuous. Moreover, is a solution of (6) if and only if is a solution of the operator equation .

We now define a function by

Lemma 2. All the zeros of are simple.

Proof. Suppose that is a double zero of ; that is, Hence, which shows that (10) cannot hold, and so has only simple zeros.

Lemma 3. Suppose that is symmetrical in . Then has no zero on and, for each , has exactly one zero on .

Proof. Since and , we have
Now, (12) implies that on ; that is, has no zero in this interval, and also
For each integer , by the symmetry of in , we have So, ; that is, .
Thus, . That is, has one zero on each interval .
For any fixed integer , suppose that has another zero on . In view of the continuity of and (13), then has the third zero on . Without loss of generality, we may assume that . We have the following three cases to consider.Consider . By (9) and (12), we have From (13) and Lemma 2, it is easy to see that , which contradicts to (15).Consider . From (13) and Lemma 2, it is easy to see that . So, we have Hence, which is a contradiction.Consider . Similar to the proof of Case , we can also lead to a contradiction.
Therefore, has exactly one zero on for each .

As the proof of Lemma 4 in [1], it is easy to obtain the following lemma.

Lemma 4. (1) For each , is one zero of if and only if is an eigenvalue of . In addition, is an eigenfunction corresponding to and .
(2) The algebraic multiplicity of each positive eigenvalue of is 1.

Lemma 5. Suppose that is symmetrical in . Then(1)there exists a subsequence of the eigenvalue sequence of such that , and the eigenfunction corresponding to is ;(2) for .

Proof. From Lemmas 3 and 4, conclusion (1) can be obtained immediately. Noticing that , , it is easy to check that for .

Define the operators and on by

and for , respectively, where the operator is defined as in (7).

It is easy to see that is completely continuous. By direct computation, we can easily get the following lemma.

Lemma 6. is a solution of BVP (1) if and only if is a solution of equation

For , by the mean-value theorem for the integral, there exists a point such that Let , for each and . Let for each , where is a constant to be defined later. The set is defined by Obviously is a closed convex set, and, for each ,

Lemma 7. Let be a positive number such that , for each . Then(1);(2)for each , .

Proof. (1) For each , let . By direct computation we have Since , then , and so is a concave function on . From (7), it is easy to see that Using the concavity of and the boundary condition , , we can see that for each and ; we have from the concavity of that By (19), we have . Hence, From the concavity of , we have for each that It follows from (25) and (26) that Then we have from the concavity of that that is, This implies that , and, therefore, conclusion (1) holds.
(2) Since , from (1), we see that For each , we have Then by (30) and (31), we have This implies that . Thus, .

3. Main Results

Theorem 8. Suppose that is symmetrical in , , and there exists such that for each . Then for each integer and each , or −, there exists an unbounded maximal subcontinuum of solutions of BVP (1) in , which meets in and satisfies (1) for each ;(2) for each .

Proof. Since , the operator equation (18) can be rewritten as Here and is defined as in (7). Obviously, it is easy to see that for near uniformly on bounded intervals. Notice that is a compact linear map on . A solution of BVP (1) is a pair . By , the known curve of solutions will henceforth be referred to as the trivial solutions. The closure of the set on nontrivial solutions of BVP (1) will be denoted by as in Theorem A.
If , then (33) becomes a linear system By Lemmas 3, 4, and 5, (34) possesses an increasing subsequence of simple eigenvalues sequence and as . Any eigenfunction corresponding to is in . Moreover, for and for .
Consider (33) as a bifurcation problem from the trivial solution. From Theorem A and , it follows that, for each integer , possess a maximal subcontinuum which can be decomposed into two subcontinua such that, for some neighborhood of , implying , where and , for near 0.
By (18) and the continuity of the operator , the set lies in and the injection is continuous. Moreover, note that . So, is also a continuum in , and the above properties hold in .
Since is open in and , we know that for sufficiently small. Then there exists such that, for , we have where is an open ball in of radius centered at . Since is open in , it can follow, similar to the proof of Proposition 4.1 in [15], that which means . Consequently, lies in .
Similarly we can obtain that lies in or . Noticing that , it can be obtained that . From Theorem A, we know that and are unbounded in .
Let be fixed. For each and , is a solution of (18), and by Lemma 7, . Thus, since where , and so
Let . Then for each , , , we have for . This implies that Thus, the conclusion holds and the proof is complete.

Immediately, from Theorem 8, we have the following result.

Theorem 9. Suppose that all the conditions of Theorem 8 hold. Then, for each , BVP (1) has at least nodal solutions in such that has zeros in and is positive near and has zeros in and is negative near .

4. An Example

Consider the following nonlinear second-order integral boundary value problem: where .

By direct computation, it is easy to see that , so, .

Next, we check that all the conditions of Theorem 9 hold. Take . It is clear that , and is symmetrical in , and is nonnegative. Since , , and . It follows from Theorem 9, when , we have , so the boundary value problem (43) has at least 10 nodal solutions in .

Conflict of Interests

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

Acknowledgment

This research is supported by the Reward Fund for Excellent Young and Middle-Aged Scientists of Shandong Province (BS2011SF022), China.