Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 418410, 8 pages
http://dx.doi.org/10.1155/2015/418410

Research Article

Positive Solutions for a Class of Fourth-Order -Laplacian Boundary Value Problem Involving Integral Conditions

Yan Sun

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 30 April 2015; Revised 19 July 2015; Accepted 27 July 2015

Academic Editor: Josef Diblík

Copyright © 2015 Yan Sun. 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.

Under some conditions concerning the first eigenvalues corresponding to the relevant linear operator, we obtain sharp optimal criteria for the existence of positive solutions for -Laplacian problems with integral boundary conditions. The main methods in the paper are constructing an available integral operator and combining fixed point index theory. The interesting point of the results is that the nonlinear term contains all lower-order derivatives explicitly. Finally, we give some examples to demonstrate the main results.

1. Introduction

In this paper, we discuss the existence of positive solutions for the following -Laplacian integral boundary value problems:

where , , , , is symmetric on the interval (i.e., and for ), is continuous, and are nonnegative symmetric on .

Boundary value problems of ordinary differential equations have become an important research field in recent years. Fourth-order -Laplacian boundary value problems arise in applied mathematics, physics, gas diffusion through porous media, engineering, elastic mechanics, electromagnetic waves of gravity driven flows, and the various areas of adiabatic tubular reactor processes, as well as biological problems; see [1–5] and the references therein.

In [6], by using the fixed point theorem for strict set contraction operator, Zhang et al. considered the existence of positive solutions for the following th-order impulsive boundary value problems with integral boundary conditions in Banach spaces:

where , , , , , , and denotes the jump of at . They also gave the upper and lower bounds for these positive solutions.

In [7], by means of the fixed point index theory, Sun and Zhang studied the following singular nonlinear Sturm-Liouville problems:

where and is allowed to be singular at and .

In [2], by making use of the theory of the spectrum, Sergejeva studied the regions of solvability for three-point boundary value problem:

where , , and is a bounded function. The solvability results are established for the problem with .

Motivated and inspired greatly by the above mentioned works, the aim of the paper is to improve, generalize, and supplement the previous results. We obtain the existence results on nonlinear fourth-order integral boundary value problems with one-dimensional -Laplacian integral boundary value problem (1). The novelty of our results is in exploring some optimal criteria for the existence of positive solutions of problem (1). The methods used in our work will depend on an application of fixed point index theory together with the first eigenvalues corresponding to the relevant linear operator. It can also be seen that the nonlinear term involves all lower-order derivatives explicitly.

The paper is organized as follows. In Section 2, we state some lemmas and several preliminary results. The main results are formulated and proved in Section 3. Examples are given in Section 4.

2. Preliminaries

Throughout the paper, we make the following assumptions:

(H ₁) and , , are continuous and also and are symmetric on for all (i.e., , ).

(H ₂) are all nonnegative and symmetric on , and is continuous on , and may be singular at and/or , and and , where

In what follows, we will consider the Banach space , , equipped with the maximum norm . Denote cone by

then, is a positive cone in .

Let for (), and

Here , , and are Green’s functions of problem (1).

Lemma 1 (see [1]). Suppose that (H ₁) and (H ₂) hold; then, for , we have

(I), , ;(II), where ;(III), , and ;(IV); , where , , , .

Define an integral operator by

It is well known that is a solution of problem (1) if and only if is a fixed point of operator equation .

Lemma 2. Suppose that (H ₁) and (H ₂) hold, if is a solution of the following integral equation:

then, is a solution of problem (1).

Lemma 3. Suppose that (H ₁) and (H ₂) hold; then, is a completely continuous operator.

Proof. First, we prove that For every , we have

Thus, , for all Hence, is nonnegative concave functional on .
By Lemma 1, for , we obtain

Hence, , and , which implies
Next, we show that is completely continuous.
For any natural number , set

Then, is continuous, and , . Let

Obviously, is a completely continuous operator. From Lemma 1, we obtain

where . Therefore, is a completely continuous operator. The proof is completed.

Now we define linear integral operator as follows:

It is easy to see that is a completely continuous linear operator, and

Lemma 4. Assume that (H ₁) and (H ₂) hold. Then, the operator defined by (15) is a completely continuous linear operator and , the spectral radius , and has a positive eigenfunction corresponding to its first eigenvalue .

Proof. A simple modification of the argument in Lemma 3 yields that is a completely continuous linear operator and . By (H ₁) and (H ₂), there is such that . Choose constants and such that and , for all . Choose a nonnegative continuous functional such that , for all . Then, for any , we have

Thus, there exists a constant such that , for all . By famous Krein-Rutman theorems, we know that the spectral radius and has a positive eigenfunction corresponding to its first eigenvalue ; that is, . This completes the proof.

Lemma 5 (see [8]). Let be a cone in Banach space , and let be a bounded open set in . Suppose that is completely continuous. If there exists , such that , for all , Then, the fixed point index

Lemma 6 (see [8]). Let be a cone in Banach space , and let be a bounded open set in with . Suppose that is a completely continuous operator. If , for all , Then, the fixed point index

Lemma 7 (see [9]). Let and be cones in Banach space , and let be a bounded open set in . Suppose that is a completely continuous operator, and has no fixed point on . If there exist linear operators which satisfy , , , and , such that

(1), for certain natural number ,(2), for all ,(3), for all ,

then the fixed point index

3. Main Results

In this section, we establish sharp optimal criteria for the existence of positive solutions to problem (1) under superlinear cases and sublinear cases, respectively.

Theorem 8. Suppose that (H ₁) and (H ₂) hold. In addition, assume that

where is the first eigenvalue of defined by (15); then, problem (1) has at least one positive solution.

To prove our main result, we need some preliminary results.

Lemma 9. Suppose that (H ₁) and (H ₂) hold. Define

where , , is the positive eigenfunction corresponding to its first eigenvalue ; that is, , and

then, is a cone in , and

Proof. It is obvious that is a cone in . For any , we obtain

Thus, . The proof is completed.

Proof of Theorem 8. It follows from (17) that there exists , such that , for is sufficiently large. In view of (H ₁), we see that there exists such that

Let , where and are defined by Lemma 9.
In the following, we prove that

where is a positive eigenfunction corresponding to its first eigenvalue .
Otherwise, there exist and , such that

Since and , we know from (24) that Therefore, it follows from (19) and (22) that

On the other hand, it follows from (24) that

which is a contradiction; thus, by Lemma 5, we see

It follows from (18) that there exists , such that

Suppose that there exist , , such that Without loss of generality, we may suppose that (otherwise, the proof is completed). Then,

Multiplying and then integrating by (29), we have

By maximum principle, and , ; then Hence, (30) implies which is a contradiction. So by Lemma 6, we have

It follows from (27) and (31) that

Then, has at least one fixed point on . This means problem (1) has at least one positive solution. The proof is completed.

Corollary 10. Suppose that (H ₁) and (H ₂) hold. In addition, assume that

Then, for any

where is the first eigenvalue of defined by (15), the following -Laplacian integral boundary value problem

has at least one positive solution.

Proof. By (34), we obtain

Therefore, it follows from Theorem 8 that Corollary 10 holds.

Theorem 11. Suppose that (H ₁) and (H ₂) hold. In addition, assume that

where is the first eigenvalue of defined by (15); then, problem (1) has at least one positive solution.

Proof. It follows from (37) that there exists such that

Let be the positive eigenfunction corresponding to its first eigenvalue . Then, ; that is,

Hence, , for all Then, is nonnegative concave on . Thus,
Let , . Then, is a completely continuous linear operator, and ,
For any , it follows from (39) that

Setting , , denotes the identical operator, and , in Lemma 7. We have

It follows from (38) that there exist and , such that

Set . Then, is a completely continuous linear operator, and Let
Now we prove that
Suppose that there exist , , such that Let ; then, from (43) we have

where Thus, , Since , we have , , for It means that , Let

For any , denote

It is easy to see that

Since , we have from (47) that , which contradicts (45). Thus, for any , , we have that Thus, from Lemma 6, we know that

From (42) and (48), we see that

Then, has at least one fixed point on . It means problem (1) has at least one positive solution. The proof is completed.

Corollary 12. Suppose that (H ₁) and (H ₂) hold. In addition, assume that

Then, for any

where is the first eigenvalue of defined by (15), problem (35) has at least one positive solution.

Proof. By (51), we obtain

It follows from Theorem 11 that Corollary 12 holds. The proof is completed.

4. Examples

Example 1. Consider the existence of positive solutions for the following -Laplacian integral boundary value problems:

where . , , , , is symmetric on the interval , for is continuous, and are nonnegative symmetric on .

Let and let Obviously, , and

All conditions of Theorem 8 are satisfied. Consequently, following from Theorem 8, we know that the boundary value problem (53) has at least one positive solution.

Example 2. Consider the existence of positive solutions for the following -Laplacian integral boundary value problems:

where , , , , is symmetric on the interval , for is continuous, and are nonnegative symmetric on . For ,

Let

Obviously, , and

All conditions of Theorem 8 are satisfied. Consequently, following from Theorem 8, we know that the boundary value problem (55) has at least one positive solution.

Conflict of Interests

The author declares that she has no competing interests.

Acknowledgments

The author is very grateful to the editor and the anonymous referee for his/her careful reading of the first draft of the paper and making very valuable comments and helpful suggestions. She would like to express her gratitude to Professor Lishan Liu and Professor R. P. Agarwal for their many valuable comments. The author was supported financially by the Foundation of Shanghai Natural Science (13ZR1430100) and the Foundation of Shanghai Municipal Education Commission (DYL201105) and the NNSF of China (no. 11371095 and no. 2013M541455).

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