Monotone Positive Solution of Nonlinear Third-Order BVP with Integral Boundary Conditions

Abstract

This paper is concerned with the following third-order boundary value problem with integral boundary conditions , where and . By using the Guo-Krasnoselskii fixed-point theorem, some sufficient conditions are obtained for the existence and nonexistence of monotone positive solution to the above problem.

1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross section, a three-layer beam, electromagnetic waves or gravity driven flows and so on [1].

Recently, third-order two-point or multipoint boundary value problems (BVPs for short) have attracted a lot of attention [2–17]. It is known that BVPs with integral boundary conditions cover multipoint BVPs as special cases. Although there are many excellent works on third-order two-point or multipoint BVPs, a little work has been done for third-order BVPs with integral boundary conditions. It is worth mentioning that, in 2007, Anderson and Tisdell [18] developed an interval of values whereby a positive solution exists for the following third-order BVP with integral boundary conditions

(1.1)

by using the Guo-Krasnoselskii fixed-point theorem. In 2008, Graef and Yang [19] studied the third-order BVP with integral boundary conditions

(1.2)

For second-order or fourth-order BVPs with integral boundary conditions, one can refer to [20–24].

In this paper, we are concerned with the following third-order BVP with integral boundary conditions

(1.3)

Throughout this paper, we always assume that and . Some sufficient conditions are established for the existence and nonexistence of monotone positive solution to the BVP (1.3). Here, a solution of the BVP (1.3) is said to be monotone and positive if , and for . Our main tool is the following Guo-Krasnoselskii fixed-point theorem [25].

Theorem 1.1.

Let be a Banach space and let be a cone in . Assume that and are bounded open subsets of such that , and let be a completely continuous operator such that either

(1) for and for , or

(2) for and for .

Then has a fixed point in .

2. Preliminaries

For convenience, we denote .

Lemma 2.1.

Let . Then for any , the BVP

(2.1)

has a unique solution

(2.2)

where

(2.3)

Proof.

Let be a solution of the BVP (2.1). Then, we may suppose that

(2.4)

By the boundary conditions in (2.1), we have

(2.5)

Therefore, the BVP (2.1) has a unique solution

(2.6)

Lemma 2.2 (see [12]).

For any ,

(2.7)

Lemma 2.3 (see [26]).

For any ,

(2.8)

In the remainder of this paper, we always assume that , and .

Lemma 2.4.

If and for , then the unique solution of the BVP (2.1) satisfies

(1), ,

(2), and , where .

Proof.

Since (1) is obvious, we only need to prove (2). By (2.2), we get

(2.9)

which indicates that for .

On the one hand, by (2.9) and Lemma 2.3, we have

(2.10)

On the other hand, in view of (2.2) and Lemma 2.2, we have

(2.11)

It follows from (2.10) and (2.11) that

(2.12)

which together with Lemma 2.2 implies that

(2.13)

Let be equipped with the norm . Then is a Banach space. If we denote

(2.14)

then it is easy to see that is a cone in . Now, we define an operator on by

(2.15)

Obviously, if is a fixed point of , then is a monotone nonnegative solution of the BVP (1.3).

Lemma 2.5.

is completely continuous.

Proof.

First, by Lemma 2.4, we know that .

Next, we assume that is a bounded set. Then there exists a constant such that for any . Now, we will prove that is relatively compact in . Suppose that . Then there exist such that . Let

(2.16)

Then for any , by Lemma 2.2, we have

(2.17)

which implies that is uniformly bounded. At the same time, for any , in view of Lemma 2.3, we have

(2.18)

which shows that is also uniformly bounded. This indicates that is equicontinuous. It follows from Arzela-Ascoli theorem that has a convergent subsequence in . Without loss of generality, we may assume that converges in . On the other hand, by the uniform continuity of , we know that for any , there exists such that for any with , we have

(2.19)

Let . Then for any , with , we have

(2.20)

which implies that is equicontinuous. Again, by Arzela-Ascoli theorem, we know that has a convergent subsequence in . Therefore, has a convergent subsequence in . Thus, we have shown that is a compact operator.

Finally, we prove that is continuous. Suppose that and . Then there exists such that for any , . Let

(2.21)

Then for any and , in view of Lemmas 2.2 and 2.3, we have

(2.22)

By applying Lebesgue Dominated Convergence theorem, we get

(2.23)

which indicates that is continuous. Therefore, is completely continuous.

3. Main Results

For convenience, we define

(3.1)

Theorem 3.1.

If , then the BVP (1.3) has at least one monotone positive solution.

Proof.

In view of , there exists such that

(3.2)

By the definition of , we may choose so that

(3.3)

Let . Then for any , in view of (3.2) and (3.3), we have

(3.4)

By integrating the above inequality on , we get

(3.5)

which together with (3.4) implies that

(3.6)

On the other hand, since , there exists such that

(3.7)

By the definition of , we may choose , so that

(3.8)

Let . Then for any , in view of (3.7) and (3.8), we have

(3.9)

which implies that

(3.10)

Therefore, it follows from (3.6), (3.10), and Theorem 1.1 that the operator has one fixed point , which is a monotone positive solution of the BVP (1.3).

Theorem 3.2.

If , then the BVP (1.3) has at least one monotone positive solution.

Proof.

The proof is similar to that of Theorem 3.1 and is therefore omitted.

Theorem 3.3.

If for and , then the BVP (1.3) has no monotone positive solution.

Proof.

Suppose on the contrary that is a monotone positive solution of the BVP (1.3). Then and for , and

(3.11)

By integrating the above inequality on , we get

(3.12)

which together with (3.11) implies that

(3.13)

This is a contradiction. Therefore, the BVP (1.3) has no monotone positive solution.

Similarly, we can prove the following theorem.

Theorem 3.4.

If for and , then the BVP (1.3) has no monotone positive solution.

Example 3.5.

Consider the following BVP:

(3.14)

Since and , if we choose , then it is easy to compute that

(3.15)

which shows that

(3.16)

So, it follows from Theorem 3.1 that the BVP (3.14) has at least one monotone positive solution.