Journal of Applied Mathematics

Volume 2013 (2013), Article ID 782363, 7 pages

http://dx.doi.org/10.1155/2013/782363

Research Article

Solvability of a Fourth-Order Boundary Value Problem with Integral Boundary Conditions

Hui Li, Libo Wang, and Minghe Pei

Department of Mathematics, Beihua University, Jilin City 132013, China

Received 21 December 2012; Accepted 13 February 2013

Academic Editor: Meng Fan

Copyright © 2013 Hui Li 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 investigate the existence of solutions and positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form , , , , , where , . By using a fixed point theorem due to D. O'Regan, the existence of solutions and positive solutions for the previous boundary value problems is obtained. Meanwhile, as applications, some examples are given to illustrate our results.

1. Introduction

It is well known that fourth-order boundary value problems (BVPs) arise in a variety of different areas of the flexibility mechanics and engineering physics and thus have been extensively studied; for instance, see [1–29] and references therein. Boundary value problems with integral boundary conditions appear in heat conduction, thermoelasticity, chemical engineering underground water flow, and plasma physics; see [12, 14, 21, 24, 26, 29] and references therein.

Motivated by the previous works and [30], in this paper, we consider fully nonlinear fourth-order differential equation

subject to the integral boundary conditions

as well as its simplified form

with the integral boundary conditions

where and are continuous functions.

We notice that if in problems (1), (2) and (3), (4), then the models are known as the one endpoint simply supported and the other one sliding clamped beam. The study of this class of problems was considered by some authors via various methods; we refer the reader to the papers [2, 5, 8, 11, 22].

The aim of this paper is to establish the existence results of solutions and positive solutions for problems (1), (2) and (3), (4), respectively. By positive solution, we mean a solution such that for . Our main tool is the fixed point theorem due to D. O'Regan [31].

2. Preliminary

In this section, we present some lemmas which are needed for our main results.

Let denote the Banach space of real-valued continuous functions on with the norm . is the Banach space of times continuously differentiable functions defined on , with the norm .

Throughout this paper, we always assume that (or ) and (or ) are continuous.

We consider a priori bound of solutions of the following one-parameter family of boundary value problem:

where . Simple computations lead to the following lemma.

Lemma 1. BVP (5), (6) with has only the trivial solution, and the corresponding Green function exists and is given by

Now, define a linear operator by

Then, we can easily show that is a Fredholm operator with index zero, and its inverse is given by

where .

Define Nemytskii operators induced by as

and induced by as

Also, define an operator as

Simple computations yield the following lemma.

Lemma 2. BVP (5), (6) is equivalent to the abstract equation

in ; that is, is a solution of BVP (5), (6) if and only if is a solution of the integral equation

where .

Let us denote the operators , as

Then, can be written as

Now, we can easily give some properties of the Green function and by direct computation.

Lemma 3. Let be as in Lemma 1 and . Then,

(1), ;

(2), , , , ; (3), , , , .

Lemma 4. Suppose that

(i)for each fixed , is nondecreasing in and ;(ii)there exists a constant such that for , ,

(iii)there exist and nondecreasing continuous function such that for , and

for all .

Then, any solution of BVP (5), (6) satisfies

where .

Proof. Let us first show that

Note that if in (5) and (6), then BVP (5), (6) has only the trivial solution, and thus (23) holds. Hence, we may assume that . Suppose now that (23) is not true. Then, there exists such that . Let

Then, , and from , we have

It is easy to see that . In fact, if , then . From (6) and (iii), it follows that for some ,

which is a contradiction, and thus . Furthermore, by definition of and (6), we have . Hence, from assumptions (i) and (ii) and (25), we have

We may assume that ; then, . Thus, by the continuity of on , there exists such that for . Since , it follows that for ; namely, is decreasing on , which contradicts the fact that attains its positive maximum value at . In summary, inequality (23) is true, which implies from that

This completes the proof of the lemma.

Remark 5. In Lemma 4, if condition (i) is replaced by

(i′), there exists a constant such that whenever and all ,

then, the conclusion of Lemma 4 remains true.

The following fixed point result due to D. O'Regan plays a crucial role.

Lemma 6 (see [31]). Let be an open set in a closed, convex set of a Banach space . Assume that , is bounded, and is given by , where is continuous and completely continuous and is a nonlinear contraction. Then, either

(A₁) has a fixed point in , or (A₂) there is a point and with .

3. Main Results

Firstly in this section, we state and prove our existence results of solutions for BVP (1), (2).

Theorem 7. Suppose that

(i)for each fixed , is nondecreasing in and ;(ii)there exists a constant such that for , ,

(iii)there exist and nondecreasing continuous function such that for , and

for all ;(iv) satisfies the Nagumo condition; that is, there exists a positive-valued continuous function on with such that

for all , where

Then, BVP (1), (2) has at least one solution.

Proof. Let be a possible solution of BVP (5), (6). We now show that

where and .

Suppose that (34) is not true. Then, there exists such that . Since , then there exists , () such that

Therefore, is positive or negative on by the continuity of the . Hence, from assumption (iv), the definition of , and Lemma 4, we can get the following contradiction:

Therefore, inequality (34) holds.

Let . It follows easily from the properties of the Green function and the continuity of that the operator is completely continuous.

We now show that is a nonlinear contraction. In fact, from assumption (iii), we have

Consequently, from Lemma 3, we have

Similarly, for all ,

Hence,

Since all possible solutions of BVP (5), (6) satisfy , it follows that there is no and such that . We conclude that (A₂) of Lemma 6 does not hold. Consequently, has a fixed point, which is a solution of BVP (1), (2). This completes the proof of the theorem.

Remark 8. In Theorem 7, if condition (i) is replaced by

(i′) there exists a constant such that whenever and all ,

then the conclusion of Theorem 7 remains true.

Remark 9. In Theorem 7, if and , then all the solutions of BVP (1), (2) are monotone and positive. This is clear because by Lemma 3 we have , and .

Next, we consider the existence of solutions and positive solutions for BVP (3), (4).

Theorem 10. Suppose that

(i)there exists being continuous and nondecreasing such that

(ii)there exists such that

(iii)there exists such that

where .

Then, BVP (3), (4) has at least one solution.

Proof. It is easy to see that is a solution of BVP (3), (4) if and only if is a solution of the integral equation (14) with . Moreover, is completely continuous, and is a nonlinear contraction.

It follows from (15), (i), and Lemma 3 that ,

Also, (16), (ii), and Lemma 3 yield

From (14), (45), and (46), we have that all possible solutions of satisfy

Let . Then, is open in , , and is bounded. Suppose that and satisfy . Then, , and (i) and (47) lead to

that is,

which contradicts (iii). Hence, (A₂) of Lemma 6 does not hold, and consequently has a fixed point which is a solution of BVP (3), (4). This completes the proof of the theorem.

Remark 11. In Theorem 10, if and , then all the solutions of BVP (3), (4) are monotone and positive.

Now, we consider BVP (3), (4) with linear boundary conditions as

where . Define

Then,

Theorem 12. Suppose that

(i);(ii)there exists being continuous and nondecreasing such that

Then, the nonlinear fourth-order differential equation (3) with boundary conditions

has at least one solution.

Proof. Notice that the existence of solutions of BVP (3), (54) is equivalent to the existence of fixed points of operator equation

As a linear operator on , from (52) and (i), we get , which implies that is invertible and its inverse is given by

Hence, we see from (55) that is a solution of BVP (3), (54) if and only if is a fixed point of the completely continuous operator .

Let us show that there exists such that any solution of operator equation () satisfies . In fact, any solution of () satisfies

and, hence, by (ii) and Lemma 3, we have

The condition implies that there exists such that for all ; that is,

Comparing (58) and (59), we see that .

Now, let . By the Leray-Schauder continuation theorem has a fixed point in , which is a solution of BVP (3), (54). This completes the proof of the theorem.

Remark 13. In Theorem 12, if and , then all the solutions of BVP (3), (54) are monotone and positive since all the solutions of BVP (3), (54) satisfy

and thus

Finally, we give some examples to illustrate our results.

Example 14. Consider the fourth-order boundary value problem

Let

It is easy to check that all the assumptions in Theorem 7 are satisfied. Hence, BVP (62), (63) has at least one solution.

Example 15. Consider the fourth-order boundary value problem

where .

Let

It is easy to check that all the assumptions in Theorem 10 and Remark 11 are satisfied. Hence, BVP (65), (66) has at least one monotone positive solution.

Example 16. Consider the fourth-order boundary value problem

where .

Let

It is easy to check that all the assumptions in Theorem 12 and Remark 13 are satisfied. Hence, BVP (68), (69) has at least one monotone positive solution.

Acknowledgment

This work was supported by NSFC (11126339).

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