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).