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

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


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  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.
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 () > 0 for  ∈ (0, 1].Our main tool is the fixed point theorem due to D. O'Regan [31].

Preliminary
In this section, we present some lemmas which are needed for our main results.
Throughout this paper, we always assume that  : We consider a priori bound of solutions of the following one-parameter family of boundary value problem: where 0 ≤  ≤ 1. Simple computations lead to the following lemma.
Let us denote the operators  1 ,  2 as Then,  −1  can be written as Now, we can easily give some properties of the Green function (, ) and () by direct computation.
Remark 5.In Lemma 4, if condition (i) is replaced by (i  ), there exists a constant  > 0 such that whenever 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 0 ∈ , () is bounded, and  :  →  is given by  =  1 +  2 , where  1 :  →  is continuous and completely continuous and  2 :  →  is a nonlinear contraction.Then, either (A 1 )  has a fixed point in , or (A 2 ) there is a point  ∈  and  ∈ (0, 1) with  = ().

Main Results
Firstly in this section, we state and prove our existence results of solutions for BVP (1), (2).
Remark 8.In Theorem 7, if condition (i) is replaced by (i  ) there exists a constant  > 0 such that whenever | 2 | >  and all (,  0 ,  1 ) ∈ then the conclusion of Theorem 7 remains true.
Next, we consider the existence of solutions and positive solutions for BVP (3), (4).
Remark 11.In Theorem 10, if  ≥ 0 ≥ ℎ and (, 0) ̸ ≡ 0, then all the solutions of BVP (3), ( 4 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 [0, 1], from (52) and (i), we get || 2 || =  0 < 1, which implies that  −  2 is invertible and its inverse is given by 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.