Positive solutions of some nonlocal fourth-order boundary value problem
Introduction
The deformation of an elastic beam in equilibrium state, whose two ends are simply supported, can be described by a fourth-order ordinary equation boundary value problem. Owing to its significance in physics, it has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point index theory and the method of upper and lower solutions, see for example, [1], [2], [3], [4], [6], [7], [8], [9], [10], [9], [10], [11], [12], [13], [14], [15], [16].
Recently, there has been much attention focused on the question of positive solution of fourth-order differential equation with one or two parameters. For example, Li [6] investigated the existence of positive solutions for the fourth-order boundary value problemunder the assumptions:
- (J1)
is continuous;
- (J2)
Chai [3] studied the above generalizing form with variable parameters. All of the results consider only an equation with two-point boundary value condition. More recently, Ma [9], [10] studied the existence of symmetric positive solutions of the nonlocal fourth-order boundary value problemMotivated by above papers we consider the following nonlocal boundary value problem
We assume the following conditions throughout
- (A1)
and
- (A2)
,
Section snippets
The preliminary lemmas
Set . Given , letBy , there hold . Denote by the Green’s functions of the following problemsThen, carefully calculation yieldDenote
The main results
Suppose are defined as in Section 2, we introduce some notations as follows: Theorem 3.1 Assume (A1), (A2) hold. Problem (1.3) has at least one positive solution, if one of the following cases holds
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