Existence of Solution for Nonlinear Fourth-order Three-point Boundary Value Problem

: In this paper, we study the existence of solution for the fourth-order three-point boundary value problem having the following form where η ∈ (0 , 1), α,β ∈ R , f ∈ C ([0 , 1] × R , R ), and f ( t, 0) 6 = 0. We give suﬃcient conditions that allow us to obtain the existence of solution. And by using the Leray-Schauder nonlinear alternative we prove the existence of at least one solution of the posed problem. As an application, we also given some examples to illustrate the results obtained.


Introduction
The study of fourth-order three-point boundary value problems (BVP) for ordinary differential equations arise in a variety of different areas of applied mathematics and physics.
Many authors studied the existence of positive solutions for nth-order m-point boundary value problems using different methods such that fixed point theorems in cones, nonlinear alternative of Leray-Schauder, and Krasnoselskii's fixed point theorem, see ( [2,3,4,5]) and the references therein.
For Some other results on fourth-order boundary value problem, we refer the reader to the papers ( [10,11,12,13,14]).
Motivated by the above works, the aim of this paper is to establish some sufficient conditions for the existence of solution for the fourth-order three-point boundary value problem (BVP) u (4) (t) + f (t, u(t)) = 0, 0 < t < 1. (1.1) where η ∈ (0, 1), α, β ∈ R, f ∈ C([0, 1] × R, R), f (t, 0) = 0, and R = (−∞, +∞). This paper is organized as follows. In section 2, we present two lemmas that will be used to prove the results. Then, in section 3, we present and prove our main results which consists of existence theorems and corollary for nontrivial solution of the BVP (1.1) − (1.2), and we establish some existence criteria of at least one solution by using the Leray-Schauder nonlinear alternative. Finally, in section 4, as an application, we give some examples to illustrate the results we obtained.

Preliminaries
Let E = C([0, 1]) with the norm y = sup t∈[0,1] |y(t)| for any y ∈ E. A solution u(t) of the BVP (1.1) − (1.2) is called nontrivial solution if u(t) = 0. To get our results, we need to provide the following lemma.

) has a solution if and only if the operator
T has a fixed point in E. So we only need to seek a fixed point of T in E. By Ascoli-Arzela theorem, we can prove that T is a completely continuous operator. Now we cite the Leray-Schauder nonlinear alternative.
. Let E be a Banach space and Ω be a bounded open subset of E, 0 ∈ Ω. T : Ω → E be a completely continuous operator. Then, either (i) there exists u ∈ ∂Ω and λ > 1 such that T (u) = λu, or (ii) there exists a fixed point u * ∈ Ω of T .

Existence of solution
In this section, we prove the existence of a nontrivial solution for the BVP Proof. Let Existence of Solution for Nonlinear Fourth-order Three-point...
This completes the proof. ✷ Theorem 3.2. Suppose that f (t, 0) = 0, ζ > 0, and there exist nonnegative func- If one of the following conditions is fulfilled (1) There exists a constant p > 1 such that (2) There exists a constant µ > −1 such that Then the BVP (1.1) − (1.2) has at least one nontrivial solution u * ∈ E.
Proof. In this case, we have Proof of this Corollary 3.3 is the same method in the proof Theorem 3.2.
Proof. In this case, we have The rest procedure is the same as for Theorem 3.4. ✷

Examples
In order to illustrate the above results, we consider some examples.
Example 4.2. Consider the following problem