On a Third-Order Three-Point Boundary Value Problem

where η ∈ 0, 1 , α, β ∈ R, f ∈ C 0, 1 × R,R . The parameters α and β are arbitrary in R such that 1 2α 2βη − 2β / 0. Our aim is to give new conditions on the nonlinearity of f , then using Leray-Schauder nonlinear alternative, we establish the existence of nontrivial solution. We only assume that f t, 0 / 0 and a generalized polynomial growth condition, that is, there exist two nonnegative functions k, h ∈ L1 0, 1 such that ∣f t, x ∣∣ ≤ k t |x| h t , ∀ t, x ∈ 0, 1 × R, 1.3


Introduction
In this work, we study the existence of nontrivial solution for the following third-order three point boundary value problem BVP : The parameters α and β are arbitrary in R such that 1 2α 2βη − 2β / 0. Our aim is to give new conditions on the nonlinearity of f, then using Leray-Schauder nonlinear alternative, we establish the existence of nontrivial solution.We only assume that f t, 0 / 0 and a generalized polynomial growth condition, that is, there exist two nonnegative functions k, h ∈ L 1 0, 1 such that where p ∈ R , so our conditions are new and more general than 1 .

International Journal of Mathematics and Mathematical Sciences
Such problems arise in the study of the equilibrium states of a heated bar.Very recently, there have been several papers on third-order boundary value problems.Graef  This paper is organized as follows.First, we list some preliminary materials to be used later.Then in Section 3, we present and prove our main results which consist in existence theorems.We end our work with some illustrating examples.

2.2
Proof.Integrating u t −y t over the interval 0, t , we see that

2.3
The constants A, B, and C are given by the three-point boundary conditions 1.2 .
We define the integral operator

2.4
International Journal of Mathematics and Mathematical Sciences 3 By Lemma 2.1, the BVP 1.1 -1.2 has a solution if and only if the operator T has a fixed point in E. By Ascoli-Arzela theorem, we prove that T is a completely continuous operator.Now we cite the Leray-Schauder as nonlinear alternative.Lemma 2.2 see 14 .Let F be a Banach space and Ω a bounded open subset of F, 0 ∈ Ω.Let T : Ω → F be a completely continuous operator.Then, either there exists x ∈ ∂Ω, λ > 1 such that T x λx, or there exists a fixed point x * ∈ Ω.

Main Results
In this section, we present and prove our main results.

International Journal of Mathematics and Mathematical Sciences
Let us define the following notation: Theorem 3.2.Under the conditions of Theorem 3.1 p ∈ R and if one of the following conditions is satisfied 1 there exist n > 1 and r > 1 such that 3 the functions k s and h s satisfy 3.17 Then the BVP 1.1 -1.2 has at least one nontrivial solution u * ∈ C 0, 1 , R .
Proof.Let M and N be defined as in the proof of Theorem 3.1.To prove Theorem 3.2, we only need to prove that M < 1/2 and N < 1/2.
1 By using H ölder inequality, we get Integrating, using 3.8 , and remarking that a > 1, we arrive at M < 1/2.Using H ölder inequality a second time, we get

3.19
Integrating and then using 3.9 , we arrive at N < 1/2. 2 Taking into account 3.10 , it yields

3.20
On the other hand, using 3.11 , we obtain 3 Using the same reasoning as in the proof of the second statement, we prove the third statement.
c 2 , 3.22 International Journal of Mathematics and Mathematical Sciences choosing ε ω 2 , then |f t, x | ≤ 2ω 2 , for all x ∈ R\ − c 2 , c 2 ; consequently, Now applying the third statement, we achieve the proof of Theorem 3.2.Applying the first statement of Theorem 3.2 for p 5, n r 2, to get Then the BVP 3.26 has at least one nontrivial solution u * in C 0, 1 .