Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 139, pp. 1-9.

Title: Solvability of a three-point nonlinear boundary-value problem

Authors: Assia Guezane-Lakoud (Univ. Badji Mokhtar, Annaba, Algeria)
         Smail Kelaiaia (Univ. Badji Mokhtar, Annaba, Algeria)

Abstract: 
 Using the Leray Schauder nonlinear alternative, we prove the
 existence of a nontrivial solution for the three-point
 boundary-value problem
 $$\displaylines{
 u''+f(t,u)= 0,\quad 0<t<1 \cr
 u(0)= \alpha u'(0),\quad u(1)=\beta u'(\eta ),
 }$$
 where $\eta \in (0,1)$, $\alpha ,\beta \in \mathbb{R}$,
 $f\in C([0,1] \times\mathbb{R},\mathbb{R})$.
 Some examples are given to illustrate the results obtained.
 
Submitted March 19, 2010. Published September 27, 2010.
Math Subject Classifications: 34B10, 34B15.
Key Words: Fixed point theorem; three-point boundary-value problem;
           non trivial solution.