Electron. J. Diff. Equ., Vol. 2010(2010), No. 139, pp. 1-9.

Solvability of a three-point nonlinear boundary-value problem

Assia Guezane-Lakoud, Smail Kelaiaia

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.

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Assia Guezane-Lakoud
Department of Mathematics, Faculty of Sciences
University Badji Mokhtar, B.P. 12, 23000, Annaba, Algeria
email: a_guezane@yahoo.fr
  Smail Kelaiaia
Department of Mathematics, Faculty of Sciences
University Badji Mokhtar, B.P. 12, 23000, Annaba, Algeria
email: kelaiaiasmail@yahoo.fr

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