Résumé:
The objective of this work is to establish the existence, the uniqueness and the positivity of a solution for a boundary value problem generated by differential equation of third order, using the Leray-Schauder nonlinear alternative, the Vanach contraction principle and theGuo-Krasnosel'skii fixed point theorem. The memory consists of an introduction and three chapters. The first chapter reminds afew basics of functional analysis that will be used later. In the second chapter, we will present some results of fixed point theory, such as: the fixed point theorem of Banach, Brouwer, Schauder and finally, we will discuss the Guo-Krasnosel'skii fixed point theorem. In the last chapter, we studied the existence, uniqueness and positivity of the solution for a third order boundary value problem, where the boundary conditions are imposed in three points. The obtained results are illustrated by examples. We conclude this memory by a bibliography.
