Contribution a l’Etude de Certains Problèmes d’Evolution

Abstract

This thesis is devoted to the existence of mild solutions for different types of second-order non-autonomous differential equations. More precisely, we study the conditions that guarantee the existence of mild solutions for a non-autonomous differential equation of second order with bounded and unbounded intervals and with local and non-local initial conditions. We also study second order non-autonomous differential equation with infinite delay. We discuss the existence of mild solutions for evolution equations of the second-order neutral type. Finalement we study the problem of functional inclusions of evolution. Our approach is based on the fixed point theorem combined with the non-compactness technique. The latter is often used in several branches of non-linear analysis. Especially this technique has proved that it is a very useful tool in the existence of solutions of several types of differential equations and integral equations.