Reduction methods for partial functional differential equations of retarded and neutral type-applications to the qualitative analysis of solutions

Abstract

This thesis is devoted to the investigation of the qualitative analysis of some dynamical systems governed by partial differential equations of retarded and neutral type. We study the asymptotic behavior of solutions, including the existence of periodic, almost periodic, and almost automorphic solutions. At the beginning, we study the existence of periodic solutions for partial differential equations of neutral type. We extend Massera criterion for nonhomogeneous equation. More precisely, we show that the existence of a bounded solution on [0, + ∞ [implies the existence of periodic solutions. In nonlinear case, we use the concept of ultimate boundedness to prove the existence of periodic solutions. The technique used here to obtain these results is the fixed point theorem for poincaré map. We continue our study by discussing the existence of almost periodic and almost automorphic solutions of this class of equations. We develop a new reduction principle, which reduces the complexity of the system. More precisely, we show that the dynamics of bounded solutions of partial differential equation of neutral type are governed by an ordinary differential equation in finite dimensional space, which gives a very powerful tool to deal with the quantitative analysis of solutions. As a consequence of this reduction principle, we show the existence of an almost periodic or an almost automorphic solution under minimal condition, namely, we prove that the existence of a bounded solution on [0, + ∞ [is enough to get an almost periodic or an almost automorphic solution. We discuss also the existence of periodic and almost periodic solutions for partial differential equations with infinite delay. We show that the ultimate boundedness of solutions implies the existence of periodic solutions, when the phase space is a uniform fading memory space. To achieve this goal, we prove that the poincaré map has a fixed point by using Horn’s fixed point theorem. For nonhomogeneous equation, we treat the existence of almost periodic solution under the stability concept. More precisely, we show that if the null solution of the homogeneous equation is BC-total stable then the considered system has an asymptotically almost periodic solution and consequently it has an almost periodic solution. To illustrate the abstract theoretical studies, we present at the end of each chapter some applications in the context of dynamical systems; we study some models arising in population’s dynamics and generally some natural phenomena.