ONE-PARAMETER OPERATOR SEMIGROUPS AND AN APPLICATION OF DYNAMICAL SYSTEMS

Abstract

This thesis consists of two parts. In the first part, which is expository, abstract theory of one-parameter operator is studied semi-groups. We develop in detail the necessary Banach space and Banach algebra theories of integration, differentiation, and series, and then give a careful rigorous proof of the exponential function characterization of continuous one-parameter operator semigroups. In the second part, which is applied and has new result, we discuss some related topics in dynamical systems. In general the linearizations give a reliable description of the non-linear orbits near the equilibrium points (the Hartman-Grobman theorem), thus illustrating the importance of linear semigroups. The aim of qualitative analysis of differential equations (DE) is to understand the qualitative behaviour (such as, for example, the long-term behaviour as $t\rightarrow \infty$) of typical solutions of the DE. The flow in the direction of increasing time defines a semigroup. As an application we study Einstein-Aether Cosmological models using dynamical systems theory.