EXISTENCE OF SOLUTIONS AND APPROXIMATE CONTROLLABILITY OF SOME EVOLUTION EQUATIONS

Abstract

The work presented in this thesis deals with the investigation of the existence of mild solutions and approximate controllability of some fractional and integer order di erential and integro-di erential equations. To prove our results, we will use semigroup theory, evolution system, xed point theorems, fractional calculus, measure of noncompactness, basic theory of functional analysis, and monotone iterative technique. The present work consists of the following eight chapters. Chapter 1 contains a brief introduction to the problems which are discussed in the consecutive chapters and provides a motivational background to study the problems which are discussed in this thesis. Further, it contains a review of relevant literature and an outline of the thesis. Chapter 2 contains some basic concepts of fractional calculus, functional analysis, semigroup theory and measure of noncompactness that will be required in the subsequent chapters. Chapter 3 concerns with the study of a fractional nonlocal neutral integrodi erential equation having ux type integral boundary conditions. The existence and uniqueness results are proved by using Banach and Leray-Schauder nonlinear alternative xed point theorems. Chapter 4 contains fractional integro-di erential equations having non-instantaneous impulses. The existence result is obtained by the help of xed point theorem and iii noncompact semigroup. Chapter 5 consists of fractional nonlocal semilinear integro-di erential equations having impulsive conditions for which the impulses are not instantaneous. The approximate controllability is proved via semigroup theory, Kuratowski measure of noncmpactness and -set contractive xed point technique, without imposing the condition of Lipschitz continuity on nonlinear term as well as the condition of compactness on impulsive functions and nonlocal function. Chapter 6 contains deformable fractional di erential equations. The results of existence and approximate controllability are obtained via semigroup theory, Schauder and Banach xed point technique. Chapter 7 considers non-autonomous semilinear di erential equations having nonlocal conditions. The existence and uniqueness are obtained via monotone iterative method with the upper and lower solutions in an ordered complete norm space, using evolution system and measure of noncompactness. Chapter 8 extends the results of chapter 7 for non-autonomous integro-di erential equations having nonlocal conditions. The relevant references are appended at the end.