Stability And Controllability Of Fractional Neutral Differential Equations

Fractional neutral differential equations is an emerging topic as the time delay phenomenon is most important and certain.However,the research on delay-related fractional differential equations is relatively exiguous.As almost all changes depend on the present and past status of the system,therefore in this dissertation we study time delay fractional differential equations.By applying Leray-Schauder fixed point theorem,contraction mapping principle,Arzela-Ascoli theorem and hybrid fixed point theorem for the sum of three operators etc.to investigate the fixed point property and existence of solutions.Controllability plays a major role in the development of modern mathematical control theory.Mainly,the problem of controllability of dynamical systems is widely used in analysis and the design of control systems.So by using Laplace transform,Mittag-Leffler matrix function and the iterative technique we establish the sufficient conditions for controllability of fractional dynamical systems with different order implicit fractional derivatives.There are six chapters in this dissertation.In Chapter 1,we introduce briefly some background and motivation as well as summarize the main results of this dissertation.In Chapter 2,we discuss the existence of a solution and then apply some sufficient conditions to obtain the unique positive solution for a class of nonlinear fractional neutral differential equations involving Caputo derivative.Banach contraction principle,Arzela-Ascoli theorem and Leray-Schauder theorems are used to study the fixed point property and existence of a solution.We establish local generalized Ulam-Hyers stability and local generalized Ulam-Hyers-Rassias stability for the same class of nonlinear fractional neutral differential equation.An example is also given to show the applicability of our results.This chapter is based on the paper[78],which is under review in FILOMAT.In Chapter 3,by using Picard operator we derived Ulam-Hyers stability,Ulam-Hyers-Rassias stability,generalized Ulam-Hyers-Rassias stability,and Ulam-Hyers-Mittag-Leffler stability results for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivative.Afterward we study two existence and uniqueness results with respect to Chebyshev and Bielecki norms for this class of differential equations.Finally,examples are provided to illustrate our results.This chapter is based on the papers[76]and[79].In Chapter 4,we construct the existence and uniqueness of the solution of the boundary value problems for a class of nonlinear fractional functional differential equations with delay involving Caputo fractional derivative.Our work relies on Schauder fixed point theorem and contraction mapping principle in a cone.Examples are also included to show the applicability of our results.This chapter is based on the paper[77].In Chapter 5,by using hybrid fixed point theorem for the sum of three operators,existence result for initial value problems for a class of hybrid fractional neutral differential equations are derived.We study uniqueness and data dependence of the solution.An example is given to illustrate the results.This chapter is based on the paper[80].In Chapter 6,we extrapolate the controllability of nonlinear fractional neutral dynamical systems with different order implicit fractional derivatives.The formula for a solution is presented which is derived by using the Laplace transform.Sufficient conditions are established for controllability using the Mittag-Leffler matrix function and the iterative technique.An example is included to show the applicability of results.This chapter is based on the paper[75].