Fixed points and some qualitative properties of neutral functional differential equations, neutral functional difference equations, and dynamic equations on time scale

Abstract

This thesis is concerned with the qualitative properties of solutions of neu- tral functional differential equations, neutral functional difference equations and dy- namic equations on time scale. Some of the equations are of the first and second order whereas some are systems of equations. All these equations are delay equa- tions with constant or variable delays. Fixed point theory is used extensively in this thesis to investigate the qualita- tive properties of solutions of neutral delay equations. In particular, the Krasnosel- skii’s fixed point theorem, the Krasnoselskii-Burton fixed point theorem and the Banach’s fixed point theorem are used in the thesis. We invert the equations and the results of the inversions are used to define suitable mappings which are then used to discuss the qualitative properties of solutions to certain classes of neutral functional equations considered. Sufficient conditions are established to discuss the qualitative properties such as periodicity, positivity, and stability of the classes of neutral equations of our focus