On Solvability of Neutral and Nonneutral Differential Equations with Variable Delay

Abstract

In this thesis, we discuss some nonlinear differential equations and some of its qualitative properties. In Chapter 1, the existence of at least one continuous solution for a nonlinear neutral differential equation via Chandrasekhar integral has been proved by the Banach fixed point theorem. Continuous dependence of the solution on the delay functions has been studied. Using the Krasnoselskii fixed point theorem, the existence of solutions and upper and lower bound estimates for solutions defined in an unbounded interval are established. Some particular cases are presented to illustrate our results. In Chapter 2, the solvability of a class of implicit fractional differential equations (IFDE for short) has been established. Also, stability of Ulam Hyers and Ulam-Hyers-Rassia-types has been investigated. The existence and local attractivity of solutions for the IFDE have been proved in an infinite interval by using some fixed point theorem. Finally, some special cases are considered.