ON CERTAIN SINGULAR BOUNDARY VALUE PROBLEMS AND INTEGRAL INEQUALITIES

Abstract

In this thesis, we are concerned with some classes of differential equations that are singular on the time axis. With the help of some sophisticated means of functional analysis and fractional calculus, like for instance, the integral inequalities theory which are very present, the fractional derivatives, the operator theory as well as the xed point theory and the well known Runge Kutta method, we study the questions of existence of solutions, the existence and uniqueness, the analysis of stabilities in the sense of Ulam-Hyers. We also present some numerical simulations on Caputo derivatives to study the second problem that is presented in this thesis. In particular, we are concerned, rst, with a more general singular problem which is combined with some sequential notions with n Caputo derivatives. Some of the above questions are studied and several examples are illustrated. Also, we study a class of singular differential equations involving fractional calculus and convergent series. Especially, we study the question of existence and uniqueness of solutions by using both xed point theory and integral inequalities. Then, we pass to study the question of stability of solutions in the sense of Ulam-Hyers. Some examples are presented in this part. At the end, we investigate the question of approximations of solutions by using some recent results on Caputo approximations and Rung Kutta numerical Method.