ÉTUDE ANALYTIQUE DES ÉQUATIONS DIFFÉRENTIELLES FRACTIONNAIRES ET APPLICATIONS

Abstract

The theory of fractional di¤erential equations plays an important role in the modeling of many processes physical, technological and biological. In recent years, a great attention has been focused on the study of the existence and uniqueness of solutions for the fractional di¤erential equations. In addition, considerable attention has recently been given for the study of the Ulam type stabilities for such fractional di¤erential equations. The main objective of this thesis is to complete the content of other fractional calculus works in these two areas of research, we obtain several results on the existence, unique- ness and Ulam-Hyers stability and the generalized Ulam-Hyers stability for the fractional nonlinear equations. The results obtained are based on the techniques of xed point. First, we presented a new existence and uniqueness results of solutions for a nonlinear fractional system. Thus, other results ensuring the existence of a solution at least for the dealt fractional problem are constructed. Some examples are built to illustrate the results. For the multiple fractional systems of dimension n, we introduced a new class using the approach of Caputo. After establishing the conditions for the existence and uniqueness of solutions for such fractional problems, some new results of the existence and uniqueness have proven. Also, other results ensuring the existence of a solution at least for the considered fractional problem are presented. To illustrate the main results, some illustrative examples were given. In addition, the proposed results for fractional problems have been extended in the case where the fractional problems are singular. Some original studies on the Ulam-Hyers stability for such fractional problem class are also presented.