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.
