Résumé:
This thesis deals with operator theory and fractional calculus. We begin by introdu-
cing new generalized metric spaces. Then, we prove some important properties of such new
structures. We also apply xed point theory to establish some new results on contraction
operators over the introduced generalized spaces. After that, we investigate the fractional
calculus theory, where we will use integro-di¤erential operators as well as the di¤erential
ones. We prove some recent results on the existence and uniqueness of solutions for some
classes of fractional di¤erential equations and systems that have not been studied. Some
generalized Fourier and Laplace transforms are also proved. Some analytic and numerical
studies for fractional equations are also discussed.
