Résumé:
This thesis contains important findings in existence and uniqueness theory, as well as Ulam
stability results for certain types of sequential differential problems involving various forms of Caputo,
Caputo-Fabrizio, and Atangana-Baleanu derivatives. Boyd and Wong’s fixed point theorems, the sum
of weakly sequentially continuous mappings, and O’Regan’s fixed point theory are used to investigate
existence and uniqueness. We also evaluate stability using the Mittag-Leffler Hyers-Ulam Rassias
concept. Afterwards, explores several categories of nonlinear multi-delay differential equations, carried
by the appearance of novel generalized Caputo-style fractional integrals and variations. We explore
existence and uniqueness using Fixed Point Theory for Operators with the Volterra Property, which
includes the Step-by-Step Contraction Theorem and Burton and Kirk’s Theorem. To enhance our studies,
we include numerical evaluations for each provided issue, allowing for a more accurate comprehension
of the responses of the aforementioned derivatives. Based on the three-step Adams-Bashforth principles,
we create novel numerical techniques and algorithms. Multiple examples illustrate the usefulness
and robustness of this suggested method, overall the results displaying excellent accuracy and strong
agreement in most problems
