Fractional order differential equation with nonlocal integral condition

Document Type : Regular research papers

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Alexandria University

Abstract

This research paper focuses on investigating the solvability of the Riemann-Liouville
di erential equation with nonlocal integral condition, study the existence of solutions
in the class of continuous functions, we use the technique of the Schauder xed point
Theorem. We drive sucient conditions for a uniqueness and the continuous depen-
dence on some functions. Additionally, we delve into the study of the Hyers{Ulam
stability. Finally, we given an examples are provided to illustrate our results.The de nition of the fractional derivative of the Riemann-Liouville type played an important
role in the development of the theory of fractional derivatives and integrals and for
its applications in pure mathematics.
However, the demands of modern technology require a certain revision of the well-established
pure mathematical approach. Applied problems require de nitions of fractional derivatives
allowing the utilization of physically interpretable initial conditions.

where RD is the refers to the fractional derivative of Riemann{Liouville of order 2 (0; 1).
Our aim here is study the existence of solutions x 2 C(I). Moreover, the continuous dependence
of the unique solution on the x0 and on the functions f, g and  will be proved.
The Hyers { Ulam stability of the problem will be given.

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