On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations

Mohamed I. Abbas and Snezhana Hristova
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Mohamed I. Abbas: Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria 21511, Egypt
Snezhana Hristova: Faculty of Mathematics and Informatics, Plovdiv University, 4000 Plovdiv, Bulgaria

Mathematics, 2021, vol. 9, issue 21, 1-10

Abstract: A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alternative and the Banach contraction principle, the main results are established. In the case of a local Lipschitz right hand side part function, the existence of a bounded mild solution is proved. Some examples illustrating the main results are provided.

Keywords: vector-order fractional derivatives; generalized proportional fractional derivatives; Leray–Schauder nonlinear alternative (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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