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

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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)

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.

Suggested Citation

  • Mohamed I. Abbas & Snezhana Hristova, 2021. "On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations," Mathematics, MDPI, vol. 9(21), pages 1-10, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2720-:d:665361
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    References listed on IDEAS

    as
    1. Tarasov, Vasily E. & Tarasova, Valentina V., 2018. "Macroeconomic models with long dynamic memory: Fractional calculus approach," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 466-486.
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