Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

Donal O’Regan, Ravi P. Agarwal, Snezhana Hristova () and Mohamed I. Abbas
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Donal O’Regan: School of Mathematical and Statistical Sciences, University of Galway, H91 TK33 Galway, Ireland
Ravi P. Agarwal: Department of Mathematics, Texas A&M University-Kingsville, Kingsville, TX 78363, USA
Snezhana Hristova: Faculty of Mathematics and Informatics, Plovdiv University “P. Hilendarski”, 4000 Plovdiv, Bulgaria
Mohamed I. Abbas: Department of Mathematics and Computer Science, Faculty of Science, Alexandria University, Alexandria 21511, Egypt

Mathematics, 2024, vol. 12, issue 2, 1-16

Abstract: An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

Keywords: variable-order fractional differential equations; generalized proportional Caputo fractional derivatives; Hyers–Ulam stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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