Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions

Author:

Kiskinov Hristo1ORCID,Milev Mariyan2ORCID,Petkova Milena1ORCID,Zahariev Andrey1ORCID

Affiliation:

1. Faculty of Mathematics and Informatics, University of Plovdiv, 4000 Plovdiv, Bulgaria

2. Faculty of Economics and Business Administration, Sofia University, 1504 Sofia, Bulgaria

Abstract

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP.

Funder

Bulgarian National Science Fund

EU-NextGeneration

Publisher

MDPI AG

Reference28 articles.

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