Abstract
We establish a class of degenerate fractional differential equations involving delay arguments in Banach spaces. The system endowed by a given background and the generalized Showalter–Sidorov conditions which are natural for degenerate type equations. We prove the results of local unique solvability by using, mainly, the method of contraction mappings. The obtained theory via its abstract results is applied to the research of initial-boundary value problems for both Scott–Blair and modified Sobolev systems of equations with delays.
Highlights
During the last decades, fractional differential equations and their potential applications have gained a lot of importance, mainly because fractional calculus has become a powerful tool with more accurate and successful results when modeling several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering [1]
Integer or fractional-order degenerate differential equations, i.e., evolution equations not solved with respect to the highest order derivative, are often used to describe various processes in science and engineering: in [9,10] certain classes of the time-fractional order partial differential equations with polynomials differential with respect to the spatial variables elliptic self-adjoint operator, which contain some equations from hydrodynamics and the filtration theory, are studied
We studied the local unique solvability of the problem with the generalized Showalter–Sidorov conditions, which is associated by a given background for degenerate fractional evolution equations in Banach spaces with delay, including the Gerasimov–Caputo derivative and a relatively bounded pair of linear operators
Summary
Laboratory of Functional Materials, South Ural State University (National Research University), Lenin Av. 76, Chelyabinsk 454080, Russia. Received: 31 August 2020; Accepted: 28 September 2020; Published: 3 October 2020
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