**Amar Debbouche 1,\* and Vladimir E. Fedorov 2,3,4**


Received: 31 August 2020; Accepted: 28 September 2020; Published: 3 October 2020

**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.

**Keywords:** Gerasimov–Caputo fractional derivative; differential equation with delay; degenerate evolution equation; fixed point theorem
