On a study of Sobolev‐type fractional functional evolution equations

Abstract

Sobolev‐type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate the existence of mild solutions for fractional‐order time‐delay evolution equation of Sobolev type with multiorders in a Banach space with the help of two various methods. First, we study abstract problem by assuming the existence and compactness of E−1 and introduce a closed‐form representation of a mild solution via a newly defined delayed Mittag–Leffler‐type function which is generated by nonpermutable and permutable linear‐bounded operators. Our second method is based on the theory of Sobolev‐type delayed resolvent families generated by the pair (A, E) and a subordination principle, and in this case, any restrictive assumptions are not required on E. Furthermore, we derive an exact analytical representation of solutions for multidimensional fractional functional dynamical systems with nonpermutable and permutable matrices. As an example, the Sobolev‐type delayed wave equation with a damping term is provided to illustrate the applications of the abstract results.