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A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. We study fractional evolution equations of order α ∈ (1 , 2) associated with the infinitesimal generator of an operator fractional cosine (sine) function generated by bounded time-dependent perturbations in a Banach space. We show that the fractional abstract Cauchy problem associated with the infinitesimal generator A of a strongly continuous fractional cosine (sine) function remains uniformly well-posed under bounded time-dependent perturbation of A. We also provide some necessary special cases by using the Laplace transform of the generators of the given operator families. [ABSTRACT FROM AUTHOR]

Perturbation theory has long been a very useful tool in the hands of mathematicians and physicists. The purpose of this paper is to prove some perturbation results for infinitesimal generators of fractional strongly continuous cosine families. That is, we impose sufficient conditions such that A is the infinitesimal generator of a fractional strongly continuous cosine family in a Banach space X , and B is a bounded linear operator in X , then A + B is also the infinitesimal generator of a fractional strongly continuous cosine family in X. Our results coincide with the classical ones when α = 2. Furthermore, depending on commutativity condition of linear bounded operators, we propose the elegant closed-form formulas for uniformly continuous perturbed fractional operator cosine and sine functions. Finally, we present an example in the context of one-dimensional perturbed fractional wave equation to demonstrate the applicability of our theoretical results and we give some comparisons with the existing literature. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A strong inspiration for studying Sobolev-type fractional evolution equations comes from the fact that have been verified to be useful tools in the modeling of many physical processes. We introduce a novel technique for solving Sobolev-type fractional evolution equations with multi-orders in a Banach space. We propose a new Mittag-Leffler-type function which is generated by linear bounded operators and investigate their properties which are productive for checking the candidate solutions for multi-term fractional differential equations. Furthermore, we propose an exact analytical representation of solutions for multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices. [ABSTRACT FROM AUTHOR]