Asymptotically almost periodic solutions of fractional relaxation inclusions with Caputo derivatives

Abstract

In the paper under review, we analyze asymptotically almost periodic solutions for a class of (semilinear) fractional relaxation inclusions with Stepanov almost periodic coefficients. As auxiliary tools, we use subordination principles, fixed point theorems and the well known results on the generation of infinitely differentiable degenerate semigroups with removable singularites at zero. Our results are well illustrated and seem to be not considered elsewhere even for fractional relaxation equations with almost sectorial operators.2010 Mathematics Subject Classification. 34G25, 47D03, 47D06, 47D99. Key words and phrases. Abstract semilinear Cauchy inclusions, Asymptotic almost periodicity, Stepanov asymptotic almost periodicity.

Impulsive Fractional Differential Inclusions and Almost Periodic Waves

“…Remark 4. The existence criteria provided by Theorem 1 also generalize the results in [11][12][13] considering impulsive perturbations, which is more natural and realistic, and, therefore, the new results offer an extended horizon for applications. It is worth noting that, if the inclusion (1) is without impulsive perturbations at some instantes or the impulsive function I k (.)…”

“…Due to their great relevance to reality and their numerous implementations, the almost periodicity is considered a very important qualitative property of solutions. However, the relevant results for fractional inclusions are few [11][12][13]. The main goal of the present paper is to contribute to the development of this area.…”

“…However, the concept of almost periodic solutions (waves) for fractional-order differential inclusions has just begun to be studied [11][12][13]. Since the investigations into the existence of non-periodic solutions and their properties are of great significance for fractional-order inclusions, the theory of almost periodic waves should be further developed.…”