Generalized Asymptotically Almost Periodic and Generalized Asymptotically Almost Automorphic Solutions of Abstract Multiterm Fractional Differential Inclusions

Abstract

The main aim of this paper is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives. We illustrate our abstract results with several examples and possible applications.

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.…”

Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions

“…In [28], G. M. N'Guérékata and M. Kostić have recently studied various classes of generalized almost periodic and generalized almost automorphic solutions of abstract multi-term fractional differential inclusions in Banach spaces. We close the paper with the observation that Proposition 4.7 can be applied in the qualitative analysis of the abstract multi-term Cauchy inclusion u(·) − g ζ+1+i * f (·)Cx + n−1 j=1 c j g αn−αj * u(·) − g ζ+1+i * f (·)Cx + j∈Nn−1\Di c j g αn−αj +i+ζ+1 * f (·)Cx ∈ A g αn−α * u (·), where 0 ≤ α 1 < · · · < α n , 0 ≤ α < α n , x ∈ X, C ∈ L(X) is injective, c j ∈ C for 1 ≤ j ≤ n − 1, CA ⊆ AC, f ∈ L 1 loc ([0, ∞) : X) and the set D i has the same meaning as in [28] (0 ≤ i ≤ ⌈α n ⌉ − 1).…”

Almost Periodic Functions and Their Applications: A Survey of Results and Perspectives

“…ere is a vast amount of articles in the existing literature which consider almost automorphic-type solutions for various classes of integrodifferential equations. Let us only mention our analysis (the joint work of the second-named author with Prof. Guérékata [74]) of the following abstract multiterm fractional differential inclusion:…”