Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Abstract

This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations D ( ) = ( ) + D −1 ( , ( ), ( )), ∈ R, where 1 < < 2, is a linear densely defined operator of sectorial type on a complex Banach space and is a bounded linear operator defined on , is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

Multi-dimensional almost periodic type functions and applications

“…We would like to note that the statements of [55,Lemma 17,Theorem 18], concerning the existence and uniqueness of almost automorphic solutions of the problem (3.9), can be straightforwardly reformulated for almost periodicity by replacing the assumptions (H4) and (H5) with the corresponding almost periodicity assumptions as well as by assuming that the function Γ(t, s) from the condition (H3) of this paper is (R, B)-almost periodic with R being the collection of all sequuences in A := {(a, a) : a ∈ R} and X ∈ B. Similarly, the statements of [55, Lemma 20,Theorem 21], concerning the existence and uniqueness of almost automorphic solutions of the problem (3.10), can be straightforwardly reformulated for almost periodicity; see also [116,Theorem 26,Theorem 27], where the same comment can be given and the recent result of J. Cao, Z. Huang and G. M. N'Guérékata [23,Theorem 3.6], where a similar modification of condition (H3) for bi-almost periodicity on bounded subsets can be made.…”

Almost Automorphic Solutions in the Sense of Besicovitch to Nonautonomous Semilinear Evolution Equations

“…Thus, (H 3 ) is verified. Moreover, U(t, s) ≤ e −2(t−s) for t ≥ s.By[14], we see that A(t) satisfies conditions (S 1 ) and (S 2 ).…”

Asymptotically almost automorphy for impulsive integrodifferential evolution equations with infinite time delay via Mönch fixed point

“…Since its inception, the theory of almost automorphic functions has undergone various developments and found applications in the study of ordinary differential equations, partial differential equations, functional differential equations, integro-differential equations, fractional differential equations, and stochastic differential equations. Notable contributions and references to this extensive body of work include [15,14,13,24,25,20,34,38,40,41,48,51]. Furthermore, the concept of almost automorphy has witnessed intriguing and powerful generalizations over time.…”

“…For example, N'Guérékata [39] introduced the notion of asymptotically almost automorphic functions, which has found numerous applications in the theory of differential equations. For additional results on this topic, readers can explore [1,13,21,22,32,44] and related references. For a comprehensive overview of the recent theory and applications of asymptotically almost automorphic functions, readers are directed to the monograph by N'Guérékata [42].…”

“…The proof of the above lemma is similar to the proof of Lemma 2.5 of [13]. Now, we introduce the Kuratowski measure of noncompactness β Z defined by β Z (S) = inf{ κ > 0 : S has a finite cover by sets of diameter ≤ κ}, for a bounded set S in any space Z.…”