Vector-valued almost automorphic distributions and vector-valued almost automorphic ultradistributions

Abstract

In this paper, we introduce the notions of a vector-valued almost automorphic distribution and a vector-valued almost automorphic ultradistribution, working in the framework of complex Banach spaces. We prove several structural characterizations for the introduced classes.

Asymptotically almost periodic and asymptotically almost automorphic vector-valued generalized functions

“…We refer the reader to [5], [10] and [25] for the basic results about vector-valued almost periodic distributions and vector-valued almost automorphic distributions. Let 1 ≤ p ≤ ∞.…”

“…This follows from the fact that this is true for the space B ′ * AP (X), resp. B ′ * AA (X) (see [24] and [25]), as well as that, for every Q ∈ B ′ * +,0 (X) and for every ultradifferential operator P (D) of * -class, we have…”

“…Suppose that f i (•) = g i (•)+ q i (•) on [0, ∞), where f i ∈ AP (R : X) and g i ∈ C 0 ([0, ∞) : X) (i = 1, 2). Then T 1 := P (D)g 1 + g 2 is an almost periodic ultradistribution of Beurling class due to [24, Theorem 2] (see [25,Theorem 3.1] for almost automorphic case), and all that we need to show is…”

“…In our recent joint research study with D. Velinov [25], we have analyzed the notions of an almost automorphic distribution and an almost automorphic ultradistribution in Banach space; the notion of an almost periodic ultradistribution in Banach space has been recently analyzed by M. Kostić [24] within the framework of Komatsu's theory of ultradistributions, with the corresponding sequences not satisfying the condition (M.3); see also the papers by I. Cioranescu [11] and M. C. Gómez-Collado [17] for first results in this direction. As mentioned in the abstract, the main aim of this paper is to introduce the notions of an asymptotically almost periodic ultradistribution and asymptotically almost automorphic ultradistribution in Banach space, as well as to provide some applications in the qualitative analysis of vector-valued distributional and vector-valued ultradistributional solutions to systems of ordinary differential equations (the notions of an asymptotically almost periodic ultradistribution seem to be not considered elsewhere even in scalar-valued case, while the notion of a vector-valued almost automorphic distribution seems to be completely new, as well).…”

Vector-Valued Almost Periodic Ultradistributions and Their Generalizations

“…The main aim of this paper is, actually, to reexamine the structural results proved in [11], [4] and [16]. The concept of almost automorphic ultradistributions will be introduced and analyzed in our follow-up research with S. Pilipović and D. Velinov [24] (cf. C. Bouzar, M. T. Khalladi, F. Z. Tchouar [5] for the notion of an almost automorphic Colombeau generalized function, C. Bouzar, Z. Tchouar [6] for the notion of an almost automorphic distribution, C. Bouzar, M. T. Khalladi [7] for the notion of an almost periodic Colombeau generalized function, and M. F. Hasler [19] for the notion of a Bloch-periodic Colombeau generalized function).…”