On Asymptotically Almost Periodic Generalized Solutions of Differential Equations

Asymptotic almost automorphy for algebras of generalized functions

“…As a by pass result, we give a Seeley type result on extention of functions in the context of the introduced generalized functions, this is needed in the proof of a fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. The papers [8] and [9] can be considered as consequences of this work. The paper aims also, as in [16], to lift a Frechet existence result of asymptotically almost automorphic solutions of differential equations to the level of neutral difference differential systems in the framework of G aaa .…”

“…In view of the result [23] on the impossibility of the multiplication of distributions, see [21] for more details, algebras of generalized functions containing spaces of Sobolev-Schwartz type distributions have been studied, see [14], [15], [1] and [20]. The concepts of almost periodicity and asymptotic almost periodicity as well as almost automorphy in the context of such algebras of generalized functions are introduced, studied and applied in the papers [6], [7] [8], [9] and [10]. So, the paper first introduces and studies a class of asymptotically almost automorphic generalized functions, denoted by G aaa .…”

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

“…The notion of a scalar-valued asymptotically almost periodic distribution has been introduced by I. Cioranescu in [12], while the notion of a vector-valued asymptotically almost periodic distribution has been considered by D. N. Cheban [9] following a different approach (see also I. K. Dontvi [14] and A. Halanay, D. Wexler [15]). The notion of a scalar-valued almost automorphic distribution and a scalarvalued almost automorphic Colombeau generalized function have been introduced by C. Bouzar, Z. Tchouar [5] and C. Bouzar, M. T. Khalladi [6]. Some contributions have been also given by B. Stanković [28]- [29].…”

“…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). In such a way, we expand and contemplate the results obtained in [5]- [6], [9], [10]- [12], [14]- [15] and [28]- [29].…”