Asymptotically Almost Periodic Solutions of Evolution Equations in Banach Spaces

Ruess, Wolfgang M.; Phóng, Vũ Qúôc

doi:10.1006/jdeq.1995.1149

Cited by 60 publications

(46 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is well known that (see Ruess, Summers, and Vũ Quôc Phóng [27,[31][32][33]), for any function f ∈ C([0, ∞) : X), the following statements are equivalent:…”

Section: Asymptotically Almost Periodic Functionsmentioning

confidence: 99%

Almost periodic and asymptotically almost periodic functions: part I

Lassoued

Shah

2018

Adv Differ Equ

View full text Add to dashboard Cite

In this paper, we review indispensable properties and characterizations of almost periodic functions and asymptotically almost periodic functions in Banach spaces. Special accent is put on the Stepanov generalizations of almost periodic functions and asymptotically almost periodic functions. We also recollect some basic results regarding equi-Weyl-almost periodic functions and Weyl-almost periodic functions. The class of asymptotically Weyl-almost periodic functions, introduced in this work, seems to be not considered elsewhere even in the scalar-valued case. We actually introduce eight new classes of asymptotically almost periodic functions and analyze relations between them. In order to make a picture as complete and clear as possible, several illustrating examples and counter-examples are given. It is worth noting that the topics dealt with in this paper seem to be of an intrinsic connection with the problem of existence and uniqueness of solutions of differential and difference equations, in both determinist and stochastic cases. MSC: 42A75

show abstract

“…It is well known that (see Ruess, Summers, and Vũ Quôc Phóng [27,[31][32][33]), for any function f ∈ C([0, ∞) : X), the following statements are equivalent:…”

Section: Asymptotically Almost Periodic Functionsmentioning

confidence: 99%

Almost periodic and asymptotically almost periodic functions: part I

Lassoued

Shah

2018

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…It was first studied intensively for bounded operators by Dalec'kiȋ and Kreȋn [4], Rosenblum [16]. For unbounded case, (3.1) was studied in [2,11,12,13], when A and B are generators of C 0 -semigroups, and in [17,19] when A and B are closed operators. We cite here some main results which will be used in the sequel.…”

Section: The Equation Ax − Xd N = Cmentioning

confidence: 99%

“…We say that u ∈ BUC(R,E) is totally ergodic if exists for all ν ∈ R. The following theorem can be found in [9] (parts (a) and (b)) and [17] (part (c)). Then u is almost periodic.…”

Section: By Theorems 31(i) and 45 P(1) Is Regularly Admissible If mentioning

confidence: 99%

“…For unbounded operator A, where the situation changes dramatically, the first question is, which solutions of (1.1) are considered as mild solutions? If A is the generator of a C 0 -semigroup T(t), t ≥ 0, it is logical to define mild solutions of (1.1) by With this definition in hand, many authors investigated the qualitative behavior of (1.3) in different ways (see [10,12,13,14,17] and references therein). The second-order differential equation u (t) = Au(t) + f (t), where A is the generator of a cosine family (C(t)), and for which mild solutions are defined by…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the mild solutions of higher‐order differential equations in Banach spaces

Lan

2003

Abstract and Applied Analysis

View full text Add to dashboard Cite

show abstract

“…Along with the development of such equations as evolution partial differential equations, retarded functional differential equations, and so forth, where the phase spaces are infinite, the theory of Banachvalued (asymptotically) almost periodic functions has been developed [2,18,20,21,22,23]. Some techniques in functional analysis and harmonic analysis are applied to such equations, for example, [3,24] apply spectrum theory to get almost periodic solutions for some linear abstract evolution differential equations.…”

Section: Introductionmentioning

confidence: 99%