Notes on almost-periodicity in topological vector spaces

AbstractThe question as to the existence of nontrivial compact or weakly compact multipliers between spaces of functions on groups has been investigated for several years. Until now, however, no general method which is applicable to a large class of function spaces seems to be knownIn this paper we prove that the existence of nontrivial compact multipliers between Banach function spaces on which a group acts is related to the existence of nonzero almost periodic functions.

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On Bohr almost periodicity

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100064434 ◽

1986 ◽

Vol 99 (3) ◽

pp. 489-493 ◽

Cited By ~ 1

Author(s):

Paul Milnes

Keyword(s):

Periodic Function ◽

Present Author ◽

Almost Periodic Functions ◽

Unusual Feature ◽

Almost Periodicity ◽

Almost Periodic Function ◽

Almost Periodic ◽

Periodic Functions ◽

Almost Automorphic

AbstractThe first examples of Bohr almost periodic functions that are not almost periodic were given by T. -S. Wu. Later, the present author showed that Bohr almost periodic functions could be distal (and not almost periodic) and even merely minimal. Here it is proved that all Bohr almost periodic functions are minimal. The proof yields an unusual feature about the orbit of a Bohr almost periodic function, one which does not characterize Bohr almost periodic functions, but can be used to show that a Bohr almost periodic function f that is point distal must be distal or, if f is almost automorphic, it must be almost periodic. Some pathologies of Bohr almost periodic functions are discussed.

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Denjoy-Bochner almost periodic functions

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700022047 ◽

1984 ◽

Vol 37 (2) ◽

pp. 205-222 ◽

Cited By ~ 1

Author(s):

B. K. Pal ◽

S. N. Mukhopadhyay

Keyword(s):

Integrable Function ◽

Almost Periodic Functions ◽

Almost Periodicity ◽

Almost Periodic ◽

Bochner Integral ◽

Periodic Functions ◽

Integrable Functions

AbstractThe special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill [3] to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in [3] a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].

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Generalized Almost Periodicity in Lebesgue Spaces with Variable Exponents

Mathematics ◽

10.3390/math8060928 ◽

2020 ◽

Vol 8 (6) ◽

pp. 928 ◽

Cited By ~ 1

Author(s):

Marko Kostić ◽

Wei-Shih Du

Keyword(s):

Banach Spaces ◽

Differential Inclusions ◽

Almost Periodic Functions ◽

Almost Periodicity ◽

Variable Exponents ◽

Almost Periodic ◽

Periodic Functions ◽

Lebesgue Spaces ◽

Recurrent Functions ◽

Convolution Products

In this paper, we introduce and analyze Stepanov uniformly recurrent functions, Doss uniformly recurrent functions and Doss almost-periodic functions in Lebesgue spaces with variable exponents. We investigate the invariance of these types of generalized almost-periodicity in Lebesgue spaces with variable exponents under the actions of convolution products, providing also some illustrative applications to the abstract semilinear integro-differential inclusions in Banach spaces.

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Composition of piecewise pseudo almost periodic functions and applications to abstract impulsive differential equations

Advances in Difference Equations ◽

10.1186/1687-1847-2013-11 ◽

2013 ◽

Vol 2013 (1) ◽

Cited By ~ 12

Author(s):

Junwei Liu ◽

Chuanyi Zhang

Keyword(s):

Differential Equations ◽

Impulsive Differential Equations ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions ◽

Pseudo Almost Periodic Functions ◽

Pseudo Almost Periodic

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Bochner-Like Transform and Stepanov Almost Periodicity on Time Scales with Applications

Symmetry ◽

10.3390/sym10110566 ◽

2018 ◽

Vol 10 (11) ◽

pp. 566 ◽

Cited By ~ 1

Author(s):

Chao-Hong Tang ◽

Hong-Xu Li

Keyword(s):

Time Scales ◽

Almost Periodic Functions ◽

Dynamic Equations ◽

Almost Periodicity ◽

Almost Periodic ◽

Periodic Functions ◽

Basic Properties ◽

Composition Theorem

By using Bochner transform, Stepanov almost periodic functions inherit some basic properties directly from almost periodic functions. Recently, this old work was extended to time scales. However, we show that Bochner transform is not valid on time scales. Then we present a revised version, called Bochner-like transform, for time scales, and prove that a function is Stepanov almost periodic if and only if its Bochner-like transform is almost periodic on time scales. Some basic properties including the composition theorem of Stepanov almost periodic functions are obtained by applying Bochner-like transform. Our results correct the recent results where Bochner transform is used on time scales. As an application, we give some results on dynamic equations with Stepanov almost periodic terms.

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Asymptotic Almost Periodic Functions with Range in a Topological Vector Space

Journal of Function Spaces and Applications ◽

10.1155/2013/965746 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Liaqat Ali Khan ◽

Saud M. Alsulami

Keyword(s):

Vector Space ◽

Topological Vector Space ◽

Almost Periodic Functions ◽

Almost Periodicity ◽

Almost Periodic ◽

Periodic Functions ◽

Range Space ◽

Finite Dimensional ◽

Locally Convex ◽

Dimensional Range

The notion of asymptotic almost periodicity wasfirst introduced by Fréchet in 1941 in the case offinite dimensional range spaces. Later, its extension to the case of Banach range spaces and locally convex range spaces has been considered by several authors. In this paper, we have generalized the concept of asymptotic almost periodicity to the case where the range space is a general topological vector space, not necessarily locally convex. Our results thus widen the scope of applications of asymptotic almost periodicity.

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