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

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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Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales

Discrete Dynamics in Nature and Society ◽

10.1155/2013/784198 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

Yan Wang ◽

Lan Li

Keyword(s):

Fixed Point ◽

Periodic Solutions ◽

Time Scales ◽

Existence And Uniqueness ◽

Second Order ◽

Almost Periodic Functions ◽

Dynamic Equations ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Periodic Functions

We firstly introduce the concept and the properties ofCmalmost periodic functions on time scales, which generalizes the concept of almost periodic functions on time scales and the concept ofC(n)-almost periodic functions. Secondly, we consider the existence and uniqueness of almost periodic solutions for second order dynamic equations on time scales by Schauder’s fixed point theorem and contracting mapping principle. At last, we obtain alternative theorems for second order dynamic equations on time scales.

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Almost Periodic Time Scales and Almost Periodic Functions on Time Scales

Journal of Applied Mathematics ◽

10.1155/2015/730672 ◽

2015 ◽

Vol 2015 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Yongkun Li ◽

Bing Li

Keyword(s):

Time Scales ◽

Time Scale ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions ◽

Basic Properties ◽

New Concepts ◽

Almost Periodic Time Scales

We propose some new concepts of almost periodic time scales and almost periodic functions on time scales and give some basic properties of these new types of almost periodic time scales and almost periodic functions on time scales. We also give some comments on a recent paper by Wang and Agarwal (2014) concerning a new almost periodic time scale.

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Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales

Abstract and Applied Analysis ◽

10.1155/2011/341520 ◽

2011 ◽

Vol 2011 ◽

pp. 1-22 ◽

Cited By ~ 43

Author(s):

Yongkun Li ◽

Chao Wang

Keyword(s):

Periodic Solutions ◽

Time Scales ◽

Almost Periodic Functions ◽

Dynamic Equations ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Periodic Functions ◽

Periodic Sequences ◽

Nonlinear Dynamic Equations ◽

Definition Of

Firstly, we propose a concept of uniformly almost periodic functions on almost periodic time scales and investigate some basic properties of them. When time scaleT=ℝorℤ, our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations. Finally, as an application of our results, we study the existence of almost periodic solutions for an almost periodic nonlinear dynamic equations on time scales.

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A Further Study of Almost Periodic Time Scales with Some Notes and Applications

Abstract and Applied Analysis ◽

10.1155/2014/267384 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 3

Author(s):

Chao Wang ◽

Ravi P. Agarwal

Keyword(s):

Neural Networks ◽

Time Scales ◽

Dynamic Equations ◽

Almost Periodicity ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Periodic Functions ◽

Cauchy Matrix ◽

Nonlinear Dynamic Equations ◽

Almost Periodic Time Scales

We introduce three equivalent concepts of almost periodic time scales as a further study of the corresponding concept proposed in Li and Wang (2011) and several examples of almost periodic time scales which are not periodic are provided. Furthermore, the concepts of almost periodic functions are redefined under the sense of this new timescale concept. Finally, almost periodicity of Cauchy matrix for dynamic equations is proved under these new definitions. Based on these results, the existence of almost periodic solutions to a class of nonlinear dynamic equations is investigated by the almost periodicity of Cauchy matrix on almost periodic time scales. Besides, as an application, we apply our results to a class of high-order Hopfield neural networks.

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Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales

Advances in Difference Equations ◽

10.1186/1687-1847-2012-77 ◽

2012 ◽

Vol 2012 (1) ◽

Cited By ~ 18

Author(s):

Yongkun Li ◽

Chao Wang

Keyword(s):

Periodic Solutions ◽

Time Scales ◽

Almost Periodic Functions ◽

Dynamic Equations ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Periodic Functions ◽

Pseudo Almost Periodic Solutions ◽

Pseudo Almost Periodic Functions ◽

Pseudo Almost Periodic

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Discontinuous generalized double-almost-periodic functions on almost-complete-closed time scales

Boundary Value Problems ◽

10.1186/s13661-019-1283-0 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

Chao Wang ◽

Ravi P. Agarwal ◽

Donal O’Regan ◽

Rathinasamy Sakthivel

Keyword(s):

Function Space ◽

Time Scales ◽

Measure Theory ◽

Almost Periodic Functions ◽

Almost Periodicity ◽

Almost Periodic ◽

Periodic Functions ◽

Closed Set ◽

New Type ◽

Piecewise Functions

Abstract In this paper, we introduce the concept of almost-complete-closed time scales (ACCTS) that allows independent variables of functions to possess almost-periodicity under translations. For this new type of time scale, a class of piecewise functions with double-almost-periodicity is proposed and studied. Based on these, concepts of weighted pseudo-double-almost-periodic functions (WPDAP) in Banach spaces and a translation-almost-closed set are introduced. Further, we prove that the function space WPDAP0 affiliated to WPDAP is a translation-almost-closed set. Then, by introducing the concept of almost-uniform convergence for piecewise functions on ACCTS and using measure theory on time scales, some composition theorems of WPDAP and the completeness of the function space are proved.

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Almost Periodic Functions and Generalizations on Complete-Closed Time Scales

Theory of Translation Closedness for Time Scales - Developments in Mathematics ◽

10.1007/978-3-030-38644-3_3 ◽

2020 ◽

pp. 169-237

Author(s):

Chao Wang ◽

Ravi P. Agarwal ◽

Donal O’Regan ◽

Rathinasamy Sakthivel

Keyword(s):

Time Scales ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions

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Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay

Open Mathematics ◽

10.1515/math-2018-0073 ◽

2018 ◽

Vol 16 (1) ◽

pp. 826-841 ◽

Cited By ~ 1

Author(s):

Chao-Hong Tang ◽

Hong-Xu Li

Keyword(s):

Time Scales ◽

Dynamic Equations ◽

Translation Invariance ◽

Almost Periodicity ◽

Almost Periodic ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Pseudo Almost Periodicity ◽

Pseudo Almost Periodic ◽

Equations With Delay

AbstractIn this paper, we introduce the concept of Sp-pseudo almost periodicity on time scales and present some basic properties of it, including the translation invariance, uniqueness of decomposition, completeness and composition theorem. Moreover, we prove the seemingly simple but nontrivial result that pseudo almost periodicity implies Stepanov-like pseudo almost periodicity. As an application of the abstract results, we present some existence and uniqueness results on the pseudo almost periodic solutions of dynamic equations with delay.

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Compactness and Almost Periodicity of Multipliers

Canadian Mathematical Bulletin ◽

10.4153/cmb-1983-010-9 ◽

1983 ◽

Vol 26 (1) ◽

pp. 58-62 ◽

Cited By ~ 2

Author(s):

G. Crombez

Keyword(s):

Large Class ◽

Function Spaces ◽

Almost Periodic Functions ◽

Almost Periodicity ◽

Almost Periodic ◽

Periodic Functions ◽

Weakly Compact ◽

Banach Function Spaces ◽

General Method

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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