Almost periodic functions on Hausdorff 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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Almost Periodic Functions on Time Scales and Applications

Discrete Dynamics in Nature and Society ◽

10.1155/2011/727068 ◽

2011 ◽

Vol 2011 ◽

pp. 1-20 ◽

Cited By ~ 30

Author(s):

Yongkun Li ◽

Chao Wang

Keyword(s):

Time Scales ◽

Sufficient Conditions ◽

Almost Periodic Solution ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions ◽

Mathematical Methods ◽

Variable Delays ◽

Definition Of ◽

Almost Periodic Time Scales

We first propose the concept of almost periodic time scales and then give the definition of almost periodic functions on almost periodic time scales, then by using the theory of calculus on time scales and some mathematical methods, some basic results about almost periodic differential equations on almost periodic time scales are established. Based on these results, a class of high-order Hopfield neural networks with variable delays are studied on almost periodic time scales, and some sufficient conditions are established for the existence and global asymptotic stability of the almost periodic solution. Finally, two examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

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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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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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Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations

Advances in Difference Equations ◽

10.1186/s13662-015-0650-0 ◽

2015 ◽

Vol 2015 (1) ◽

Cited By ~ 11

Author(s):

Chao Wang ◽

Ravi P Agarwal

Keyword(s):

Time Scales ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions ◽

Dense Sets

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Weighted pseudo-almost periodic functions on time scales with applications to cellular neural networks with discrete delays

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.4107 ◽

2016 ◽

Cited By ~ 4

Author(s):

Yongkun Li ◽

Lili Zhao

Keyword(s):

Neural Networks ◽

Time Scales ◽

Cellular Neural Networks ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions ◽

Pseudo Almost Periodic Functions ◽

Discrete Delays ◽

Pseudo Almost Periodic

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THE CONNECTION BETWEEN PSEUDO ALMOST PERIODIC FUNCTIONS DEFINED ON TIME SCALES AND ON THE REAL LINE

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972717000041 ◽

2017 ◽

Vol 95 (3) ◽

pp. 482-494 ◽

Cited By ~ 1

Author(s):

CHAO-HONG TANG ◽

HONG-XU LI

Keyword(s):

Time Scales ◽

Almost Periodic Functions ◽

Almost Periodic Function ◽

Almost Periodic ◽

Necessary And Sufficient Condition ◽

Periodic Functions ◽

Pseudo Almost Periodic Functions ◽

Necessary And Sufficient ◽

Pseudo Almost Periodic ◽

Uniformly Continuous

A necessary and sufficient condition for a continuous function $g$ to be almost periodic on time scales is the existence of an almost periodic function $f$ on $\mathbb{R}$ such that $f$ is an extension of $g$. Our aim is to study this question for pseudo almost periodic functions. We prove the necessity of the condition for pseudo almost periodic functions. An example is given to show that the sufficiency of the condition does not hold for pseudo almost periodic functions. Nevertheless, the sufficiency is valid for uniformly continuous pseudo almost periodic functions. As applications, we give some results on the connection between the pseudo almost periodic (or almost periodic) solutions of dynamic equations on time scales and of the corresponding differential equations.

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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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