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

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
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.

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

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
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.

scholarly journals Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
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.

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

Symmetry ◽  
2018 ◽  
Vol 10 (11) ◽  
pp. 566 ◽  
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.

scholarly journals The Existence and Global Exponential Stability of Almost Periodic Solutions for Neutral-Type CNNs on Time Scales

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 321 ◽  
Author(s):  
Bing Li ◽  
Yongkun Li ◽  
Xiaofang Meng
Keyword(s):  
Neural Networks ◽  
Exponential Stability ◽  
Periodic Solutions ◽  
Time Scales ◽  
Global Exponential Stability ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Leakage Delays

In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained.

scholarly journals C1-Almost Periodic Solutions of BAM Neural Networks with Time-Varying Delays on Time Scales

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Yongkun Li ◽  
Lili Zhao ◽  
Li Yang
Keyword(s):  
Neural Networks ◽  
Periodic Solutions ◽  
Time Scales ◽  
Sufficient Conditions ◽  
Time Varying ◽  
Bam Neural Networks ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
New Type ◽  
Inequality Techniques

On a new type of almost periodic time scales, a class of BAM neural networks is considered. By employing a fixed point theorem and differential inequality techniques, some sufficient conditions ensuring the existence and global exponential stability ofC1-almost periodic solutions for this class of networks with time-varying delays are established. Two examples are given to show the effectiveness of the proposed method and results.

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

2018 ◽  
Vol 16 (1) ◽  
pp. 826-841 ◽  
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.

scholarly journals ALMOST PERIODIC SOLUTIONS OF IMPULSIVE INTEGRO‐DIFFERENTIAL NEURAL NETWORKS

2010 ◽  
Vol 15 (4) ◽  
pp. 505-516 ◽  
Author(s):  
Gani Tr. Stamov ◽  
Jehad O. Alzabut
Keyword(s):  
Neural Networks ◽  
Periodic Solutions ◽  
Contraction Mapping ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Contraction Mapping Principle ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Cauchy Matrix ◽  
Bellman’S Inequality

In this paper, sufficient conditions are established for the existence of almost periodic solutions for system of impulsive integro‐differential neural networks. Our approach is based on the estimation of the Cauchy matrix of linear impulsive differential equations. We shall employ the contraction mapping principle as well as Gronwall‐Bellman's inequality to prove our main result. The research of Gani Tr. Stamov is partially supported by the Grand 100ni087–16 from Technical University–Sofia

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