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 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 Almost Periodic Functions on the Quantum Time Scale and Applications

2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Yongkun Li
Keyword(s):  
Neural Networks ◽  
Periodic Solutions ◽  
Time Scale ◽  
Lyapunov Method ◽  
Almost Periodic Functions ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Quantum Time ◽  
Definition Of

In this paper, we first propose two types of concepts of almost periodic functions on the quantum time scale. Secondly, we study some basic properties of almost periodic functions on the quantum time scale. Thirdly, based on these, we study the existence and uniqueness of almost periodic solutions of dynamic equations on the quantum time scale by Lyapunov method. Then, we give an equivalent definition of almost periodic functions on the quantum time scale. Finally, as an application, we propose a class of high-order Hopfield neural networks on the quantum time scale and establish the existence and global exponential stability of almost periodic solutions of this class of neural networks.

scholarly journals Almost Periodic Functions on Time Scales and Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
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.

Values of limit periodic sequences and functions

2016 ◽  
Vol 66 (1) ◽  
Author(s):  
Michal Veselý ◽  
Petr Hasil
Keyword(s):  
Periodic Solutions ◽  
Periodic Sequence ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Periodic Sequences ◽  
Totally Bounded ◽  
Difference Systems

AbstractWe consider limit periodic sequences and functions with values in pseudometric spaces. We construct limit periodic sequences and functions with given values. For any totally bounded countable set, we find a limit periodic sequence which attains each value from this set periodically. A corresponding result concerning such a construction of limit periodic functions is proved as well. For any totally bounded countable set which is dense in itself, we construct a limit periodic bijective map from the integers into this set. As corollaries, we obtain new results about non-almost periodic solutions of almost periodic transformable difference systems.

scholarly journals Existence of periodic and almost periodic solutions in a resonant model on stem cell dynamics

2021 ◽  
Author(s):  
Carlos Alliera
Keyword(s):  
Stem Cell ◽  
Periodic Solutions ◽  
Implicit Function Theorem ◽  
Almost Periodic Functions ◽  
Implicit Function ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Cell Dynamics ◽  
Existence Of Periodic Solutions

This work deals with the existence of almost periodic solutions in a biological model, the model proposed by VG Nazarenko and E.E. Sel’kov of stem cell dynamics. This article demonstrates the existence of almost periodic solutions, for this purpose, the constant parameters of the system were changed to almost periodic functions which allows greater adaptability in biological cases such as this. This kind of changes have already been raised in other biological systems. In this case we will use the implicit function theorem to prove the existence of periodic solutions.

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 Almost Periodic Solutions for Some Convolution Equations

2013 ◽  
Vol 21 (3) ◽  
pp. 73-80
Author(s):  
Silvia-Otilia Corduneanu
Keyword(s):  
Fourier Series ◽  
Periodic Solutions ◽  
Almost Periodic Functions ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Complex Valued ◽  
Convolution Equations

Abstract We use the theory of Fourier series for almost periodic functions to looking for complex-valued functions f which are almost periodic on R and satisfy the following equation In this context g and h are almost periodic functions on ℝ and ϕ belongs to L1 (ℝ).

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