scholarly journals δ-Almost Periodic Functions and Applications to Dynamic Equations

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 525 ◽  
Author(s):  
Chao Wang ◽  
Ravi P. Agarwal ◽  
Donal O’Regan
Keyword(s):  
Dynamic Equation ◽  
Exponential Dichotomy ◽  
Homogeneous Equation ◽  
Almost Periodic Solution ◽  
Dynamic Equations ◽  
Almost Periodic Solutions ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Delay Dynamic Equations

In this paper, by employing matched spaces for time scales, we introduce a δ -almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence are presented. In the sense of admitting exponential dichotomy for the homogeneous equation, the expression of a δ -almost periodic solution for a type of nonhomogeneous dynamic equation is obtained and the existence of δ -almost periodic solutions for new delay dynamic equations is considered. The results in this paper are valid for delay q-difference equations and delay dynamic equations whose delays may be completely separated from the time scale T .

scholarly journals On the Spectrum of almost periodic solutions of an abstract differential equation

1974 ◽  
Vol 18 (4) ◽  
pp. 385-387
Author(s):  
Aribindi Satyanarayan Rao ◽  
Walter Hengartner
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Periodic Solution ◽  
Linear Operator ◽  
Periodic Function ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic Function ◽  
Almost Periodic ◽  
Abstract Differential Equation

AbstractIf a linear operator A in a Banach space satisfies certain conditions, then the spectrum of any almost periodic solution of the differential equation u′ = Au + f is shown to be identical with the spectrum of f, where f is a Stepanov almost periodic function.

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 Almost periodic solutions of Volterra difference systems

2017 ◽  
Vol 50 (1) ◽  
pp. 320-329
Author(s):  
Halis Can Koyuncuoglu ◽  
Murat Adıvar
Keyword(s):  
Periodic Solution ◽  
Exponential Dichotomy ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Difference Systems ◽  
Volterra Systems

Abstract We study the existence of an almost periodic solution of discrete Volterra systems by means of fixed point theory. Using discrete variant of exponential dichotomy, we provide sufficient conditions for the existence of an almost periodic solution. Hence, we provide an alternative solution for the open problem proposed in the literature.

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

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Yongkun Li ◽  
Chao Wang
Keyword(s):  
Periodic Solution ◽  
Time Scales ◽  
Existence And Uniqueness ◽  
Exponential Dichotomy ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Dynamic Equations ◽  
Almost Periodic ◽  
Linear Dynamic ◽  
Expression Form

We first introduce the concept of admitting an exponential dichotomy to a class of linear dynamic equations on time scales and study the existence and uniqueness of almost periodic solution and its expression form to this class of linear dynamic equations on time scales. Then, as an application, using these concepts and results, we establish sufficient conditions for the existence and exponential stability of almost periodic solution to a class of Hopfield neural networks with delays. Finally, two examples and numerical simulations given to illustrate our results are plausible and meaningful.

scholarly journals On the spectrum of weakly almost periodic solutions of certain abstract differential equations

1985 ◽  
Vol 8 (1) ◽  
pp. 109-112 ◽  
Author(s):  
Aribindi Satyanarayan Rao ◽  
L. S. Dube
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Periodic Solution ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic Function ◽  
Dual Operator ◽  
Almost Periodic ◽  
Abstract Differential Equations ◽  
Weakly Almost Periodic

In a sequentially weakly complete Banach space, if the dual operator of a linear operatorAsatisfies certain conditions, then the spectrum of any weakly almost periodic solution of the differential equationu′=Au+fis identical with the spectrum offexcept at the origin, wherefis a weakly almost periodic function.

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 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.

Almost Periodic Solutions for a Class of Stochastic Differential Equations

2013 ◽  
Vol 8 (4) ◽  
Author(s):  
Yongjian Liu ◽  
Aimin Liu
Keyword(s):  
Linear Operator ◽  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Exponential Dichotomy ◽  
Stochastic Equations ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Uniform Exponential Stability ◽  
New Criteria

In this paper, the existence and uniqueness of the square-mean almost periodic solutions to a class of the semilinear stochastic equations is studied. In particular, the condition of the uniform exponential stability of the linear operator is essentially removed, only using the exponential dichotomy of the linear operator. Some new criteria ensuring the existence and uniqueness of the square-mean almost periodic solution for the system are presented. Finally, an example of a kind of the stochastic cellular neural networks is given. These obtained results are important in signal processing and the in design of networks.

scholarly journals Some Oscillation Criteria for Higher-Order Neutral Emden-Fowler Delay Dynamic Equations on Time Scales

2018 ◽  
Vol 228 ◽  
pp. 01003
Author(s):  
Ying Sui ◽  
Yulong Shi ◽  
Yibin Sun ◽  
Shurong Sun
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Higher Order ◽  
Delay Equation ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Oscillation Criteria ◽  
Delay Dynamic Equations

New oscillation criteria are established for higher-order Emdn-Fowler dynamic equation $ q(v)x^{\beta } (\delta (v)) + (r(v)(z^{{\Delta ^{{n - 1}} }} (v))^{\alpha } )^{\Delta } = 0 $ on time scales, $ z(v): = p(v)x(\tau (v)) + x(v) $ Our results extend and supplement those reported in literatures in the sense that we study a more generalized neutral delay equation and do not require $ r^{\Delta } (v) \ge 0 $ and the commutativity of the jump and delay operators.

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