Continuity properties of compact right topological groups

1979 ◽  
Vol 86 (3) ◽  
pp. 427-435 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Groups ◽  
Almost Automorphic ◽  
Almost Automorphic Function ◽  
Almost Automorphic Functions ◽  
Continuity Properties ◽  
Enveloping Semigroups ◽  
The One ◽  
Automorphic Functions

AbstractCompact right topological groups appear naturally in topological dynamics. Some continuity properties of the one arising as an enveloping semigroup from the distal function are considered here (and, by way of comparison, the enveloping semigroups arising from two almost automorphic functions are discussed). The continuity properties are established either explicitly or by citing a theorem which is proved here and gives some characterizations of almost periodic functions. One characterization is proved using the result (essentially due to W. A. Veech) that a distal, almost automorphic function is almost periodic. A proof of this last result is also given.

scholarly journals The Space of Continuous Periodic Functions Is a Set of First Category inAP(X)

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Zhe-Ming Zheng ◽  
Hui-Sheng Ding ◽  
Gaston M. N'Guérékata
Keyword(s):  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
First Category ◽  
Set Of First Category ◽  
Automorphic Functions

We prove that the space of continuous periodic functions is a set of first category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of first category in the space of almost automorphic functions.

Weighted pseudo δ-almost automorphic functions and abstract dynamic equations

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Chao Wang ◽  
Ravi P. Agarwal ◽  
Donal O’Regan
Keyword(s):  
Time Scales ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Automorphic Function ◽  
Dynamic Equations ◽  
Almost Automorphic ◽  
Almost Automorphic Function ◽  
Almost Automorphic Functions ◽  
Automorphic Functions

Abstract In this paper, we propose the concept of a weighted pseudo δ-almost automorphic function under the matched space for time scales and we present some properties. Also, we obtain sufficient conditions for the existence of weighted pseudo δ-almost automorphic mild solutions to a class of semilinear dynamic equations under the matched spaces for time scales.

scholarly journals n0-Order Weighted Pseudo Δ-Almost Automorphic Functions and Abstract Dynamic Equations

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 775 ◽  
Author(s):  
Wang ◽  
Agarwal ◽  
O’Regan ◽  
N’Guérékata
Keyword(s):  
Time Scales ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Automorphic Function ◽  
Dynamic Equations ◽  
Open Problems ◽  
Almost Automorphic ◽  
Almost Automorphic Function ◽  
Almost Automorphic Functions ◽  
Automorphic Functions

In this paper, we introduce the concept of a n 0 -order weighted pseudo Δ n 0 δ -almost automorphic function under the matched space for time scales and we present some properties. The results are valid for q-difference dynamic equations among others. Moreover, we obtain some sufficient conditions for the existence of weighted pseudo Δ n 0 δ -almost automorphic mild solutions to a class of semilinear dynamic equations under the matched space. Finally, we end the paper with a further discussion and some open problems of this topic.

scholarly journals Bochner definition of Stepanov-like almost automorphic functions on time scales and an application to cellular neural networks with delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Li-Li Zhang ◽  
Xu-Dong Yang
Keyword(s):  
Neural Networks ◽  
Time Scales ◽  
Dynamic Equation ◽  
Cellular Neural Networks ◽  
Automorphic Function ◽  
Almost Automorphic ◽  
Almost Automorphic Function ◽  
Almost Automorphic Functions ◽  
Definition Of ◽  
Automorphic Functions

AbstractThe definition of Stepanov-like almost automorphic functions on time scales had been proposed in the literature, but at least one result was incorrect, which involved Bochner transform. In our work, we give the Bochner definition of Stepanov-like almost automorphic functions on time scales, and prove that a function is Stepanov-like almost automorphic if and only if it satisfies Bochner definition of Stepanov-like almost automorphic function on time scales. The Bochner definition of Stepanov-like almost automorphic functions on time scales corrects the faulty result, and perfects the definition of Stepanov-like almost automorphic functions. As applications, we discuss the almost automorphy of a certain dynamic equation and some cellular neural networks with delays on time scales.

scholarly journals Almost Automorphic Functions on the Quantum Time Scale and Applications

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Yongkun Li
Keyword(s):  
Time Scales ◽  
Time Scale ◽  
Dynamic Equations ◽  
Almost Periodic ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Quantum Time ◽  
Almost Periodic Time Scales ◽  
Automorphic Functions ◽  
General Time

We first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers; by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.

scholarly journals (&omega,c)- PSEUDO ALMOST PERIODIC FUNCTIONS, (&omega,c)- PSEUDO ALMOST AUTOMORPHIC FUNCTIONS AND APPLICATIONS

2021 ◽  
pp. 165
Author(s):  
Mohammed Taha Khalladi ◽  
Marko Kostić ◽  
Abdelkader Rahmani ◽  
Daniel Velinov
Keyword(s):  
Banach Spaces ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
First Order ◽  
Almost Automorphic Functions ◽  
Pseudo Almost Periodic Functions ◽  
Pseudo Almost Automorphic ◽  
Pseudo Almost Periodic ◽  
Automorphic Functions

In this paper, we introduce the classes of $(\omega, c)$-pseudo almost periodicfunctions and $(\omega, c)$-pseudo almost automorphicfunctions. These collections include $(\omega, c)$-pseudo periodicfunctions, pseudo almost periodic functions and their automorphic analogues.We present an application to the abstract semilinear first-order Cauchy inclusions in Banach spaces.

scholarly journals Pseudo almost automorphic solutions of class r in α-norm under the light of measure theory

2020 ◽  
Vol 7 (1) ◽  
pp. 81-101
Author(s):  
Issa Zabsonre ◽  
Djendode Mbainadji
Keyword(s):  
Phase Space ◽  
Spectral Decomposition ◽  
Measure Theory ◽  
New Approach ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
Pseudo Almost Automorphic ◽  
Automorphic Functions

AbstractUsing the spectral decomposition of the phase space developed in Adimy and co-authors, we present a new approach to study weighted pseudo almost automorphic functions in the α-norm using the measure theory.

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