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.

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 S-Almost Automorphic Solutions for Impulsive Evolution Equations on Time Scales in Shift Operators

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1028
Author(s):  
Chao Wang ◽  
Rathinasamy Sakthivel ◽  
Gaston M. N’Guérékata
Keyword(s):  
Time Scales ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Dynamic Equations ◽  
Shift Operators ◽  
Almost Automorphic ◽  
Almost Automorphic Functions ◽  
New Concepts ◽  
Impulsive Evolution Equations ◽  
Automorphic Functions

In this paper, based on the concept of complete-closed time scales attached with shift direction under non-translational shifts (or S-CCTS for short), as a first attempt, we develop the concepts of S-equipotentially almost automorphic sequences, discontinuous S-almost automorphic functions and weighted piecewise pseudo S-almost automorphic functions. More precisely, some novel results about their basic properties and some related theorems are obtained. Then, we apply the introduced new concepts to investigate the existence of weighted piecewise pseudo S-almost automorphic mild solutions for the impulsive evolution equations on irregular hybrid domains. The obtained results are valid for q-difference partial dynamic equations and can also be extended to other dynamic equations on more general time scales. Finally, some heat dynamic equations on various hybrid domains are provided as applications to illustrate the obtained theory.

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