Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations

Chao Wang; Ravi P Agarwal

doi:10.3934/dcdsb.2019267

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2020, Volume 25, Issue 2: 781-798. Doi: 10.3934/dcdsb.2019267

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Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations

Chao Wang^1, and
Ravi P Agarwal^2,3, ,

1.
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
2.
Department of Mathematics, Texas A & M University-Kingsville, 700 University Blvd., TX 78363-8202, Kingsville, TX, USA
3.
Distinguished University Professor of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA

^* Corresponding author: Ravi P Agarwal
^* Corresponding author: Ravi P Agarwal

Received: November 30, 2018

Revised: March 31, 2019

Early access: November 2019

Published: February 2020

This work is supported by Youth Fund of NSFC (No. 11601470), Tian Yuan Fund of NSFC (No. 11526181), Dong Lu youth excellent teachers development program of Yunnan University (No. wx069051), IRTSTYN and Joint Key Project of Yunnan Provincial Science and Technology Department of Yunnan University (No. 2018FY001(-014))

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Abstract

In this paper, we introduce the concepts of Bochner and Bohr almost automorphic functions on the semigroup induced by complete-closed time scales and their equivalence is proved. Particularly, when $ \Pi = \mathbb{R}^{+} $ (or $ \Pi = \mathbb{R}^{-} $), we can obtain the Bochner and Bohr almost automorphic functions on continuous semigroup, which is the new almost automorphic case on time scales compared with the literature [20] (W.A. Veech, Almost automorphic functions on groups, Am. J. Math., Vol. 87, No. 3 (1965), pp 719-751) since there may not exist inverse element in a semigroup. Moreover, when $ \Pi = h\mathbb{Z}^{+},\,h>0 $ (or $ \Pi = h\mathbb{Z}^{-},\,h>0 $), the corresponding automorphic functions on discrete semigroup can be obtained. Finally, we establish a theorem to guarantee the existence of Bochner (or Bohr) almost automorphic mild solutions of dynamic equations on semigroups induced by time scales.

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Mathematics Subject Classification: Primary: 34N05, 43A60; Secondary: 26E70.

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