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February  2020, 25(2): 781-798. doi: 10.3934/dcdsb.2019267

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

Received  December 2018 Revised  April 2019 Published  November 2019

Fund Project: 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)).

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.

Keywords: Almost automorphic functions, semigroup, almost automorphic mild solutions, dynamic equations, time scales.
Mathematics Subject Classification: Primary: 34N05, 43A60; Secondary: 26E70.
Citation: Chao Wang, Ravi P Agarwal. Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 781-798. doi: 10.3934/dcdsb.2019267
References:
[1]

R. P. Agarwal and D. O'Regan, Some comments and notes on almost periodic functions and changing-periodic time scales, Electron. J. Math. Anal. Appl., 6 (2018), 125-136.   Google Scholar

[2]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[3]

S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Univ. e Politec. Torino Rend. Sem. Mat., 15 (2019), 225-253.   Google Scholar

[4]

S. Bochner, Uniform convergence of monotone sequences of functions, Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 582-585.  doi: 10.1073/pnas.47.4.582.  Google Scholar

[5]

S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  doi: 10.1073/pnas.48.12.2039.  Google Scholar

[6]

M. Bohner and J. G. Mesquita, Almost periodic functions in quantum calculus, Electron. J. Differential Equations, 2018, 1–11.  Google Scholar

[7]

Y. K. Chang and T. W. Feng, Properties on measure pseudo almost automorphic functions and applications to fractional differential equations in Banach spaces, Electron. J. Differential Equations, 2018, 1–14.  Google Scholar

[8]

Y. K. Chang and S. Zheng, Weighted pseudo almost automorphic solutions to functional differential equations with infinite delay, Electron. J. Differential Equations, 2016, 1–19.  Google Scholar

[9]

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013. doi: 10.1007/978-3-319-00849-3.  Google Scholar

[10]

T. Diagana and G. M. N'Guérékata, Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal., 86 (2007), 723-733.  doi: 10.1080/00036810701355018.  Google Scholar

[11]

H. S. Ding, T. J. Xiao and J. Liang, Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math. Anal. Appl., 338 (2008), 141-151.  doi: 10.1016/j.jmaa.2007.05.014.  Google Scholar

[12]

H. S. Ding and S. M. Wan, Asymptotically almost automorphic solutions of differential equations with piecewise constant argument, Open Math., 15 (2017), 595-610.  doi: 10.1515/math-2017-0051.  Google Scholar

[13]

M. Kéré and G. M. N'Guérékata, Almost automorphic dynamic systems on time scales, PanAmer. Math. J., 28 (2018), 19-37.   Google Scholar

[14]

A. Milcé and J. C. Mado, Almost automorphic solutions of some semilinear dynamic equations on time scales, Int. J. Evol. Equ., 9 (2014), 217-229.   Google Scholar

[15]

G. Mophou, G. M. N'Guérékata and A. Milce, Almost automorphic functions of order n and applications to dynamic equations on time scales, Discrete Dyn. Nat. Soc., (2014), 1–13. doi: 10.1155/2014/410210.  Google Scholar

[16]

J. von Neumann, Almost periodic functions in a group, I, Trans. Amer. Math. Soc., 36 (1934), 445-492.  doi: 10.1090/S0002-9947-1934-1501752-3.  Google Scholar

[17]

G. M. N'Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.  Google Scholar

[18]

G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.  Google Scholar

[19]

G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68 (2008), 2658-2667.  doi: 10.1016/j.na.2007.02.012.  Google Scholar

[20]

W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071.  Google Scholar

[21]

C. Wang and R. P. Agarwal, Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive $\nabla$-dynamic equations on time scales, Adv. Difference Equ., 153 (2014), 1-29.  doi: 10.1186/1687-1847-2014-153.  Google Scholar

[22]

C. Wang, R. P. Agarwal and D. O'Regan, n0-order $\Delta$-almost periodic functions and dynamic equations, Appl. Anal., 97 (2018), 2626-2654.  doi: 10.1080/00036811.2017.1382689.  Google Scholar

[23]

C. Wang, R. P. Agarwal and D. O'Regan, Periodicity, almost periodicity for time scales and related functions, Nonauton. Dyn. Syst., 3 (2016), 24-41.  doi: 10.1515/msds-2016-0003.  Google Scholar

[24]

C. Wang, R. P. Agarwal, D. O'Regan, C. Wang and R. P. Agarwal, Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations, Adv. Difference Equ., 312 (2015), 1-9.  doi: 10.1186/s13662-015-0650-0.  Google Scholar

[25]

C. Wang, R. P Agarwal and D. O'Regan, A matched space for time scales and applications to the study on functions, Adv. Difference Equ., 305 (2017), 1-28.  doi: 10.1186/s13662-017-1366-0.  Google Scholar

[26]

C. Wang, R. P Agarwal and D. O'Regan, Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations, J. Nonlinear Sci. Appl., 10 (2017), 3863-3886.  doi: 10.22436/jnsa.010.07.41.  Google Scholar

[27]

C. Wang, R. P Agarwal and D. O'Regan, Π-semigroup for invariant under translations time scales and abstract weighted pseudo almost periodic functions with applications, Dynam. Systems Appl., 25 (2016), 1-28.   Google Scholar

[28]

C. Wang and R. P. Agarwal, Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Appl. Math. Lett., 70 (2017), 58-65.  doi: 10.1016/j.aml.2017.03.009.  Google Scholar

[29]

T. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum, 76 (2008), 518-524.  doi: 10.1007/s00233-007-9011-y.  Google Scholar

[30]

M. Zaki, Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl., 101 (1974), 91-114.  doi: 10.1007/BF02417100.  Google Scholar

[31]

Z. M. Zheng and H. S. Ding, On completeness of the space of weighted pseudo almost automorphic functions, J. Funct. Anal., 268 (2015), 3211-3218.  doi: 10.1016/j.jfa.2015.02.012.  Google Scholar

show all references

References:
[1]

R. P. Agarwal and D. O'Regan, Some comments and notes on almost periodic functions and changing-periodic time scales, Electron. J. Math. Anal. Appl., 6 (2018), 125-136.   Google Scholar

[2]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-1-4612-0201-1.  Google Scholar

[3]

S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Univ. e Politec. Torino Rend. Sem. Mat., 15 (2019), 225-253.   Google Scholar

[4]

S. Bochner, Uniform convergence of monotone sequences of functions, Proc. Nat. Acad. Sci. U.S.A., 47 (1961), 582-585.  doi: 10.1073/pnas.47.4.582.  Google Scholar

[5]

S. Bochner, A new approach to almost periodicity, Proc. Nat. Acad. Sci. U.S.A., 48 (1962), 2039-2043.  doi: 10.1073/pnas.48.12.2039.  Google Scholar

[6]

M. Bohner and J. G. Mesquita, Almost periodic functions in quantum calculus, Electron. J. Differential Equations, 2018, 1–11.  Google Scholar

[7]

Y. K. Chang and T. W. Feng, Properties on measure pseudo almost automorphic functions and applications to fractional differential equations in Banach spaces, Electron. J. Differential Equations, 2018, 1–14.  Google Scholar

[8]

Y. K. Chang and S. Zheng, Weighted pseudo almost automorphic solutions to functional differential equations with infinite delay, Electron. J. Differential Equations, 2016, 1–19.  Google Scholar

[9]

T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract Spaces, Springer, Cham, 2013. doi: 10.1007/978-3-319-00849-3.  Google Scholar

[10]

T. Diagana and G. M. N'Guérékata, Stepanov-like almost automorphic functions and applications to some semilinear equations, Appl. Anal., 86 (2007), 723-733.  doi: 10.1080/00036810701355018.  Google Scholar

[11]

H. S. Ding, T. J. Xiao and J. Liang, Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math. Anal. Appl., 338 (2008), 141-151.  doi: 10.1016/j.jmaa.2007.05.014.  Google Scholar

[12]

H. S. Ding and S. M. Wan, Asymptotically almost automorphic solutions of differential equations with piecewise constant argument, Open Math., 15 (2017), 595-610.  doi: 10.1515/math-2017-0051.  Google Scholar

[13]

M. Kéré and G. M. N'Guérékata, Almost automorphic dynamic systems on time scales, PanAmer. Math. J., 28 (2018), 19-37.   Google Scholar

[14]

A. Milcé and J. C. Mado, Almost automorphic solutions of some semilinear dynamic equations on time scales, Int. J. Evol. Equ., 9 (2014), 217-229.   Google Scholar

[15]

G. Mophou, G. M. N'Guérékata and A. Milce, Almost automorphic functions of order n and applications to dynamic equations on time scales, Discrete Dyn. Nat. Soc., (2014), 1–13. doi: 10.1155/2014/410210.  Google Scholar

[16]

J. von Neumann, Almost periodic functions in a group, I, Trans. Amer. Math. Soc., 36 (1934), 445-492.  doi: 10.1090/S0002-9947-1934-1501752-3.  Google Scholar

[17]

G. M. N'Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.  Google Scholar

[18]

G. M. N'Guérékata, Almost Automorphic and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4757-4482-8.  Google Scholar

[19]

G. M. N'Guérékata and A. Pankov, Stepanov-like almost automorphic functions and monotone evolution equations, Nonlinear Anal., 68 (2008), 2658-2667.  doi: 10.1016/j.na.2007.02.012.  Google Scholar

[20]

W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751.  doi: 10.2307/2373071.  Google Scholar

[21]

C. Wang and R. P. Agarwal, Weighted piecewise pseudo almost automorphic functions with applications to abstract impulsive $\nabla$-dynamic equations on time scales, Adv. Difference Equ., 153 (2014), 1-29.  doi: 10.1186/1687-1847-2014-153.  Google Scholar

[22]

C. Wang, R. P. Agarwal and D. O'Regan, n0-order $\Delta$-almost periodic functions and dynamic equations, Appl. Anal., 97 (2018), 2626-2654.  doi: 10.1080/00036811.2017.1382689.  Google Scholar

[23]

C. Wang, R. P. Agarwal and D. O'Regan, Periodicity, almost periodicity for time scales and related functions, Nonauton. Dyn. Syst., 3 (2016), 24-41.  doi: 10.1515/msds-2016-0003.  Google Scholar

[24]

C. Wang, R. P. Agarwal, D. O'Regan, C. Wang and R. P. Agarwal, Relatively dense sets, corrected uniformly almost periodic functions on time scales, and generalizations, Adv. Difference Equ., 312 (2015), 1-9.  doi: 10.1186/s13662-015-0650-0.  Google Scholar

[25]

C. Wang, R. P Agarwal and D. O'Regan, A matched space for time scales and applications to the study on functions, Adv. Difference Equ., 305 (2017), 1-28.  doi: 10.1186/s13662-017-1366-0.  Google Scholar

[26]

C. Wang, R. P Agarwal and D. O'Regan, Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations, J. Nonlinear Sci. Appl., 10 (2017), 3863-3886.  doi: 10.22436/jnsa.010.07.41.  Google Scholar

[27]

C. Wang, R. P Agarwal and D. O'Regan, Π-semigroup for invariant under translations time scales and abstract weighted pseudo almost periodic functions with applications, Dynam. Systems Appl., 25 (2016), 1-28.   Google Scholar

[28]

C. Wang and R. P. Agarwal, Almost periodic solution for a new type of neutral impulsive stochastic Lasota-Wazewska timescale model, Appl. Math. Lett., 70 (2017), 58-65.  doi: 10.1016/j.aml.2017.03.009.  Google Scholar

[29]

T. Xiao, J. Liang and J. Zhang, Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces, Semigroup Forum, 76 (2008), 518-524.  doi: 10.1007/s00233-007-9011-y.  Google Scholar

[30]

M. Zaki, Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl., 101 (1974), 91-114.  doi: 10.1007/BF02417100.  Google Scholar

[31]

Z. M. Zheng and H. S. Ding, On completeness of the space of weighted pseudo almost automorphic functions, J. Funct. Anal., 268 (2015), 3211-3218.  doi: 10.1016/j.jfa.2015.02.012.  Google Scholar

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