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

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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 February 2020 Early access 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 and 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.
[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.
[3]	S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Univ. e Politec. Torino Rend. Sem. Mat., 15 (2019), 225-253.
[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.
[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.
[6]	M. Bohner and J. G. Mesquita, Almost periodic functions in quantum calculus, Electron. J. Differential Equations, 2018, 1–11.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	M. Kéré and G. M. N'Guérékata, Almost automorphic dynamic systems on time scales, PanAmer. Math. J., 28 (2018), 19-37.
[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.
[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.
[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.
[17]	G. M. N'Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.
[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.
[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.
[20]	W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751. doi: 10.2307/2373071.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[30]	M. Zaki, Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl., 101 (1974), 91-114. doi: 10.1007/BF02417100.
[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.

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.
[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.
[3]	S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Univ. e Politec. Torino Rend. Sem. Mat., 15 (2019), 225-253.
[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.
[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.
[6]	M. Bohner and J. G. Mesquita, Almost periodic functions in quantum calculus, Electron. J. Differential Equations, 2018, 1–11.
[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.
[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.
[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.
[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.
[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.
[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.
[13]	M. Kéré and G. M. N'Guérékata, Almost automorphic dynamic systems on time scales, PanAmer. Math. J., 28 (2018), 19-37.
[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.
[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.
[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.
[17]	G. M. N'Guérékata, Topics in Almost Automorphy, Springer-Verlag, New York, 2005.
[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.
[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.
[20]	W. A. Veech, Almost automorphic functions on groups, Amer. J. Math., 87 (1965), 719-751. doi: 10.2307/2373071.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[30]	M. Zaki, Almost automorphic solutions of certain abstract differential equations, Ann. Mat. Pura Appl., 101 (1974), 91-114. doi: 10.1007/BF02417100.
[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.

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