American Institute of Mathematical Sciences

Advanced
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 & 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

[1]

Rui Zhang, Yong-Kui Chang, G. M. N'Guérékata. Weighted pseudo almost automorphic mild solutions to semilinear integral equations with $S^{p}$-weighted pseudo almost automorphic coefficients. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5525-5537. doi: 10.3934/dcds.2013.33.5525
[2]

Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857
[3]

Gaston Mandata N ' Guerekata. Remarks on almost automorphic differential equations. Conference Publications, 2001, 2001 (Special) : 276-279. doi: 10.3934/proc.2001.2001.276
[4]

Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078
[5]

Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026
[6]

Aníbal Coronel, Christopher Maulén, Manuel Pinto, Daniel Sepúlveda. Almost automorphic delayed differential equations and Lasota-Wazewska model. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1959-1977. doi: 10.3934/dcds.2017083
[7]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745
[8]

Yongkun Li, Pan Wang. Almost periodic solution for neutral functional dynamic equations with Stepanov-almost periodic terms on time scales. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 463-473. doi: 10.3934/dcdss.2017022
[9]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51
[10]

Gaston N'Guerekata. On weak-almost periodic mild solutions of some linear abstract differential equations. Conference Publications, 2003, 2003 (Special) : 672-677. doi: 10.3934/proc.2003.2003.672
[11]

Ruichao Guo, Yong Li, Jiamin Xing, Xue Yang. Existence of periodic solutions of dynamic equations on time scales by averaging. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 959-971. doi: 10.3934/dcdss.2017050
[12]

Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
[13]

Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161
[14]

Yuval Z. Flicker. Automorphic forms on PGSp(2). Electronic Research Announcements, 2004, 10: 39-50.

[15]

Reinhard Farwig, Yasushi Taniuchi. Uniqueness of backward asymptotically almost periodic-in-time solutions to Navier-Stokes equations in unbounded domains. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1215-1224. doi: 10.3934/dcdss.2013.6.1215
[16]

Marko Kostić. Almost periodic type functions and densities. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021008
[17]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[18]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113
[19]

Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671
[20]

Demou Luo, Qiru Wang. Dynamic analysis on an almost periodic predator-prey system with impulsive effects and time delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3427-3453. doi: 10.3934/dcdsb.2020238

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (368)
  • HTML views (157)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy