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On an optimal control problem of time-fractional advection-diffusion equation
Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations
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 |
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 [
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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