American Institute of Mathematical Sciences

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doi: 10.3934/eect.2021006

$ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations

Michal Fečkan 1,2, , Kui Liu 3,4, and  JinRong Wang 3,5,,

1. 

Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina 842 48, Bratislava, Slovakia
2. 

Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49,814 73 Bratislava, Slovakia
3. 

Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P. R. China
4. 

College of Science, Guizhou Institute of Technology, Guiyang, Guizhou 550025, P. R. China
5. 

School of Mathematical Sciences, Qufu Normal University, , Qufu, Shandong 273165, P. R. China

* Corresponding author: JinRong Wang

Received  September 2020 Revised  November 2020 Published  January 2021

In this paper, we study $ (\omega,\mathbb{T}) $-periodic impulsive evolution equations via the operator semigroups theory in Banach spaces $ X $, where $ \mathbb{T}: X\rightarrow X $ is a linear isomorphism. Existence and uniqueness of $ (\omega,\mathbb{T}) $-periodic solutions results for linear and semilinear problems are obtained by Fredholm alternative theorem and fixed point theorems, which extend the related results for periodic impulsive differential equations.

Keywords: Impulsive evolution equations, $ (\omega \mathbb{T}) $-periodic solutions, existence and uniqueness.
Mathematics Subject Classification: 34A37, 93B05, 93C25.
Citation: Michal Fečkan, Kui Liu, JinRong Wang. $ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2021006
References:
[1]

N. U. Ahmed, K. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis: TMA, 54 (2003), 907-925.  doi: 10.1016/S0362-546X(03)00117-2.  Google Scholar

[2]

E. Alvarez, A. Gómez and M. Pinto, $(\omega, c)$-periodic functions and mild solutions to abstract fractional integro-differential equations, Electron. J. Qual. Theory Differ. Equ., (2018), 16–24. doi: 10.14232/ejqtde.2018.1.16.  Google Scholar

[3]

M. Agaoglou, M. Fečkan and A. Panagiotidou, Existence and uniqueness of $(\omega, c)$-periodic solutions of semilinear evolution equations, Int. J. Dynamical Systems and Differential Equations, 10 (2020), 149-166.  doi: 10.1504/IJDSDE.2020.106027.  Google Scholar

[4]

E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)-$Pseudo periodic functions, first order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells, Bound. Value Prob., (2019), 106–126. doi: 10.1186/s13661-019-1217-x.  Google Scholar

[5]

E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)-$asymptotically periodic functions, first-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Meth. Appl. Sci., 43 (2020), 305-319.  doi: 10.1002/mma.5880.  Google Scholar

[6]

D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, vol. 28. Singapore, World Scientifc, 1995. doi: 10.1142/9789812831804.  Google Scholar

[7]

D. Bainov and P. Simeonov, Oscillation Theory of Impulsive Differential Equations, Interna-tional Publications, 1998.  Google Scholar

[8]

C. Cooke and J. Kroll, The existence of periodic solutions to certain impulsive differential equations, Comput. Math. Appl., 44 (2002), 667-676.  doi: 10.1016/S0898-1221(02)00181-5.  Google Scholar

[9]

X. Chang and Y. Li, Rotating periodic solutions of second order dissipative dynamical systems, Discret. Contin. Dyn. Syst., 36 (2016), 633-652.   Google Scholar

[10]

M. Fečkan, J. Wang and Y. Zhou, Existence of periodic solutions for nonlinear evolution equations with non- instantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101.   Google Scholar

[11]

M. Fečkan, R. Ma and B. Thompson, Forced symmetric oscillations, Bull. Belg. Math. Soc., 14 (2007), 73-85.  doi: 10.36045/bbms/1172852245.  Google Scholar

[12]

Y. Li, F. Cong, Z. Lin and W. liu, Periodic solutions for evolution equations, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 275-293.  doi: 10.1016/S0362-546X(97)00626-3.  Google Scholar

[13]

X. Li, B. Martin and C. Wang, Impulsive differential equations: Periodic solutions and applications, Automatica, 52 (2015), 173-178.  doi: 10.1016/j.automatica.2014.11.009.  Google Scholar

[14]

M. Li, J. Wang and M. Fečkan, $(\omega, c)$-periodic solutions for impulsive differential systems, Communications Mathematical Analysis, 21 (2018), 35-46.   Google Scholar

[15]

K. Liu, J. Wang, D. O'Regan and M. Fečkan, A new class of $(\omega, c)$-periodic non-instantaneous impulsive differential equations, Mediterr. J. Math., 17 (2020), 155-177.  doi: 10.1007/s00009-020-01574-8.  Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[17]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664.  Google Scholar

[18]

C. Wang, Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2828-2842.  doi: 10.1016/j.cnsns.2013.12.038.  Google Scholar

[19]

J. Wang and M. Fečkan, A general class of impulsive evolution equations, Topol. Meth. Nonlinear Anal., 46 (2015), 915-934.   Google Scholar

[20]

J. Wang, X. Xiang and W. Wei, Linear impulsive periodic system with time-varying generating operators on Banach space, Adv. Differ. Equ., (2007), 26196, 16 pp. doi: 10.1155/2007/26196.  Google Scholar

show all references

References:
[1]

N. U. Ahmed, K. L. Teo and S. H. Hou, Nonlinear impulsive systems on infinite dimensional spaces, Nonlinear Analysis: TMA, 54 (2003), 907-925.  doi: 10.1016/S0362-546X(03)00117-2.  Google Scholar

[2]

E. Alvarez, A. Gómez and M. Pinto, $(\omega, c)$-periodic functions and mild solutions to abstract fractional integro-differential equations, Electron. J. Qual. Theory Differ. Equ., (2018), 16–24. doi: 10.14232/ejqtde.2018.1.16.  Google Scholar

[3]

M. Agaoglou, M. Fečkan and A. Panagiotidou, Existence and uniqueness of $(\omega, c)$-periodic solutions of semilinear evolution equations, Int. J. Dynamical Systems and Differential Equations, 10 (2020), 149-166.  doi: 10.1504/IJDSDE.2020.106027.  Google Scholar

[4]

E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)-$Pseudo periodic functions, first order Cauchy problem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red cells, Bound. Value Prob., (2019), 106–126. doi: 10.1186/s13661-019-1217-x.  Google Scholar

[5]

E. Alvarez, S. Castillo and M. Pinto, $(\omega, c)-$asymptotically periodic functions, first-order Cauchy problem, and Lasota-Wazewska model with unbounded oscillating production of red cells, Math. Meth. Appl. Sci., 43 (2020), 305-319.  doi: 10.1002/mma.5880.  Google Scholar

[6]

D. Bainov and P. Simeonov, Theory of Impulsive Differential Equations, Series on Advances in Mathematics for Applied Sciences, vol. 28. Singapore, World Scientifc, 1995. doi: 10.1142/9789812831804.  Google Scholar

[7]

D. Bainov and P. Simeonov, Oscillation Theory of Impulsive Differential Equations, Interna-tional Publications, 1998.  Google Scholar

[8]

C. Cooke and J. Kroll, The existence of periodic solutions to certain impulsive differential equations, Comput. Math. Appl., 44 (2002), 667-676.  doi: 10.1016/S0898-1221(02)00181-5.  Google Scholar

[9]

X. Chang and Y. Li, Rotating periodic solutions of second order dissipative dynamical systems, Discret. Contin. Dyn. Syst., 36 (2016), 633-652.   Google Scholar

[10]

M. Fečkan, J. Wang and Y. Zhou, Existence of periodic solutions for nonlinear evolution equations with non- instantaneous impulses, Nonauton. Dyn. Syst., 1 (2014), 93-101.   Google Scholar

[11]

M. Fečkan, R. Ma and B. Thompson, Forced symmetric oscillations, Bull. Belg. Math. Soc., 14 (2007), 73-85.  doi: 10.36045/bbms/1172852245.  Google Scholar

[12]

Y. Li, F. Cong, Z. Lin and W. liu, Periodic solutions for evolution equations, Nonlinear Analysis: Theory, Methods and Applications, 36 (1999), 275-293.  doi: 10.1016/S0362-546X(97)00626-3.  Google Scholar

[13]

X. Li, B. Martin and C. Wang, Impulsive differential equations: Periodic solutions and applications, Automatica, 52 (2015), 173-178.  doi: 10.1016/j.automatica.2014.11.009.  Google Scholar

[14]

M. Li, J. Wang and M. Fečkan, $(\omega, c)$-periodic solutions for impulsive differential systems, Communications Mathematical Analysis, 21 (2018), 35-46.   Google Scholar

[15]

K. Liu, J. Wang, D. O'Regan and M. Fečkan, A new class of $(\omega, c)$-periodic non-instantaneous impulsive differential equations, Mediterr. J. Math., 17 (2020), 155-177.  doi: 10.1007/s00009-020-01574-8.  Google Scholar

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[17]

A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. doi: 10.1142/9789812798664.  Google Scholar

[18]

C. Wang, Almost periodic solutions of impulsive BAM neural networks with variable delays on time scales, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 2828-2842.  doi: 10.1016/j.cnsns.2013.12.038.  Google Scholar

[19]

J. Wang and M. Fečkan, A general class of impulsive evolution equations, Topol. Meth. Nonlinear Anal., 46 (2015), 915-934.   Google Scholar

[20]

J. Wang, X. Xiang and W. Wei, Linear impulsive periodic system with time-varying generating operators on Banach space, Adv. Differ. Equ., (2007), 26196, 16 pp. doi: 10.1155/2007/26196.  Google Scholar

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