April  2022, 11(2): 415-437. doi: 10.3934/eect.2021006

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

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  April 2022 Early access  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.

Citation: Michal Fečkan, Kui Liu, JinRong Wang. $ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations. Evolution Equations and Control Theory, 2022, 11 (2) : 415-437. doi: 10.3934/eect.2021006
References:
[1]

N. U. AhmedK. 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.

[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.

[3]

M. AgaoglouM. 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.

[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.

[5]

E. AlvarezS. 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.

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[12]

Y. LiF. CongZ. 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.

[13]

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

[14]

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

[15]

K. LiuJ. WangD. 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.

[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.

[17]

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

[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.

[19]

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

[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.

show all references

References:
[1]

N. U. AhmedK. 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.

[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.

[3]

M. AgaoglouM. 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.

[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.

[5]

E. AlvarezS. 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.

[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.

[7]

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

[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.

[9]

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

[10]

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

[11]

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

[12]

Y. LiF. CongZ. 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.

[13]

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

[14]

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

[15]

K. LiuJ. WangD. 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.

[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.

[17]

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

[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.

[19]

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

[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.

[1]

Pallavi Bedi, Anoop Kumar, Thabet Abdeljawad, Aziz Khan. S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations. Evolution Equations and Control Theory, 2021, 10 (4) : 733-748. doi: 10.3934/eect.2020089

[2]

Pablo Amster, Mariel Paula Kuna, Dionicio Santos. Stability, existence and non-existence of $ T $-periodic solutions of nonlinear delayed differential equations with $ \varphi $-Laplacian. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2723-2737. doi: 10.3934/cpaa.2022070

[3]

E. Compaan. A note on global existence for the Zakharov system on $ \mathbb{T} $. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2473-2489. doi: 10.3934/cpaa.2019112

[4]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ Ⅱ: Discrete torus bifurcations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1847-1874. doi: 10.3934/cpaa.2020081

[5]

Christian Aarset, Christian Pötzsche. Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 1-60. doi: 10.3934/dcdsb.2020231

[6]

Yang Liu, Chunyou Sun. Inviscid limit for the damped generalized incompressible Navier-Stokes equations on $ \mathbb{T}^2 $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4383-4408. doi: 10.3934/dcdss.2021124

[7]

Ziqing Yuan, Jianshe Yu. Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $ \mathbb R^N $$ ^\diamondsuit $. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3285-3303. doi: 10.3934/dcdss.2020281

[8]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[9]

Pablo Amster, Alberto Déboli, Manuel Pinto. Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3019-3037. doi: 10.3934/dcdsb.2021171

[10]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations and Control Theory, 2022, 11 (3) : 621-633. doi: 10.3934/eect.2021017

[11]

Lu Chen, Guozhen Lu, Yansheng Shen. Sharp subcritical Sobolev inequalities and uniqueness of nonnegative solutions to high-order Lane-Emden equations on $ \mathbb{S}^n $. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2799-2817. doi: 10.3934/cpaa.2022073

[12]

Fengshuang Gao, Yuxia Guo. Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5591-5616. doi: 10.3934/dcds.2020239

[13]

Qianqian Han, Bo Deng, Xiao-Song Yang. The existence of $ \omega $-limit set for a modified Nosé-Hoover oscillator. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022043

[14]

Changguang Dong, Adam Kanigowski. Rigidity of a class of smooth singular flows on $ \mathbb{T}^2 $. Journal of Modern Dynamics, 2020, 16: 37-57. doi: 10.3934/jmd.2020002

[15]

Gyu Eun Lee. Local wellposedness for the critical nonlinear Schrödinger equation on $ \mathbb{T}^3 $. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2763-2783. doi: 10.3934/dcds.2019116

[16]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051

[17]

Mohamed Karim Hamdani, Lamine Mbarki, Mostafa Allaoui, Omar Darhouche, Dušan D. Repovš. Existence and multiplicity of solutions involving the $ p(x) $-Laplacian equations: On the effect of two nonlocal terms. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022129

[18]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete and Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

[19]

Yinbin Deng, Wei Shuai. Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3139-3168. doi: 10.3934/dcds.2018137

[20]

Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $ t $-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030

2021 Impact Factor: 1.169

Metrics

  • PDF downloads (412)
  • HTML views (489)
  • Cited by (0)

Other articles
by authors

[Back to Top]