-
Previous Article
Dynamic observers for unknown populations
- DCDS-B Home
- This Issue
-
Next Article
Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency
Periodic solutions to non-autonomous evolution equations with multi-delays
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China |
In this paper, we provide some sufficient conditions for the existence, uniqueness and asymptotic stability of time $ \omega $-periodic mild solutions for a class of non-autonomous evolution equation with multi-delays. This work not only extend the autonomous evolution equation with multi-delays studied in [
References:
[1] |
P. Acquistapace,
Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457.
|
[2] |
P. Acquistapace and B. Terreni,
A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107.
|
[3] |
H. Amann, Periodic solutions of semilinear parabolic equations, in: Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe (eds. L. Cesari, R. Kannan and R. Weinberger), Academic Press, New York, (1978), 1–29. |
[4] |
H. Amann,
Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.
doi: 10.1016/0022-0396(88)90156-8. |
[5] |
T. Burton, Stability and Periodic Solutions of Ordinary Differential Equations and Functional Differential Equations, Academic Press, Orlando, FL, 1985.
![]() |
[6] |
T. Burton and B. Zhang,
Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations, 90 (1991), 357-396.
doi: 10.1016/0022-0396(91)90153-Z. |
[7] |
A. Caicedo, C. Cuevas, G. Mophou and G. N'Guérékata,
Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces, J. Franklin Inst., 349 (2012), 1-24.
doi: 10.1016/j.jfranklin.2011.02.001. |
[8] |
X. Chen and J. S. Guo,
Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.
doi: 10.1006/jdeq.2001.4153. |
[9] |
P. Chen, X. Zhang and Y. Li,
Cauchy problem for fractional non-autonomous evolution equations, Banach J. Math. Anal., 14 (2020), 559-584.
doi: 10.1007/s43037-019-00008-2. |
[10] |
P. Chen, X. Zhang and Y. Li,
A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.
doi: 10.3934/cpaa.2018094. |
[11] |
P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), Paper No. 118, 14 pp.
doi: 10.1007/s00009-019-1384-0. |
[12] |
P. Chen, X. Zhang and Y. Li,
Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ.Oper. Appl., 10 (2019), 955-973.
doi: 10.1007/s11868-018-0257-9. |
[13] |
P. Chen, Y. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B, 2020.
doi: 10.3934/dcdsb.2020171. |
[14] |
W. E. Fitzgibbon,
Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.
doi: 10.1016/0022-0396(78)90037-2. |
[15] |
X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15 pp. |
[16] |
X. Fu and Y. Zhang,
Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 747-757.
doi: 10.1016/S0252-9602(13)60035-1. |
[17] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet,
Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981. |
[20] |
Y. Li,
Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261 (2011), 1309-1324.
doi: 10.1016/j.jfa.2011.05.001. |
[21] |
D. Li and Y. Wang,
Asymptotic behavior of gradient systems with small time delays, Nonlinear Anal. Real World Appl., 11 (2010), 1627-1633.
doi: 10.1016/j.nonrwa.2009.03.015. |
[22] |
J. Liang, J. H. Liu and T. J. Xiao,
Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.
doi: 10.3934/cpaa.2006.5.529. |
[23] |
Y. Liu and Z. Li,
Schaefer type theorem and periodic solutions of evolution equations, J. Math. Anal. Appl., 316 (2006), 237-255.
doi: 10.1016/j.jmaa.2005.04.045. |
[24] |
Z. Ouyang,
Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61 (2011), 860-870.
doi: 10.1016/j.camwa.2010.12.034. |
[25] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[26] |
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[27] |
S. H. Saker,
Oscillation and global attractivity in hematopoiesis model with delay time, Appl. Math. Comput., 136 (2003), 241-250.
doi: 10.1016/S0096-3003(02)00035-8. |
[28] |
J. H. So, J. Wu and X. Zou,
Structured population on two patches: Modeling desperal and delay, J. Math. Biol., 43 (2001), 37-51.
doi: 10.1007/s002850100081. |
[29] |
H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997. |
[30] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second ed., Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[31] |
R. N. Wang and P. X. Zhu,
Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013), 180-191.
doi: 10.1016/j.na.2013.02.026. |
[32] |
R. N. Wang, K. Ezzinbi and P. X. Zhu,
Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.
doi: 10.1216/JIE-2014-26-2-275. |
[33] |
Z. Wang, Y. Liu and X. Liu,
On global asymptotic stability of neural networks with discrete and distributed delays, Physics Lett. A, 345 (2005), 299-308.
doi: 10.1016/j.physleta.2005.07.025. |
[34] |
M. Wazewska-Czyzevsia and A. Lasota, Mathematical problems of dynamics of red blood cell system, Ann. Polish Math. Soc. Ser. 3 Appl. Math., 17 (1976), 23-40. Google Scholar |
[35] |
J. Wu, Theory and Application of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[36] |
X. Xiang and N. U. Ahmed,
Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear Anal., 18 (1992), 1063-1070.
doi: 10.1016/0362-546X(92)90195-K. |
[37] |
J. Zhu, Y. Liu and Z. Li,
The existence and attractivity of time periodic solutions for evolution equations with delays, Nonlinear Anal. Real World Appl., 9 (2008), 842-851.
doi: 10.1016/j.nonrwa.2007.01.004. |
[38] |
B. Zhu, L. Liu and Y. Wu,
Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations, Fract. Calc. Appl. Anal., 20 (2017), 1338-1355.
doi: 10.1515/fca-2017-0071. |
[39] |
B. Zhu, L. Liu and Y. Wu,
Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79.
doi: 10.1016/j.aml.2016.05.010. |
show all references
References:
[1] |
P. Acquistapace,
Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457.
|
[2] |
P. Acquistapace and B. Terreni,
A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107.
|
[3] |
H. Amann, Periodic solutions of semilinear parabolic equations, in: Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe (eds. L. Cesari, R. Kannan and R. Weinberger), Academic Press, New York, (1978), 1–29. |
[4] |
H. Amann,
Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269.
doi: 10.1016/0022-0396(88)90156-8. |
[5] |
T. Burton, Stability and Periodic Solutions of Ordinary Differential Equations and Functional Differential Equations, Academic Press, Orlando, FL, 1985.
![]() |
[6] |
T. Burton and B. Zhang,
Periodic solutions of abstract differential equations with infinite delay, J. Differential Equations, 90 (1991), 357-396.
doi: 10.1016/0022-0396(91)90153-Z. |
[7] |
A. Caicedo, C. Cuevas, G. Mophou and G. N'Guérékata,
Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces, J. Franklin Inst., 349 (2012), 1-24.
doi: 10.1016/j.jfranklin.2011.02.001. |
[8] |
X. Chen and J. S. Guo,
Existence and asymptotic stability of traveling waves of discrete quasilinear monostable equations, J. Differential Equations, 184 (2002), 549-569.
doi: 10.1006/jdeq.2001.4153. |
[9] |
P. Chen, X. Zhang and Y. Li,
Cauchy problem for fractional non-autonomous evolution equations, Banach J. Math. Anal., 14 (2020), 559-584.
doi: 10.1007/s43037-019-00008-2. |
[10] |
P. Chen, X. Zhang and Y. Li,
A blowup alternative result for fractional nonautonomous evolution equation of Volterra type, Commun. Pure Appl. Anal., 17 (2018), 1975-1992.
doi: 10.3934/cpaa.2018094. |
[11] |
P. Chen, X. Zhang and Y. Li, Non-autonomous evolution equations of parabolic type with non-instantaneous impulses, Mediterr. J. Math., 16 (2019), Paper No. 118, 14 pp.
doi: 10.1007/s00009-019-1384-0. |
[12] |
P. Chen, X. Zhang and Y. Li,
Fractional non-autonomous evolution equation with nonlocal conditions, J. Pseudo-Differ.Oper. Appl., 10 (2019), 955-973.
doi: 10.1007/s11868-018-0257-9. |
[13] |
P. Chen, Y. Li and X. Zhang, Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families, Discrete Contin. Dyn. Syst. Ser. B, 2020.
doi: 10.3934/dcdsb.2020171. |
[14] |
W. E. Fitzgibbon,
Semilinear functional equations in Banach space, J. Differential Equations, 29 (1978), 1-14.
doi: 10.1016/0022-0396(78)90037-2. |
[15] |
X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15 pp. |
[16] |
X. Fu and Y. Zhang,
Exact null controllability of non-autonomous functional evolution systems with nonlocal conditions, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 747-757.
doi: 10.1016/S0252-9602(13)60035-1. |
[17] |
W. S. C. Gurney, S. P. Blythe and R. M. Nisbet,
Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[18] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. |
[19] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981. |
[20] |
Y. Li,
Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261 (2011), 1309-1324.
doi: 10.1016/j.jfa.2011.05.001. |
[21] |
D. Li and Y. Wang,
Asymptotic behavior of gradient systems with small time delays, Nonlinear Anal. Real World Appl., 11 (2010), 1627-1633.
doi: 10.1016/j.nonrwa.2009.03.015. |
[22] |
J. Liang, J. H. Liu and T. J. Xiao,
Nonlocal Cauchy problems for nonautonomous evolution equations, Commun. Pure Appl. Anal., 5 (2006), 529-535.
doi: 10.3934/cpaa.2006.5.529. |
[23] |
Y. Liu and Z. Li,
Schaefer type theorem and periodic solutions of evolution equations, J. Math. Anal. Appl., 316 (2006), 237-255.
doi: 10.1016/j.jmaa.2005.04.045. |
[24] |
Z. Ouyang,
Existence and uniqueness of the solutions for a class of nonlinear fractional order partial differential equations with delay, Comput. Math. Appl., 61 (2011), 860-870.
doi: 10.1016/j.camwa.2010.12.034. |
[25] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[26] |
J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993.
doi: 10.1007/978-3-0348-8570-6. |
[27] |
S. H. Saker,
Oscillation and global attractivity in hematopoiesis model with delay time, Appl. Math. Comput., 136 (2003), 241-250.
doi: 10.1016/S0096-3003(02)00035-8. |
[28] |
J. H. So, J. Wu and X. Zou,
Structured population on two patches: Modeling desperal and delay, J. Math. Biol., 43 (2001), 37-51.
doi: 10.1007/s002850100081. |
[29] |
H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997. |
[30] |
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Second ed., Springer-Verlag, New York, 1997.
doi: 10.1007/978-1-4612-0645-3. |
[31] |
R. N. Wang and P. X. Zhu,
Non-autonomous evolution inclusions with nonlocal history conditions: Global integral solutions, Nonlinear Anal., 85 (2013), 180-191.
doi: 10.1016/j.na.2013.02.026. |
[32] |
R. N. Wang, K. Ezzinbi and P. X. Zhu,
Non-autonomous impulsive Cauchy problems of parabolic type involving nonlocal initial conditions, J. Integral Equations Appl., 26 (2014), 275-299.
doi: 10.1216/JIE-2014-26-2-275. |
[33] |
Z. Wang, Y. Liu and X. Liu,
On global asymptotic stability of neural networks with discrete and distributed delays, Physics Lett. A, 345 (2005), 299-308.
doi: 10.1016/j.physleta.2005.07.025. |
[34] |
M. Wazewska-Czyzevsia and A. Lasota, Mathematical problems of dynamics of red blood cell system, Ann. Polish Math. Soc. Ser. 3 Appl. Math., 17 (1976), 23-40. Google Scholar |
[35] |
J. Wu, Theory and Application of Partial Functional Differential Equations, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[36] |
X. Xiang and N. U. Ahmed,
Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear Anal., 18 (1992), 1063-1070.
doi: 10.1016/0362-546X(92)90195-K. |
[37] |
J. Zhu, Y. Liu and Z. Li,
The existence and attractivity of time periodic solutions for evolution equations with delays, Nonlinear Anal. Real World Appl., 9 (2008), 842-851.
doi: 10.1016/j.nonrwa.2007.01.004. |
[38] |
B. Zhu, L. Liu and Y. Wu,
Local and global existence of mild solutions for a class of semilinear fractional integro-differential equations, Fract. Calc. Appl. Anal., 20 (2017), 1338-1355.
doi: 10.1515/fca-2017-0071. |
[39] |
B. Zhu, L. Liu and Y. Wu,
Local and global existence of mild solutions for a class of nonlinear fractional reaction-diffusion equations with delay, Appl. Math. Lett., 61 (2016), 73-79.
doi: 10.1016/j.aml.2016.05.010. |
[1] |
Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020383 |
[2] |
Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171 |
[3] |
Divine Wanduku. Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021005 |
[4] |
Yanan Li, Zhijian Yang, Na Feng. Uniform attractors and their continuity for the non-autonomous Kirchhoff wave models. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021018 |
[5] |
Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080 |
[6] |
Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242 |
[7] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[8] |
Michal Fečkan, Kui Liu, JinRong Wang. $ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021006 |
[9] |
Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020376 |
[10] |
Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075 |
[11] |
Maika Goto, Kazunori Kuwana, Yasuhide Uegata, Shigetoshi Yazaki. A method how to determine parameters arising in a smoldering evolution equation by image segmentation for experiment's movies. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 881-891. doi: 10.3934/dcdss.2020233 |
[12] |
Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265 |
[13] |
Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077 |
[14] |
Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 |
[15] |
Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119 |
[16] |
Erica Ipocoana, Andrea Zafferi. Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020289 |
[17] |
Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020 doi: 10.3934/dcds.2020400 |
[18] |
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104 |
[19] |
Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 |
[20] |
Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2020 doi: 10.3934/nhm.2021002 |
2019 Impact Factor: 1.27
Tools
Article outline
[Back to Top]