doi: 10.3934/dcdsb.2020211

Periodic solutions to non-autonomous evolution equations with multi-delays

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received  March 2020 Revised  April 2020 Published  July 2020

Fund Project: Research supported by National Natural Science Foundations of China (No. 11501455, No. 11661071), Project of NWNU-LKQN2019-3 and China Scholarship Council (No. 201908625016)

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 [37] to non-autonomous cases, but also greatly weaken the condition presented in [37] even for the case $ a(t)\equiv a $ by establishing a general abstract framework to find time $ \omega $-periodic mild solutions for non-autonomous evolution equation with multi-delays. Finally, one illustrating example is supplied.

Citation: Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020211
References:
[1]

P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457.   Google Scholar

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] T. Burton, Stability and Periodic Solutions of Ordinary Differential Equations and Functional Differential Equations, Academic Press, Orlando, FL, 1985.   Google Scholar
[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.  Google Scholar

[7]

A. CaicedoC. CuevasG. 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.  Google Scholar

[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.  Google Scholar

[9]

P. ChenX. 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.  Google Scholar

[10]

P. ChenX. 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.  Google Scholar

[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.  Google Scholar

[12]

P. ChenX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15 pp.  Google Scholar

[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.  Google Scholar

[17]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.  Google Scholar

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. LiangJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[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.  Google Scholar

[28]

J. H. SoJ. Wu and X. Zou, Structured population on two patches: Modeling desperal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081.  Google Scholar

[29]

H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. N. WangK. 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.  Google Scholar

[33]

Z. WangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

J. ZhuY. 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.  Google Scholar

[38]

B. ZhuL. 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.  Google Scholar

[39]

B. ZhuL. 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.  Google Scholar

show all references

References:
[1]

P. Acquistapace, Evolution operators and strong solution of abstract parabolic equations, Differential Integral Equations, 1 (1988), 433-457.   Google Scholar

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear parabolic equations, Rend. Semin. Mat. Univ. Padova, 78 (1987), 47-107.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5] T. Burton, Stability and Periodic Solutions of Ordinary Differential Equations and Functional Differential Equations, Academic Press, Orlando, FL, 1985.   Google Scholar
[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.  Google Scholar

[7]

A. CaicedoC. CuevasG. 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.  Google Scholar

[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.  Google Scholar

[9]

P. ChenX. 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.  Google Scholar

[10]

P. ChenX. 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.  Google Scholar

[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.  Google Scholar

[12]

P. ChenX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

X. Fu, Existence of solutions for non-autonomous functional evolution equations with nonlocal conditions, Electron. J. Differential Equations, 2012 (2012), 15 pp.  Google Scholar

[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.  Google Scholar

[17]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.  Google Scholar

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988.  Google Scholar

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

J. LiangJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Prüss, Evolutionary Integral Equations and Applications, Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.  Google Scholar

[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.  Google Scholar

[28]

J. H. SoJ. Wu and X. Zou, Structured population on two patches: Modeling desperal and delay, J. Math. Biol., 43 (2001), 37-51.  doi: 10.1007/s002850100081.  Google Scholar

[29]

H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Marcel Dekker, New York, USA, 1997.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. N. WangK. 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.  Google Scholar

[33]

Z. WangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

J. ZhuY. 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.  Google Scholar

[38]

B. ZhuL. 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.  Google Scholar

[39]

B. ZhuL. 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.  Google Scholar

[1]

Mahesh G. Nerurkar. Spectral and stability questions concerning evolution of non-autonomous linear systems. Conference Publications, 2001, 2001 (Special) : 270-275. doi: 10.3934/proc.2001.2001.270

[2]

K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020038

[3]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032

[4]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[5]

Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020076

[6]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[7]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020171

[8]

Saulo R.M. Barros, Antônio L. Pereira, Cláudio Possani, Adilson Simonis. Spatially periodic equilibria for a non local evolution equation. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 937-948. doi: 10.3934/dcds.2003.9.937

[9]

Abdelaziz Rhandi, Roland Schnaubelt. Asymptotic behaviour of a non-autonomous population equation with diffusion in $L^1$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 663-683. doi: 10.3934/dcds.1999.5.663

[10]

Hongmei Cheng, Rong Yuan. Existence and asymptotic stability of traveling fronts for nonlocal monostable evolution equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 3007-3022. doi: 10.3934/dcdsb.2017160

[11]

Xianlong Fu. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026

[12]

Tôn Việt Tạ. Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4507-4542. doi: 10.3934/dcds.2017193

[13]

Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473

[14]

Pengyu Chen, Xuping Zhang. Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020308

[15]

Aníbal Rodríguez-Bernal, Alejandro Vidal–López. Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 537-567. doi: 10.3934/dcds.2007.18.537

[16]

Tomás Caraballo, Antonio M. Márquez-Durán, Rivero Felipe. Asymptotic behaviour of a non-classical and non-autonomous diffusion equation containing some hereditary characteristic. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1817-1833. doi: 10.3934/dcdsb.2017108

[17]

Fatih Bayazit, Ulrich Groh, Rainer Nagel. Floquet representations and asymptotic behavior of periodic evolution families. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4795-4810. doi: 10.3934/dcds.2013.33.4795

[18]

Hongbin Chen, Yi Li. Existence, uniqueness, and stability of periodic solutions of an equation of duffing type. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 793-807. doi: 10.3934/dcds.2007.18.793

[19]

Zhijian Yang, Yanan Li. Criteria on the existence and stability of pullback exponential attractors and their application to non-autonomous kirchhoff wave models. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2629-2653. doi: 10.3934/dcds.2018111

[20]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020233

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (37)
  • HTML views (130)
  • Cited by (0)

Other articles
by authors

[Back to Top]