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June  2021, 26(6): 2921-2939. doi: 10.3934/dcdsb.2020211

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

Pengyu Chen

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.

Keywords: Non-autonomous evolution equation with multi-delays, Periodic solution, Existence and uniqueness, Asymptotic stability, Evolution family.
Mathematics Subject Classification: Primary 34K13; Secondary 35K57, 47J35.
Citation: Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. 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

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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

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[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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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

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

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

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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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. Gurney, S. 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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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