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

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

    Mathematics Subject Classification: Primary 34K13; Secondary 35K57, 47J35.


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  • [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. BurtonStability 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. 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.
    [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. 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.
    [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.
    [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. 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.
    [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. GurneyS. 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. 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.
    [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. 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.
    [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. 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.
    [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.
    [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. 
    [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. 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.
    [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.
    [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.
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