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September  2020, 9(3): 753-772. doi: 10.3934/eect.2020032

Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay

Qiang Li , and  Mei Wei ,

Department of Mathematics, Shanxi Normal University, Linfen 041000, China

* Corresponding author. Qiang Li and Mei Wei

Received  July 2019 Revised  October 2019 Published  September 2020 Early access  December 2019

Fund Project: Research supported by NNSF of China (11261053) and NSF of Gansu Province (1208RJZA129)

In this paper, we are devoted to consider the periodic problem for the neutral evolution equation with delay in Banach space. By using operator semigroups theory and fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equation. In addition, with the aid of a new integral inequality with delay, we present essential conditions on the nonlinear function to guarantee that the equation has an asymptotically stable periodic mild solution.

Keywords: Neutral evolution equation, periodic mild solutions, compact analytic semigroup, existence and uniqueness, asymptotic stability.
Mathematics Subject Classification: 34K30, 47H07, 47H08.
Citation: Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
References:
[1]

M. Adimy, H. Bouzahir and K. Ezzinbi, Existence and stability for some partial neutral functional differential equations with infinite delay, J. Math. Anal. Appl., 294 (2004), 438-461.  doi: 10.1016/j.jmaa.2004.02.033.

[2]

M. Adimy and K. Ezzinbi, Existence and stability in the $\alpha$-norm for partial functional equations of neutral type, Ann. Mat. Pura Appl., 185 (2006), 437-460.  doi: 10.1007/s10231-005-0162-8.

[3]

R. Benkhalti and K. Ezzinbi, Periodic solutions for some partial functional differential equations, J. Appl. Math. Stochastic Anal., 2004 (2004), 9-18.  doi: 10.1155/S1048953304212011.

[4]

R. Benkhalti, A. Elazzouzi and K. Ezzinbi, Periodic solutions for some partial neutral functional differential equations, Electron, J. Differ. Equ., 2006 (2006), 1-14. 

[5]

R. Benkhalti, A. Elazzouzi and K. Ezzinbi, Periodic solutions for some nonlinear partial neutral functional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 545-555.  doi: 10.1142/S0218127410025600.

[6]

T. A. Burton and B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Diffe. Equ., 90 (1991), 357-396.  doi: 10.1016/0022-0396(91)90153-Z.

[7]

P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, NoDEA Nonlinear Differential Equations. Appl., 10 (2003), 399-430.  doi: 10.1007/s00030-003-1004-2.

[8]

Y. Chen, The existence of periodic solutions for a class of neutral differential difference equations, J. Austral. Math. Soc. Ser. B, 33 (1992), 507-516.  doi: 10.1017/S0334270000007190.

[9]

K. Ezzinbi and J. Lui, Periodic solutions of non-densely defined delay evolutions equations, J. Appl. Math. Stochastic Anal., 15 (2002), 113-123. 

[10]

K. Ezzinbi and S. Ghnimib, Existence and regularity of solutions for neutral partial functional integrodifferential equations, Nonlinear Anal. Real World Appl., 11 (2010), 2335-2344.  doi: 10.1016/j.nonrwa.2009.07.007.

[11]

K. Ezzinbi, B. A. Kyelem and S. Ouaro, Periodicity in the $\alpha$-norm for partial functional differential equations in fading memory spaces, Nonlinear Anal., 97 (2014), 30-54.  doi: 10.1016/j.na.2013.10.026.

[12]

X. Fu and X. Liu, Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay, J. Math. Anal. Appl., 325 (2007), 249-267.  doi: 10.1016/j.jmaa.2006.01.048.

[13]

X. Fu, Existence of solutions and periodic solutions for abstract neutral equations with unbounded delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 17-35. 

[14]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

[15]

J. Hale, Partial neutral functional-differential equations, Rev. Roumaine Math. Pures Appl., 39 (1994), 339-344. 

[16]

E. Hernández and H. Henríquez, Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay, J. Math. Anal. Appl., 221 (1998), 499-522.  doi: 10.1006/jmaa.1997.5899.

[17]

E. Hernández, Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292 (2004), 194-210. 

[18]

E. Hernández and M. L. Pelicer, Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations, Appl. Math. Lett., 18 (2005), 1265-1272.  doi: 10.1016/j.aml.2005.02.015.

[19]

N. Huy and N. Dang, Dichotomy and periodic solutions to partial functional differential equations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3127-3144.  doi: 10.3934/dcdsb.2017167.

[20]

Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Functional Anal., 261 (2011), 1309-1324.  doi: 10.1016/j.jfa.2011.05.001.

[21]

Q. Li, Y. Li and P. Chen, Existence and uniqueness of periodic solutions for parabolic equation with nonlocal delay, Kodai Mathematical Journal, 39 (2016), 276-289.  doi: 10.2996/kmj/1467830137.

[22]

J. Liang, Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces, Anal. Appl., 15 (2017), 457-476.  doi: 10.1142/S0219530515500281.

[23]

J. Liang, J. H. Liu and T. J. Xiao, Condensing operators and periodic solutions of infinite delay impulsive evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 475-485.  doi: 10.3934/dcdss.2017023.

[24]

J. Liu, Bounded and periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 286 (2003), 705-712.  doi: 10.1016/S0022-247X(03)00512-2.

[25]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066.

[26]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

J. Wu and H. Xia, The existence of periodic solutions to integro-differential equations of neutral type via limiting equations, Math. Proc. Cambridge Philos. Soc., 112 (1992), 403-418.  doi: 10.1017/S0305004100071073.

[28]

J. Wu, Theory and Application of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[29]

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.

[30]

K. Yosida, Functional Analysis, (Sixth Edition), Springer-Verlag, Berlin, 1980.

show all references

References:
[1]

M. Adimy, H. Bouzahir and K. Ezzinbi, Existence and stability for some partial neutral functional differential equations with infinite delay, J. Math. Anal. Appl., 294 (2004), 438-461.  doi: 10.1016/j.jmaa.2004.02.033.

[2]

M. Adimy and K. Ezzinbi, Existence and stability in the $\alpha$-norm for partial functional equations of neutral type, Ann. Mat. Pura Appl., 185 (2006), 437-460.  doi: 10.1007/s10231-005-0162-8.

[3]

R. Benkhalti and K. Ezzinbi, Periodic solutions for some partial functional differential equations, J. Appl. Math. Stochastic Anal., 2004 (2004), 9-18.  doi: 10.1155/S1048953304212011.

[4]

R. Benkhalti, A. Elazzouzi and K. Ezzinbi, Periodic solutions for some partial neutral functional differential equations, Electron, J. Differ. Equ., 2006 (2006), 1-14. 

[5]

R. Benkhalti, A. Elazzouzi and K. Ezzinbi, Periodic solutions for some nonlinear partial neutral functional differential equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 545-555.  doi: 10.1142/S0218127410025600.

[6]

T. A. Burton and B. Zhang, Periodic solutions of abstract differential equations with infinite delay, J. Diffe. Equ., 90 (1991), 357-396.  doi: 10.1016/0022-0396(91)90153-Z.

[7]

P. Cannarsa and D. Sforza, Global solutions of abstract semilinear parabolic equations with memory terms, NoDEA Nonlinear Differential Equations. Appl., 10 (2003), 399-430.  doi: 10.1007/s00030-003-1004-2.

[8]

Y. Chen, The existence of periodic solutions for a class of neutral differential difference equations, J. Austral. Math. Soc. Ser. B, 33 (1992), 507-516.  doi: 10.1017/S0334270000007190.

[9]

K. Ezzinbi and J. Lui, Periodic solutions of non-densely defined delay evolutions equations, J. Appl. Math. Stochastic Anal., 15 (2002), 113-123. 

[10]

K. Ezzinbi and S. Ghnimib, Existence and regularity of solutions for neutral partial functional integrodifferential equations, Nonlinear Anal. Real World Appl., 11 (2010), 2335-2344.  doi: 10.1016/j.nonrwa.2009.07.007.

[11]

K. Ezzinbi, B. A. Kyelem and S. Ouaro, Periodicity in the $\alpha$-norm for partial functional differential equations in fading memory spaces, Nonlinear Anal., 97 (2014), 30-54.  doi: 10.1016/j.na.2013.10.026.

[12]

X. Fu and X. Liu, Existence of periodic solutions for abstract neutral non-autonomous equations with infinite delay, J. Math. Anal. Appl., 325 (2007), 249-267.  doi: 10.1016/j.jmaa.2006.01.048.

[13]

X. Fu, Existence of solutions and periodic solutions for abstract neutral equations with unbounded delay, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15 (2008), 17-35. 

[14]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-1-4612-4342-7.

[15]

J. Hale, Partial neutral functional-differential equations, Rev. Roumaine Math. Pures Appl., 39 (1994), 339-344. 

[16]

E. Hernández and H. Henríquez, Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay, J. Math. Anal. Appl., 221 (1998), 499-522.  doi: 10.1006/jmaa.1997.5899.

[17]

E. Hernández, Existence results for partial neutral integrodifferential equations with unbounded delay, J. Math. Anal. Appl., 292 (2004), 194-210. 

[18]

E. Hernández and M. L. Pelicer, Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations, Appl. Math. Lett., 18 (2005), 1265-1272.  doi: 10.1016/j.aml.2005.02.015.

[19]

N. Huy and N. Dang, Dichotomy and periodic solutions to partial functional differential equations, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 3127-3144.  doi: 10.3934/dcdsb.2017167.

[20]

Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Functional Anal., 261 (2011), 1309-1324.  doi: 10.1016/j.jfa.2011.05.001.

[21]

Q. Li, Y. Li and P. Chen, Existence and uniqueness of periodic solutions for parabolic equation with nonlocal delay, Kodai Mathematical Journal, 39 (2016), 276-289.  doi: 10.2996/kmj/1467830137.

[22]

J. Liang, Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces, Anal. Appl., 15 (2017), 457-476.  doi: 10.1142/S0219530515500281.

[23]

J. Liang, J. H. Liu and T. J. Xiao, Condensing operators and periodic solutions of infinite delay impulsive evolution equations, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 475-485.  doi: 10.3934/dcdss.2017023.

[24]

J. Liu, Bounded and periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 286 (2003), 705-712.  doi: 10.1016/S0022-247X(03)00512-2.

[25]

A. Lunardi, On the linear heat equation with fading memory, SIAM J. Math. Anal., 21 (1990), 1213-1224.  doi: 10.1137/0521066.

[26]

A. Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

J. Wu and H. Xia, The existence of periodic solutions to integro-differential equations of neutral type via limiting equations, Math. Proc. Cambridge Philos. Soc., 112 (1992), 403-418.  doi: 10.1017/S0305004100071073.

[28]

J. Wu, Theory and Application of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[29]

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.

[30]

K. Yosida, Functional Analysis, (Sixth Edition), Springer-Verlag, Berlin, 1980.

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