Asymptotically periodic solutions of neutral partial  differential
equations with infinite delay

Hernán R. Henríquez; Claudio Cuevas; Alejandro Caicedo

doi:10.3934/cpaa.2013.12.2031

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September 2013, 12(5): 2031-2068. doi: 10.3934/cpaa.2013.12.2031

Asymptotically periodic solutions of neutral partial differential equations with infinite delay

Hernán R. Henríquez ^1, , Claudio Cuevas ^2, and Alejandro Caicedo ^2,

1.	Departamento de Matemática, Universidad de Santiago, USACH, Casilla 307, Correo 2, Santiago, Chile
2.	Departamento de Matemática, Centro de Ciências Exatas e da Natureza, Universidade Federal de Pernambuco, Av. Jornalista Anibal Fernandes S/N, Cidade Universitária, CEP 50740-560, Recife-PE, Brazil, Brazil

Received February 2012 Revised September 2012 Published January 2013

Abstract
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In this paper we discuss the existence and uniqueness of asymptotically almost automorphic and $S$-asymptotically $\omega$-periodic mild solutions to some abstract nonlinear integro-differential equation of neutral type with infinite delay. We apply our results to neutral partial differential equations with infinite delay.

Keywords: asymptotically almost automorphic functions, Neutral functional differential equations, integro-differential equations, $S$-asymptotically $\omega$-periodic functions..

Mathematics Subject Classification: 34K05; 34A12; 34A4.

Citation: Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

References:

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[23]	C. Cuevas, E. Hernández and M. Rabelo, The existence of solutions for impulsive neutral functional differential equations, Comput. Math. Appl., 58 (2009), 774-757. doi: 10.1016/j.camwa.2009.04.008. Google Scholar
[24]	C. Cuevas and C. Lizama, $S$-asymptotically $\omega$-periodic solutions for semilinear Volterra equations, Math. Meth. Appl. Sci., 33 (2010), 1628-1636. doi: 10.1002/mma.1284. Google Scholar
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[29]	B. de Andrade and C. Cuevas, $S$-asymptotically $\omega$-periodic and asymptotically $\omega$-periodic solutions to semilinear Cauchy problems with non dense domain, Nonlin. Anal., 72 (2010), 3190-3208. doi: 10.1016/j.na.2009.12.016. Google Scholar
[30]	W. Desch, R. Grimmer and W. Schappacher, Well-posedness and wave propagation for a class of integrodifferential equations in Banach space, J. Differential Equations, 74 (1988), 391-411. Google Scholar
[31]	T. Diagana, H. R. Henríquez and E. Hernández, Almost automorphic mild solutions of some partial neutral functional differential equations and applications, Nonlin. Anal., 69 (2008), 1485-1493. doi: 10.1016/j.na.2007.06.048. Google Scholar
[32]	T. Diagana, E. Hernández and J. P. C. dos Santos, Existence of asymptotically almost automorphic solutions to some abstract partial neutral integro-differential equations, Nonlin. Anal., 71 (2009), 248-257. doi: 10.1016/j.na.2008.10.046. Google Scholar
[33]	T. Diagana and R. P. Agarwal, Existence of pseudo almost automorphic solutions for the heat equation with $S^p$-pseudo almost automorphic coefficients, Boundary Value Problems, 2009 (2009), Article ID 182527, 19 pages. doi:10.1155/2009/182527 doi: 10.1155/2009/182527. Google Scholar
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[35]	H. S. Ding, T. J. Xiao and J. Liang, Asymptotically almost automorphic solutions for some integrodifferential equations with nonlocal initial conditions, J. Math. Anal. Appl., 338 (2008), 141-151. doi: 10.1016/j.jmaa.2007.05.014. Google Scholar
[36]	J. P. C. dos Santos and C. Cuevas, Asymptotically almost automorphic solutions of abstract fractional integro-differential neutral equations, Appl. Math. Lett., 23 (2010), 960-965. doi: 10.1016/j.aml.2010.04.016. Google Scholar
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[39]	K. Ezzinbi and G. M. N'Guérékata, Almost automorphic solutions for some partial functional differential equations, J. Math. Anal. Appl., 328 (2007), 344-358. doi: 10.1016/j.jmaa.2006.05.036. Google Scholar
[40]	K. Ezzinbi, S. Fatajou and G. M. N'Guérékata, Pseudo-almost-automorphic solutions to some neutral partial functional differential equations in Banach spaces, Nonlin. Anal., 70 (2009), 1641-1647. doi: 10.1016/j.na.2008.02.039. Google Scholar
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