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

Asymptotically periodic solutions of neutral partial differential equations with infinite delay

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

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.
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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References:
[1]

S. Abbas and D. Bahuguna, Almost periodic solutions of neutral functional differential equations,, {Comp. Math. Appl., 55 (2008), 2593.  doi: 10.1016/j.camwa.2007.00.011.  Google Scholar

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M. Adimy and K. Ezzinbi, Existence and linearized stability for partial neutral functional differential equations with nondense domains,, {Differential Equations and Dynamical Systems, 7 (1999), 371.   Google Scholar

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[4]

M. Adimy, A. Elazzouzi and K. Ezzinbi, Bohr-Neugebauer type theorem for some partial neutral functional differential equations,, {Nonlin. Anal., 66 (2007), 1145.  doi: 10.1016/j.na.2006.01.011.  Google Scholar

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R. P. Agarwal, B. de Andrade and C. Cuevas, On type of periodicity and ergodicity to a class of integral equations with infinite delay,, {J. Nonlin. Convex Anal., 11 (2010), 309.   Google Scholar

[6]

R. P. Agarwal, T. Diagana and E. Hernández, Weighted pseudo almost periodic solutions to some partial neutral functional differential equations,, {J. Nonlin. Convex Anal., 8 (2007), 397.   Google Scholar

[7]

R. P. Agarwal, B. de Andrade and C. Cuevas, On type of periodicity and ergodicity to a class of fractional order differential equations,, {Adv. Difference Equ., 2010 (2010).  doi: 10.1155/2010/179750.  Google Scholar

[8]

R. P. Agarwal, B. de Andrade and C. Cuevas, Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations,, {Nonlin. Anal.: Real World Appl., 11 (2010), 3532.  doi: 10.1016/j.nonrwa.2010.01.002.  Google Scholar

[9]

R. P. Agarwal, M. Benchohra and S. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions,, {Acta Appl. Math., 109 (2010), 973.  doi: 10.1007/s10440-008-9356-6.  Google Scholar

[10]

M. Alia, K. Ezzinbi and S. Fatajou, Exponential dichotomy and pseudo almost automorphy for partial neutral functional differential equations,, {Nonlin. Anal., 71 (2009), 2210.  doi: 10.1016/j.na.2009.01.057.  Google Scholar

[11]

E. G. Bazhlekova, "Fractional Evolution Equations in Banach Spaces,", Thesis (Dr.) Technische Universiteit Eindhoven (The Netherlands), (2001).   Google Scholar

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[13]

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[14]

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[16]

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[17]

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[18]

A. Caicedo, C. Cuevas and H. Henríquez, Asymptotic periodicity for a class of partial integro-differential equations,, ISRN Mathematical Analysis, 2011 (2011).  doi: 10.5402/2011/537890.  Google Scholar

[19]

A. Caicedo, C. Cuevas, G. M. Mophou and G. M. N'Guérékata, Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces,, {J. Franklin Institute, 349 (2012), 1.  doi: 10.1016/j.jfranklin.2011.02.001.  Google Scholar

[20]

C. Chen, Control and stabilization for the wave equation in a bounded domain,, {SIAM J. Control, 17 (1979), 66.   Google Scholar

[21]

E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations,, Discr. Contin. Dyn. Syst., (2007), 277.   Google Scholar

[22]

C. Cuevas, G. N'Guérékata and M. Rabelo, Mild solutions for impulsive neutral functional differential equations with state-dependent delay,, Semigroup Forum, 80 (2010), 375.  doi: 10.1007/s00233-010-9213-6.  Google Scholar

[23]

C. Cuevas, E. Hernández and M. Rabelo, The existence of solutions for impulsive neutral functional differential equations,, Comput. Math. Appl., 58 (2009), 774.  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.  doi: 10.1002/mma.1284.  Google Scholar

[25]

C. Cuevas and J. C. de Souza, $S$-asymptotically $\omega$-periodic solutions of semilinear fractional integro-differential equations,, {Appl. Math. Lett., 22 (2009), 865.  doi: 10.1016/j.aml.2008.07.013.  Google Scholar

[26]

C. Cuevas and J. C. de Souza, Existence of $S$-asymptotically $\omega$-periodic solutions for fractional order functional integro-differential equations with infinite delay,, {Nonlin. Anal., 72 (2010), 1683.  doi: 10.1016/j.na.2009.09.007.  Google Scholar

[27]

C. Cuevas and C. Lizama, Almost automorphic solutions to integral equations on the line,, {Semigroup Forum, 79 (2009), 461.  doi: 10.1007/s00233-009-9154-0.  Google Scholar

[28]

C. Cuevas and C. Lizama, Almost automorphic solutions to a class of semilinear fractional differential equations,, {Appl. Math. Lett., 21 (2008), 1315.  doi: 10.1016/j.aml.2008.02.001.  Google Scholar

[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.  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.   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.  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.  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).  doi: 10.1155/2009/182527.  Google Scholar

[34]

H.-S. Ding, J. Liang, G. N'Guérékata and T.-J. Xiao, Existence of positive almost automorphic solutions to neutral nonlinear integral equations,, {Nonlin. Anal., 69 (2008), 1188.  doi: 10.1016/j.na.2007.06.017.  Google Scholar

[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.  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.  doi: 10.1016/j.aml.2010.04.016.  Google Scholar

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K. Ezzinbi and G. M. N'Guérékata, Massera type theorem for almost automorphic solutions of functional differential equations of neutral type,, {J. Math. Anal. Appl., 316 (2006), 707.  doi: http://dx.doi.org/10.1016/j.jmaa.2005.04.074.  Google Scholar

[39]

K. Ezzinbi and G. M. N'Guérékata, Almost automorphic solutions for some partial functional differential equations,, {J. Math. Anal. Appl., 328 (2007), 344.  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.  doi: 10.1016/j.na.2008.02.039.  Google Scholar

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