# American Institute of Mathematical Sciences

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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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 [12] M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay,, {J. Math. Anal. Appl.}, 338 (2008), 1340. doi: 10.1016/j.jmaa.2007.06.021. Google Scholar [13] A. Berger, S. Siegmund and Y. Yi, On almost automorphic dynamics in symbolic lattices,, {Ergodic Theory Dynam. Syst.}, 24 (2004), 677. doi: 10.1017/S0143385703000609. Google Scholar [14] S. Boulite, L. Maniar and G. M. N'Guérékata, Almost automorphic solutions for hyperbolic semilinear evolution equations,, {Semigroup Forum, 71 (2005), 231. doi: 10.1007/s00233-005-0524-y. Google Scholar [15] H. Bounit and S. Hadd, Regular linear systems governed by neutral FDEs,, {J. Math. Anal. Appl.}, 320 (2006), 836. doi: 10.1016/j.jmaa.2005.07.048. Google Scholar [16] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Springer, (2011). Google Scholar [17] A. Caicedo and C. Cuevas, $S$-asymptotically $\omega$-periodic solutions of abstract partial neutral integro-differential equations,, {Functional Differential Equations, 17 (2010), 387. Google Scholar [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 [37] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000). Google Scholar [38] 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 [41] F. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order,, in, (1997), 223. Google Scholar [42] R. C. Grimmer, Resolvent operators for integral equations in a Banach space,, {Trans. Amer. Math. Soc., 273 (1982), 333. Google Scholar [43] R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces. Hyperbolic partial differential equations II,, {Comput. Math. Appl., 11 (1985), 189. Google Scholar [44] G. Gripenberg, S.-O. Londen and O. Staffans, "Volterra Integral and Functional Equations,", Cambridge University Press, (1990). Google Scholar [45] S. Guo, Equivariant normal forms for neutral functional differential equations,, {Nonlinear Dynam., 61 (2010). doi: 10.1007/s11071-009-9651-4. Google Scholar [46] S. Hadd, Singular functional differential equations of neutral type in Banach spaces,, {J. Funct. Anal., 254 (2008), 2069. doi: 10.1016/j.jfa.2008.01.011. Google Scholar [47] J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer Verlag, (1993). Google Scholar [48] J. K. Hale, Partial neutral functional differential equations,, {Rev. Roumaine Math. Pures Appl., 39 (1994), 339. Google Scholar [49] J. Hale, Coupled oscillators on a circle,, {Resenhas do Instituto de Matem\'atica e Estat\'{\i}stica da Universidade de S\ ao Paulo}, 1 (1994), 441. Google Scholar [50] H. R. Henríquez, E. Hernández and J. C. dos Santos, Asymptotically almost periodic and almost periodic solutions for partial neutral integrodifferential equations,, {Zeitschrift f\, 26 (2007), 363. Google Scholar [51] H. R. Henríquez, M. Pierri and P. Táboas, Existence of $S$-asymptotically $\omega$-periodic solutions for abstract neutral functional differential equations,, {Bull. Austral. Math. Soc., 78 (2008), 365. Google Scholar [52] H. R. Henríquez, Periodic solutions of abstract neutral functional differential equations with infinite delay,, {Acta Math. Hungar., 121 (2008), 203. doi: 10.1007/s10474-008-7009-x. Google Scholar [53] H. R. Henríquez, M. Pierri and P. Táboas, On $S-$asymptotically $\omega$-periodic functions on Banach spaces and applications,, {J. Math. Anal. Appl., 343 (2008), 1119. doi: 10.1016/j.jmaa.2008.02.023. Google Scholar [54] H. R. Henríquez and C. Lizama, Compact almost automorphic solutions to integral equations with infinite delay,, {Nonlin. Anal., 71 (2009), 6029. doi: 10.1016/j.na.2009.05.042. Google Scholar [55] E. Hernández and H. R. Henríquez, Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay,, {J. Math. Anal. Appl., 221 (1998), 499. doi: 10.1006/jmaa.1997.5899. Google Scholar [56] E. Hernández and H. R. Henríquez, Existence results for partial neutral functional equations with unbounded delay,, {J. Math. Anal. Appl., 221 (1998), 452. doi: 10.1006/jmaa.1997.5875. Google Scholar [57] E. Hernández, M. Rabelo and H. R. Henríquez, Existence of solutions for impulsive partial neutral functional differential equations,, {J. Math. Anal. 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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 [2] 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 [3] M. Adimy, K. Ezzinbi and M. Laklach, Spectral decomposition for partial neutral functional differential equations,, {Canadian Applied Math. Quart., 9 (2001), 1. Google Scholar [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 [5] 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 [12] M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay,, {J. Math. Anal. Appl.}, 338 (2008), 1340. doi: 10.1016/j.jmaa.2007.06.021. Google Scholar [13] A. Berger, S. Siegmund and Y. Yi, On almost automorphic dynamics in symbolic lattices,, {Ergodic Theory Dynam. Syst.}, 24 (2004), 677. doi: 10.1017/S0143385703000609. Google Scholar [14] S. Boulite, L. Maniar and G. M. N'Guérékata, Almost automorphic solutions for hyperbolic semilinear evolution equations,, {Semigroup Forum, 71 (2005), 231. doi: 10.1007/s00233-005-0524-y. Google Scholar [15] H. Bounit and S. Hadd, Regular linear systems governed by neutral FDEs,, {J. Math. Anal. Appl.}, 320 (2006), 836. doi: 10.1016/j.jmaa.2005.07.048. Google Scholar [16] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Springer, (2011). Google Scholar [17] A. Caicedo and C. Cuevas, $S$-asymptotically $\omega$-periodic solutions of abstract partial neutral integro-differential equations,, {Functional Differential Equations, 17 (2010), 387. Google Scholar [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 [37] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000). Google Scholar [38] 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 [41] F. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order,, in, (1997), 223. Google Scholar [42] R. C. Grimmer, Resolvent operators for integral equations in a Banach space,, {Trans. Amer. Math. Soc., 273 (1982), 333. Google Scholar [43] R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces. Hyperbolic partial differential equations II,, {Comput. Math. Appl., 11 (1985), 189. Google Scholar [44] G. Gripenberg, S.-O. Londen and O. Staffans, "Volterra Integral and Functional Equations,", Cambridge University Press, (1990). Google Scholar [45] S. Guo, Equivariant normal forms for neutral functional differential equations,, {Nonlinear Dynam., 61 (2010). doi: 10.1007/s11071-009-9651-4. Google Scholar [46] S. Hadd, Singular functional differential equations of neutral type in Banach spaces,, {J. Funct. Anal., 254 (2008), 2069. doi: 10.1016/j.jfa.2008.01.011. Google Scholar [47] J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer Verlag, (1993). Google Scholar [48] J. K. Hale, Partial neutral functional differential equations,, {Rev. 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