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:
[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.

[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.

[3]

M. Adimy, K. Ezzinbi and M. Laklach, Spectral decomposition for partial neutral functional differential equations,, {Canadian Applied Math. Quart., 9 (2001), 1.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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[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.

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H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Springer, (2011).

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A. Caicedo and C. Cuevas, $S$-asymptotically $\omega$-periodic solutions of abstract partial neutral integro-differential equations,, {Functional Differential Equations, 17 (2010), 387.

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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.

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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.

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

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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.

[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.

[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.

[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.

[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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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.

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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.

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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.

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S. Guo, Equivariant normal forms for neutral functional differential equations,, {Nonlinear Dynam., 61 (2010). doi: 10.1007/s11071-009-9651-4.

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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.

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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.

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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.

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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.

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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.

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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.

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show all references

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.

[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.

[3]

M. Adimy, K. Ezzinbi and M. Laklach, Spectral decomposition for partial neutral functional differential equations,, {Canadian Applied Math. Quart., 9 (2001), 1.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[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.

[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.

[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.

[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.

[16]

H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations,", Springer, (2011).

[17]

A. Caicedo and C. Cuevas, $S$-asymptotically $\omega$-periodic solutions of abstract partial neutral integro-differential equations,, {Functional Differential Equations, 17 (2010), 387.

[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.

[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.

[20]

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

[21]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[37]

K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations,", Springer-Verlag, (2000).

[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.

[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.

[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.

[41]

F. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order,, in, (1997), 223.

[42]

R. C. Grimmer, Resolvent operators for integral equations in a Banach space,, {Trans. Amer. Math. Soc., 273 (1982), 333.

[43]

R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces. Hyperbolic partial differential equations II,, {Comput. Math. Appl., 11 (1985), 189.

[44]

G. Gripenberg, S.-O. Londen and O. Staffans, "Volterra Integral and Functional Equations,", Cambridge University Press, (1990).

[45]

S. Guo, Equivariant normal forms for neutral functional differential equations,, {Nonlinear Dynam., 61 (2010). doi: 10.1007/s11071-009-9651-4.

[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.

[47]

J. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Springer Verlag, (1993).

[48]

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

[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.

[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.

[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.

[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.

[53]

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