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 and 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), 2593-2601. 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), 371-417. [3] M. Adimy, K. Ezzinbi and M. Laklach, Spectral decomposition for partial neutral functional differential equations, Canadian Applied Math. Quart., 9 (2001), 1-34. [4] M. Adimy, A. Elazzouzi and K. Ezzinbi, Bohr-Neugebauer type theorem for some partial neutral functional differential equations, Nonlin. Anal., 66 (2007), 1145-1160. 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), 309-335. [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), 397-415. [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), article ID 179750, 25 pp. 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), 3532-3534. 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(3) (2010), 973-1033. 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), 2210-2226. 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), 1340-1350. 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), 677-696. 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), 231-240. 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), 836-858. doi: 10.1016/j.jmaa.2005.07.048. [16] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Springer, New York, 2011. [17] A. Caicedo and C. Cuevas, $S$-asymptotically $\omega$-periodic solutions of abstract partial neutral integro-differential equations, Functional Differential Equations, 17 (2010), 387-405. [18] A. Caicedo, C. Cuevas and H. Henríquez, Asymptotic periodicity for a class of partial integro-differential equations, ISRN Mathematical Analysis, 2011 (2011), Article ID 537890, 18 pp., doi: 10.5402/2011/537890. 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), 1-24. 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), 66-81. [21] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discr. Contin. Dyn. Syst., (Supplement) (2007), 277-285. [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), 375-390. 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), 774-757. 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), 1628-1636. 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), 865-870. 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), 1683-1689. 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), 461-472. 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), 1315-1319. 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), 3190-3208. 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), 391-411. [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. [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. [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. [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), 1188-1199. 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), 141-151. 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), 960-965. doi: 10.1016/j.aml.2010.04.016. [37] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 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), 707-721. 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), 344-358. 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), 1641-1647. doi: 10.1016/j.na.2008.02.039. [41] F. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, in "Fractals and Fractional Calculus in Continuum Mechanics" (A. Carpinteri and F. Mainardi eds.), Springer, New York, 1997, 223-276. [42] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349. [43] R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces. 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References:
 [1] S. Abbas and D. Bahuguna, Almost periodic solutions of neutral functional differential equations, Comp. Math. Appl., 55 (2008), 2593-2601. 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), 371-417. [3] M. Adimy, K. Ezzinbi and M. Laklach, Spectral decomposition for partial neutral functional differential equations, Canadian Applied Math. Quart., 9 (2001), 1-34. [4] M. Adimy, A. Elazzouzi and K. Ezzinbi, Bohr-Neugebauer type theorem for some partial neutral functional differential equations, Nonlin. Anal., 66 (2007), 1145-1160. 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), 309-335. [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), 397-415. [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), article ID 179750, 25 pp. 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), 3532-3534. 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(3) (2010), 973-1033. 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), 2210-2226. 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), 1340-1350. 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), 677-696. 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), 231-240. 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), 836-858. doi: 10.1016/j.jmaa.2005.07.048. [16] H. Brezis, "Functional Analysis, Sobolev Spaces and Partial Differential Equations," Springer, New York, 2011. [17] A. Caicedo and C. Cuevas, $S$-asymptotically $\omega$-periodic solutions of abstract partial neutral integro-differential equations, Functional Differential Equations, 17 (2010), 387-405. [18] A. Caicedo, C. Cuevas and H. Henríquez, Asymptotic periodicity for a class of partial integro-differential equations, ISRN Mathematical Analysis, 2011 (2011), Article ID 537890, 18 pp., doi: 10.5402/2011/537890. 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), 1-24. 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), 66-81. [21] E. Cuesta, Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations, Discr. Contin. Dyn. Syst., (Supplement) (2007), 277-285. [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), 375-390. 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), 774-757. 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), 1628-1636. 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), 865-870. 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), 1683-1689. 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), 461-472. 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), 1315-1319. 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), 3190-3208. 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), 391-411. [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. [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. [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. [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), 1188-1199. 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), 141-151. 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), 960-965. doi: 10.1016/j.aml.2010.04.016. [37] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer-Verlag, New York, 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), 707-721. 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), 344-358. 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), 1641-1647. doi: 10.1016/j.na.2008.02.039. [41] F. Gorenflo and F. Mainardi, Fractional Calculus: Integral and Differential Equations of Fractional Order, in "Fractals and Fractional Calculus in Continuum Mechanics" (A. Carpinteri and F. Mainardi eds.), Springer, New York, 1997, 223-276. [42] R. C. Grimmer, Resolvent operators for integral equations in a Banach space, Trans. Amer. Math. Soc., 273 (1982), 333-349. [43] R. Grimmer and J. Prüss, On linear Volterra equations in Banach spaces. 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