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Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions
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 |
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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. |
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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.
|
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M. Adimy, K. Ezzinbi and M. Laklach, Spectral decomposition for partial neutral functional differential equations,, {Canadian Applied Math. Quart., 9 (2001), 1.
|
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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. |
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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. |
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M. Alia, K. Ezzinbi and S. Fatajou, Exponential dichotomy and pseudo almost automorphy for partial neutral functional differential equations,, {Nonlin. Anal., 71 (2009), 2210.
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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.
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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.
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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. |
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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. |
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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.
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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. |
[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). 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. |
[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.
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