Abstract and Applied Analysis

Variational Methods for Almost Periodic Solutions of a Class of Neutral Delay Equations

M. Ayachi and J. Blot

Source: Abstr. Appl. Anal. Volume 2008 (2008), 13 pages.

Abstract

We provide new variational settings to study the a.p. (almost periodic) solutions of a class of nonlinear neutral delay equations. We extend Shu and Xu (2006) variational setting for periodic solutions of nonlinear neutral delay equation to the almost periodic settings. We obtain results on the structure of the set of the a.p. solutions, results of existence of a.p. solutions, results of existence of a.p. solutions, and also a density result for the forced equations.

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aaa/1220969145
Digital Object Identifier: doi:10.1155/2008/153285
Mathematical Reviews number (MathSciNet): MR2393110
Zentralblatt MATH identifier: 1149.34042

References

C. Corduneau, Almost Periodic Functions, Chelsea, New York, NY, USA, 2nd edition, 1989.
A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, Cambridge, UK, 1932.
X.-B. Shu and Y.-T. Xu, ``Multiple periodic solutions for a class of second-order nonlinear neutral delay equations,'' Abstract and Applied Analysis, vol. 2006, Article ID 10252, 9 pages, 2006.
Mathematical Reviews (MathSciNet): MR2211659
Zentralblatt MATH: 1145.34039
Digital Object Identifier: doi:10.1155/AAA/2006/10252
L. È. Èl'sgol'c, Qualitative Methods in Mathematical Analysis, Translations of Mathematical Monographs, Vol. 12, American Mathematical Society, Providence, RI, USA, 1964.
Mathematical Reviews (MathSciNet): MR0170048
Zentralblatt MATH: 0133.37102
D. K. Hughes, ``Variational and optimal control problems with delayed argument,'' Journal of Optimization Theory and Applications, vol. 2, no. 1, pp. 1--14, 1968.
Mathematical Reviews (MathSciNet): MR0243403
Zentralblatt MATH: 0153.41201
Digital Object Identifier: doi:10.1007/BF00927159
L. D. Sabbagh, ``Variational problems with lags,'' Journal of Optimization Theory and Applications, vol. 3, pp. 34--51, 1969.
Mathematical Reviews (MathSciNet): MR0257855
Zentralblatt MATH: 0169.13701
Digital Object Identifier: doi:10.1007/BF00929540
F. Colonius, Optimal Periodic Control, vol. 1313 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988.
Mathematical Reviews (MathSciNet): MR947339
Zentralblatt MATH: 0663.49011
J. Blot, ``Calculus of variations in mean and convex Lagrangians,'' Journal of Mathematical Analysis and Applications, vol. 134, no. 2, pp. 312--321, 1988.
Mathematical Reviews (MathSciNet): MR961340
Zentralblatt MATH: 0655.49011
Digital Object Identifier: doi:10.1016/0022-247X(88)90025-X
J. Blot, ``Une approche variationnelle des orbites quasi-périodiques des systèmes hamiltoniens,'' Annales des Sciences Mathématiques du Québec, vol. 13, no. 2, pp. 7--32, 1990.
Mathematical Reviews (MathSciNet): MR1038364
Zentralblatt MATH: 0698.70015
J. Blot, ``Calculus of variations in mean and convex Lagrangians. II,'' Bulletin of the Australian Mathematical Society, vol. 40, no. 3, pp. 457--463, 1989.
Mathematical Reviews (MathSciNet): MR1037643
Zentralblatt MATH: 0679.49022
Digital Object Identifier: doi:10.1017/S0004972700017524
J. Blot, ``Une méthode hilbertienne pour les trajectoires presque-périodiques,'' Comptes Rendus de l'Académie des Sciences. Série I, vol. 313, no. 8, pp. 487--490, 1991.
Mathematical Reviews (MathSciNet): MR1131860
Zentralblatt MATH: 0755.47048
J. Blot, ``Almost-periodic solutions of forced second order Hamiltonian systems,'' Annales de la Faculté des Sciences de Toulouse Mathématiques, vol. 12, no. 3, pp. 351--363, 1991.
Mathematical Reviews (MathSciNet): MR1189445
Zentralblatt MATH: 0761.34037
J. Blot, ``Oscillations presque-périodiques forcées d'équations d'Euler-Lagrange,'' Bulletin de la Société Mathématique de France, vol. 122, no. 2, pp. 285--304, 1994.
Mathematical Reviews (MathSciNet): MR1273905
Zentralblatt MATH: 0801.34043
T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, vol. 14 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1975.
Mathematical Reviews (MathSciNet): MR0466797
Zentralblatt MATH: 0304.34051
J. Blot, P. Cieutat, and J. Mawhin, ``Almost-periodic oscillations of monotone second-order systems,'' Advances in Differential Equations, vol. 2, no. 5, pp. 693--714, 1997.
Mathematical Reviews (MathSciNet): MR1751424
Zentralblatt MATH: 1023.34503
P. Cieutat, ``Solutions presque-périodiques d'équations d'évolution et de systèmes nonlinéaires,'' Doctorat Thesis, Université Paris 1 Panthéon-Sorbonne, Paris, France, 1996.
L. Schwartz, Théorie des Distributions, Hermann, Paris, France, 1966.
Mathematical Reviews (MathSciNet): MR0209834
Zentralblatt MATH: 0149.09501
V. Alexéev, V. Tikhomirov, and S. Fomine, Commande Optimale, Mir, Moscow, Russia, French edition, 1982.
Mathematical Reviews (MathSciNet): MR728225
Zentralblatt MATH: 0923.93003
H. Brezis, Analyse Fonctionnelle, Théorie et applications, Masson, Paris, France, 1983.
Mathematical Reviews (MathSciNet): MR697382
Zentralblatt MATH: 0511.46001
K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
Mathematical Reviews (MathSciNet): MR787404
Zentralblatt MATH: 0559.47040

2009 © Hindawi Publishing Corporation