Communications in Mathematical Analysis

Variational Methods and Almost Periodic Solutions of Second Order Functional Differential Equations with Infinite Delay

Moez Ayachi

Full-text: Open access

Abstract

By means of variational methods, we study the existence and uniqueness of almost periodic solutions for a class of second order neutral functional differential equations with infinite delay.

Article information

Source
Commun. Math. Anal., Volume 9, Number 1 (2010), 15-31.

Dates
First available in Project Euclid: 21 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.cma/1271890715

Mathematical Reviews number (MathSciNet)
MR2569958

Zentralblatt MATH identifier
1195.34107

Subjects
Primary: 34K14: Almost and pseudo-periodic solutions
Secondary: 65K10: Optimization and variational techniques [See also 49Mxx, 93B40]

Keywords
variational principle almost periodic solution functional differential equations infinite delay

Citation

Ayachi, Moez. Variational Methods and Almost Periodic Solutions of Second Order Functional Differential Equations with Infinite Delay. Commun. Math. Anal. 9 (2010), no. 1, 15--31. https://projecteuclid.org/euclid.cma/1271890715


Export citation

References

  • N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space. I, Dover Publications, Mineola, New York, 1993.
  • V. M. Alexeev, V. M. Tihomirov and S. V. Fomin, Commande Optimale. French Edition, Mir, Moscow, 1982.
  • M. Ayachi and J. Blot, Variational Methods for Almost Periodic Solutions of a Class of Neutral Delay Equations Abstract and Applied Analysis. 2008, Article ID 153285, 13 pages, 2008. doi: 1155/2008/153285.
  • A. S. Besicovitch, Almost Periodic Functions. Cambridge University Press, Cambridge, 1932.
  • A. S. Besicovitch and H. Bohr, Almost Periodicity and General Trigonometric Series. Acta Math. 57 (1931), pp 203-292.
  • J. Blot, Calculus of Variations in Mean and Convex Lagrangians. J. Math. Anal. Appl. 134 (1988), $\rm{n^{o}}$2, pp 312-321.
  • J. Blot, Une Approche Variationnelle des Orbites Quasi-périodiques des Systèmes Hamiltoniens. Ann. Sc. Math. Québec 13 (1989), $\rm{n^{o}}$2, pp 7-32 (French).
  • J. Blot, Calculus of Variations in Mean and Convex Lagrangians, II. Bull. Austral. Math. Soc. 40 (1989), pp 457-463.
  • J. Blot, Une Approche Hilbertienne Pour les Trajectoires Presque-périodiques. Notes CR. Acad. Sc. Paris 313 (1991), Série I, pp 487-490 (French).
  • J. Blot, Almost Periodic Solutions of Forced Second Order Hamiltonian Systems. Ann. Fac. Sc. Toulouse XIII (1991), $\rm{n^{o}}$3, pp 351-363.
  • J. Blot, Oscillations Presque-périodiques Forcées d'équations d'Euler-lagrange. Bull. Soc. Math. 122 (194), pp 285-304. (French).
  • H. Bohr, Almost Periodic Functions, Chelsea, New York, 1956.
  • H. Bohr and E. Foelner, One Some Types of Functional Spaces. Acta Math. 76 (1945), pp 31-155.
  • H. Brezis, Analyse Fonctionnelle. Théorie et Applications. Masson, Paris, 1983. (French).
  • P. Cieutat, Solutions Presque-périodiques d'équations d'évolution et de Systèmes Non Linéaires. Doctorat Thesis, Université Paris 1 Panthéon-Sorbonne, (1996). (French).
  • T. Diagana, H. Henriquez, and E. Hernandez, Asymptotically Almost Periodic Solutions to Some Classes of Second-Order Functional Differential Equations. Diff. and Int. Eq. 21, Numb 5-6(2008), pp 575-600.
  • L. E. Elsgolc, Qualitative Methods in Mathematical Analysis. Translation of Mathematical Monograph, Am. Math. Soc., Providence, Rhode Island, (1964).
  • J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equa- tions. Applied Mathematical Sciences 99, Springer-Verlag, New York, 1993.
  • H. R. Henriquez and C. Vasquez, Almost Periodic Solutions of Abstract Retarded Func- tional Differential Equations With Unbounded Delay. Acta Appl. Math. 57 (1999), pp 105-132.
  • H. R. Henriquez and C. Vasquez, Differentiability of Solutions of Second-Order Functional Differential Equations With Unbounded delay. J. Math. Appl. 280 (2003), pp 284-312.
  • E. Hernandez and M. McKibben, Some Comments On: Existence of Solutions of Abstract Nonlinear Second-Order Neutral Functional Integro-differential Equations. Comput. Math. Appl. 50 (2005), pp 655-669.
  • Y. Hino, S. Murakami and T. Naito, Functional Differential Equations With Infinite Delay, Lecture Notes in Mathematics, 1473, Springer-Verlag, 1991.
  • D. K. Hughes, Variational and Optimal Control Problems with Delayed Argument. J. Optim. Th. Appl. 2 (1968), $\rm{n^{o}}$1, pp 1-14.
  • M. Krasnoselsky, Topological Methods in the Theory of Nonlinear Integral Equations GITTL. Moscow (1956), English trans. Macmillan, New York 1964.
  • V. Lakshmikantham, L. Wen and B. Zhang, Theory of Differential Equations with Unbounded Delay. Kluwer Acad. Publ., Dordrecht, 1994.
  • B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations. Cambridge Univ. Press, Cambridge, 1982.
  • Y. Li, Existence of Periodic Solutions for Second-Order Neutral Differential Equations. Electron. J. Diff. Equ. (2005), $\rm{n^{o}.}$26, pp 1-5.
  • N.-V. Minh, T. Naito and J.-S. Shin, Periodic and Almost Periodic Solutions of Fuctional Differential Equations With Infinite Delay. Nonlinear Anal. 47 (2001), $\rm{n^{o}}$6.
  • A. A. Pankov, Bounded and Almost Periodic Solutions of Nonlinear Operator Differential Equations. Kuwer Acad. Publ., Dordrecht, 1990.
  • L. D. Sabbagh, Variational Problems with Lags. J. Optim. Th. Appl. 3 (1969), $\rm{n^{o}}$1, pp 34-51.
  • X.-B. Shu and Y.-T. Xu, Multiple Periodic Solutions for a Class of Second-Order Nonlinear Neutral Delay Equations. Abstract and Applied Analysis (2006), Article ID 10252, 9 pages, 2006. doi: 10.1155/2006/10252.
  • T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Springer-Verlag, New York, Inc., 1975.