Ayachi, M.; Blot, J. Variational methods for almost periodic solutions of a class of neutral delay equations. (English) Zbl 1149.34042 Abstr. Appl. Anal. 2008, Article ID 153285, 13 p. (2008). Summary: We provide new variational settings to study a.p. (almost periodic) solutions of a class of nonlinear neutral delay equations. We extend the variational setting for periodic solutions of nonlinear neutral delay equation by Shu and Xu (2006) 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 on existence of a.p. solutions, and also a density result for the forced equations. Cited in 8 Documents MSC: 34K14 Almost and pseudo-almost periodic solutions to functional-differential equations 34K40 Neutral functional-differential equations 47J30 Variational methods involving nonlinear operators PDFBibTeX XMLCite \textit{M. Ayachi} and \textit{J. Blot}, Abstr. Appl. Anal. 2008, Article ID 153285, 13 p. (2008; Zbl 1149.34042) Full Text: DOI EuDML References: [1] C. Corduneau, Almost Periodic Functions, Chelsea, New York, NY, USA, 2nd edition, 1989. [2] A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, Cambridge, UK, 1932. · Zbl 0004.25303 [3] 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. · Zbl 1145.34039 [4] L. È. Èl/sgol/c, Qualitative Methods in Mathematical Analysis, Translations of Mathematical Monographs, Vol. 12, American Mathematical Society, Providence, RI, USA, 1964. · Zbl 0133.37102 [5] 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. · Zbl 0153.41201 [6] L. D. Sabbagh, “Variational problems with lags,” Journal of Optimization Theory and Applications, vol. 3, pp. 34-51, 1969. · Zbl 0169.13701 [7] F. Colonius, Optimal Periodic Control, vol. 1313 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1988. · Zbl 0663.49011 [8] J. Blot, “Calculus of variations in mean and convex Lagrangians,” Journal of Mathematical Analysis and Applications, vol. 134, no. 2, pp. 312-321, 1988. · Zbl 0655.49011 [9] 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. · Zbl 0698.70015 [10] 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. · Zbl 0679.49022 [11] 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. · Zbl 0755.47048 [12] 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. · Zbl 0761.34037 [13] 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. · Zbl 0801.34043 [14] 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. · Zbl 0304.34051 [15] 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. · Zbl 1023.34503 [16] P. Cieutat, Solutions presque-périodiques d/équations d/évolution et de systèmes nonlinéaires, Doctorat Thesis, Université Paris 1 Panthéon-Sorbonne, Paris, France, 1996. [17] L. Schwartz, Théorie des Distributions, Hermann, Paris, France, 1966. · Zbl 0149.09501 [18] V. Alexéev, V. Tikhomirov, and S. Fomine, Commande Optimale, Mir, Moscow, Russia, French edition, 1982. [19] H. Brezis, Analyse Fonctionnelle, Théorie et applications, Masson, Paris, France, 1983. · Zbl 0511.46001 [20] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. · Zbl 0559.47040 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.