Bahri, Abbas; Berestycki, Henri Existence of forced oscillations for some nonlinear differential equations. (English) Zbl 0588.34028 Commun. Pure Appl. Math. 37, 403-442 (1984). This article studies the existence of T-periodic solutions for systems of nonlinear second order ordinary differential equations of the type ẍ\(+V'(x)=f(t)\). Here, \(x: R\to R^ N\), \(V\in C^ 2(R^ N,R)\) and \(f: R\to R^ N\) is a given T-periodic forcing term \((T>0\) is given). Assuming V to be superquadratic, it is shown that this system possesses infinitely many T-periodic solutions. The proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial. Some more general results concerning systems of the type ẍ\(+\hat V'\!_ x(t,x)=0\) are also presented here. Cited in 2 ReviewsCited in 43 Documents MSC: 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 34C25 Periodic solutions to ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Keywords:nonlinear second order ordinary differential equations; homotopy groups PDFBibTeX XMLCite \textit{A. Bahri} and \textit{H. Berestycki}, Commun. Pure Appl. Math. 37, 403--442 (1984; Zbl 0588.34028) Full Text: DOI References: [1] ThëGse de Doctorat d’Etat. Université P. & M. Curie Paris VI, 1981. [2] Groupes d’homotopie des ensembles de niveaux pour certaines fonctionnelles à gradient Fredholm, to appear. [3] Bahri, Trans. Amer. Math. Soc. 267 pp 1– (1981) [4] Bahri, C. R. Acad. Sc. Paris 291 pp 189– (1980) [5] Bahri, C. R. Acad. Sc. Paris 292 pp 315– (1981) [6] Bahri, Acta Math [7] Benci, Comm. Pure Appl. Math. 33 pp 147– (1980) [8] Benci, Trans. Amer. Math. Soc [9] Berestycki, J. Functional Anal. 40 pp 1– (1981) [10] Cesari, Contr. Diff. Equations 1 pp 149– (1963) [11] Ehrmann, Math. Ann. 134 pp 167– (1957) [12] Solvability of Nonlinear Equations and Boundary Value Problems, D. Reidel, Boston, 1980. [13] Fučik, Casopic Pěst. Mat., Prague, roč 100 pp 160– (1975) [14] Hartman, Amer. J. of Math. 26 pp 37– (1977) [15] Jacobowitz, J. Diff. Eqn. 20 pp 37– (1976) [16] Marino, Boll. U. M. I. 11 pp 1– (1975) [17] Micheletti, Ann. Univ. Ferrara 12 pp 103– (1967) [18] Nehari, Acta Math. 105 pp 141– (1961) [19] Rabinowitz, Indiana Univ. Math. J. 23 pp 729– (1974) [20] Rabinowitz, Comm. Pure Appl. Math. 31 pp 157– (1978) [21] Rabinowitz, Comm. Pure Appl. Math. 33 pp 609– (1980) [22] Variational methods for nonlinear eigenvalue problems, in Eigenvalues of Nonlinear Problems, Roma, 1974, Ediz. Cremonese. [23] On periodic solutions of large norm of some ordinary and partial differential equations, in Ergodic Theory and Dynamical Systems, Proc. Symp. Univ. Maryland 79–80, A. Katok ed., to appear. [24] Multiple critical points of perturbed symmetric functionals, MRC Report # 2233 and paper to appear. [25] Nonlinear Functional Analysis, Gordon and Breach, New York, 1969. [26] Wolkowiski, J. Diff. Eqn. 11 pp 385– (1972) [27] and , Nonlinear oscillations and boundary value problems for Hamiltonian systems, to appear. 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.