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Nonlinear oscillations and boundary value problems for Hamiltonian systems. (English) Zbl 0514.34032

##### MSC:
 34C15 Nonlinear oscillations, coupled oscillators (ODE) 34C25 Periodic solutions of ODE 34B15 Nonlinear boundary value problems for ODE
##### References:
 [1] H. Amann & E. Zehnder, Multiple solutions for a class of nonresonance problems and applications to differential equations, to appear. [2] J. P. Aubin & I. Ekeland, Second-order evolution equations associated with convex Hamiltonians, Cahiers Mathématiques de la Décision, No. 7825, to appear. [3] F. H. Clarke, The Euler-Lagrange differential inclusion, J. Differential Equations 19 (1975), 80–90. · Zbl 0323.49021 · doi:10.1016/0022-0396(75)90020-0 [4] F. H. Clarke, Periodic solutions to Hamiltonian inclusions, J. Differential Equations 40 (1981) 1–6. · Zbl 0461.34030 · doi:10.1016/0022-0396(81)90007-3 [5] F. H. Clarke, Multiple integrals of Lipschitz functions in the calculus of variations, Proc. Amer. Math. Soc. 64 (1977), 260–264. · doi:10.1090/S0002-9939-1977-0451156-3 [6] F. H. Clarke & I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Applied Math. 33 (1980), 103–116. · Zbl 0428.70029 · doi:10.1002/cpa.3160330202 [7] I. Ekeland, Periodic Hamiltonian trajectories and a theorem of Rabinowitz, J. Differential Equations 34 (1979), 523–534. · Zbl 0446.70019 · doi:10.1016/0022-0396(79)90034-2 [8] I. Ekeland & J. M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Annals of Math. 112 (1980) 283–319. · Zbl 0449.70014 · doi:10.2307/1971148 [9] I. Ekeland & R. Temam, ”Convex Analysis and Variational Problems,” North-Holland (1976). [10] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Applied Math. 31 (1978), 157–184. · Zbl 0369.70017 · doi:10.1002/cpa.3160310203 [11] P. H. Rabinowitx, A variational method for finding periodic solutions of differential equations, in ”Nonlinear Evolution Equations,” edited by M. Crandall, Academic Press (1980). [12] R. T. Rockafellar, ”Convex Analysis,” Princeton University Press (1970). [13] A. Weinstein, Periodic orbits for convex Hamiltonian systems, Annals of Math. 108 (1978), 507–518. · Zbl 0403.58001 · doi:10.2307/1971185 [14] V. Benci & P. Rabinowitz, ”Critical point theorems for indefinite functional,” Inventiones Math., 52 (1979), 241–273. · Zbl 0465.49006 · doi:10.1007/BF01389883 [15] H. Brezis & L. Nirenberg, ”Characterization of the range of some nonlinear operators and applications to boundary value problems,” Annali Scuola Norm. Sup. Pisa, 5 (1978), 225–326. [16] H. Brezis & A. Bahri, ”Periodic solutions of a nonlinear wave equation,” to appear. [17] J. M. Coron, ”Resolution de l’équation Au+Bu=f, ou A est linéaire autoadjoint et B dérivé d’un potentiel convexe,” to appear. [18] F. H. Clarke, ”Solutions périodiques des équations hamiltoniennes”, C. R. Acad. Sci. Paris 287 (1978), 951–952.