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Dato un sistema Hamiltoniano, nel quale la funzione Hamiltoniana è la somma di un termine quadratico definito positivo e di un termine superquadratico convesso, si dimostra l'esistenza di soluzioni periodiche di periodo minimo T fissato, per ogni T ε (0, 2π/ΩN), dove ΩN è il massimo autovalore della forma quadratica. Si utilizzano alcune tecniche relative alla teoria dell'indice di Morse, introdutte in [13], [15], [16] ed una opportuna formulazione del principio di dualità di Clarke ed Ekeland (vedi [10], [11]).
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References
H. Amann -E. Zehnder,Nontrivial solutions for a class of nonresonance problems and applications to nonlinear partial differential equations, Ann. Sc. Norm. Sup. di Pisa,7 (1980), pp. 539–603.
H. Amann -E. Zehnder,Periodic solutions of asymptotically linear Hamiltonian equations, Manuscripta Mathematica,32 (1980), pp. 149–189.
A. Ambrosetti -G. Mancini,Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann.,255 (1981), pp. 405–421.
A.Ambrosetti - V.Coti Zelati,Solutions with minimal period for Hamiltonian systems in a potential well, preprint.
A.Ambrosetti - V.Coti Zelati - I.Ekeland,Symmetry breaking in critical point theory, preprint CEREMADE, Paris, 1985.
A. Ambrosetti -P. Rabinowitz,Dual variational methods in critical point theory and applications, J. Funct. Anal.,44 (1973), pp. 349–381.
J. P.Aubin - I.Ekeland,Applied nonlinear analysis, Wiley, 1984.
V. Benci -A. Capozzi -D. Fortunato,Periodic solutions for a class of Hamiltonian systems, Springer Verlag Lect. Notes in Math.,964 (1982), pp. 86–94.
H. Berestycki -J. M. Lasry -G. Mancini -B. Ruf,Existence of multiple periodic orbits on star-shaped Hamilton surfaces, Comm. Pure Appl. Math.,38 (1985), pp. 253–289.
F. Clarke,Periodic solutions of Hamiltonian inclusions, J. Diff. Eq.,40 (1981), pp. 1–6.
F. Clarke,Periodic solutions of Hamilton's equations and local minima of the dual action, Trans. of the A.M.S.,287 (1985), pp. 239–251.
F. Clarke -I. Ekeland,Hamiltonian trajectories having prescribed minimal period, Comm. Pure and Appl. Math.,33 (1980), pp. 103–116.
C.Conley - E.Zehnder,Morse type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure and Appl. Math.
I. Ekeland,Une théorie de Morse pour les systèmes Hamiltoniens convexes, Ann. IHP, Analyse non linéaire,1 (1984), pp. 19–78.
I. Ekeland,Periodic solutions to Hamiltonian equations and a theorem of P. Rabinowitz, J. Diff. Eq.,34 (1979), pp. 523–534.
I. Ekeland,An index theory for periodic solutions of convex Hamiltonian systems, Proc. of Symposia in Pure Math.,45 (1986), pp. 395–423.
I.Ekeland - H.Hofer,Periodic solutions with prescribed period for convex autonomous Hamiltonian systems, preprint CEREMADE no. 8421, Paris (1984).
I.Ekeland - R.Temam,Analyse convexe et problèmes variationnelles, Dunod-Gauthier-Villard, 1974.
D. Gromoll -W. Meyer,On differentiable functions with isolated critical points, Topology,8 (1969), pp. 361–369.
M. Girardi -M. Matzeu,Some results on solutions of minimal period to superquadratic Hamiltonian systems, Nonlinear Analysis,7, no. 5 (1983), pp. 475–482.
M. Girardi -M. Matzeu,Solutions of prescribed minimal period to convex and nonconvex Hamiltonian syst n ms, Boll. U.M.I., (6)4-B (1985), pp. 951–967.
M. Girardi -M. Matzeu,Solutions of minimal period for a class of nonconvex Hamiltonian systems and application to 33 e fixed energy problem, Nonlinear Analysis,10, no. 4, (1986), pp. 371–382.
M.Girardi - M.Matzeu,Some results on solutions of minimal period to Hamiltonian systems, Nonlinear Oscillations for Conservative Systems, Proceedings, Venice, 9–12 January 1985.
H.Hofer,A geometric description of the neighbourhood of a critical point given by the mountain pass theorem, J. London Math. Soc., to appear.
H.Hofer,The topological degree at a critical point of mountain pass type, Proc. AMS Summer Institute on Nonlinear Functional Analysis (Berkeley, 1983), to appear.
M. A.Krasnoselskii,Topological methods in the theory of nonlinear integral equations, English translation, Pergamon Press, 1963.
R. Palais,Ljusiernik-Schnirelman, theory on Banach manifolds, Topology,5 (1966), pp. 115–132.
P. Rabinowitz,Periodic solutions of Hamiltonian systems, Comm. Pure and Appl. Math.,34 (1978), pp. 157–184.
P. Rabinowitz,On subharmonic solutions of Hamiltonian systems, Comm. Pure and Appl. Math.,33 (1980), pp. 609–633.
C.Viterbo,Une théorie de Morse pour les systèmes hamiltoniens étoilés, thse de III cycle, Univ. Paris Dauphine, 1985.
V.Yakubovich - V.Starzhinskii,Linear differential equations with periodic coefficients, Halsted Press, Wiley.
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Girardi, M., Matzeu, M. Periodic solutions of convex autonomous Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity. Annali di Matematica pura ed applicata 147, 21–72 (1987). https://doi.org/10.1007/BF01762410
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DOI: https://doi.org/10.1007/BF01762410