zbMATH — the first resource for mathematics

Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems. (English) Zbl 0594.58035
In 1977, Rabinowitz proved the existence of a nonconstant periodic solution of any prescribed period \(T>0\) for the Hamiltonian systems under some reasonable conditions. However, he does not claim that the T- periodic solution he finds actually has minimal period T. Therefore, it challenges one’s attention to prove minimality of the period for that T- periodic solution.
The first researchers to make progress on this question were A. Ambrosetti and G. Mancini, and then M. Girardi and M. Matzeu. This paper is a similar one devoted to the same question for convex autonomous Hamiltonian systems with some (more or less) weaker conditions. The authors combine two new tools: An index theory for periodic solutions of Hamiltonian systems, and a complete description of \(C^ 2\) functionals near critical points of mountain pass type. The main result is that the Ambrosetti-Rabinowitz theorem, when applied to the dual action functional, will actually yield solutions with minimal period T.
Reviewer: Ding Tongren

37G99 Local and nonlocal bifurcation theory for dynamical systems
34C25 Periodic solutions to ordinary differential equations
Full Text: DOI EuDML
[1] Amann, H., Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Sc. Norm. Super. Pisa7, 539-603 (1980) · Zbl 0452.47077
[2] Amann, H., Zehnder, E.: Periodic solutions of asymptotically linear Hamiltonian equations. Manuscr. Math.32, 149-189 (1980) · Zbl 0443.70019 · doi:10.1007/BF01298187
[3] Ambrosetti, A., Mancini, G.: Solutions of minimal period for a class of convex Hamiltonian systems. Math. Ann.255, 405-421 (1981) · Zbl 0466.70022 · doi:10.1007/BF01450713
[4] Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal.14, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[5] Aubin, J.P., Ekeland, I.: Applied nonlinear analysis. New York: Wiley 1984 · Zbl 0641.47066
[6] Cambini, A.: Sul lemma di Morse. Boll. Unione Met. Ital.7, 87-93 (1973) · Zbl 0267.58007
[7] Clarke, F.: Solutions périodiques des équations hamiltoniennes. C.R. Acad. Sci., Paris287, 951-952 (1978) · Zbl 0422.35005
[8] Clarke, F.: Periodic solutions of Hamiltonian inclusions. J. Differ. Equations40, 1-6 (1981) · Zbl 0461.34030 · doi:10.1016/0022-0396(81)90007-3
[9] Clarke, F.: Periodic solutions of Hamiltonian’s equations and local minima of the dual action (To appear)
[10] Clarke, F., Ekeland, I.: Hamiltonian trajectories having prescribed minimal period. Comm. Pure Appl. Math.33, 103-116 (1980) · Zbl 0428.70029 · doi:10.1002/cpa.3160330202
[11] Conley, C., Zehnder, E.: Morse type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. (To appear) · Zbl 0559.58019
[12] Ekeland I.: Nonconvex minimization problems. Bull. Am. Math. Soc.1, New Series 443-474 (1979) · Zbl 0441.49011 · doi:10.1090/S0273-0979-1979-14595-6
[13] Ekeland, I.: Une théorie de Morse pour les systèmes hamiltoniens convexes. Ann. Inst. Henri Poincaré: Analyse non linéaire.1, 19-78 (1984) · Zbl 0537.58018
[14] Ekeland, I.: Periodic solutions to Hamiltonian equations and a theorem of P. Rabinowitz. Differ. Equations34, 523-534 (1979) · Zbl 0446.70019 · doi:10.1016/0022-0396(79)90034-2
[15] Ekeland, I.: An index theory for periodic solutions of convex Hamiltonian systems. Proc. Am. Math. Soc. Summer Institute on Nonlinear Functional Analysis (Berkeley, 1983, (To appear) · Zbl 0476.34035
[16] Ekeland, I.: Hypersurfaces pincées et systèmes hamiltoniens. Note C.R. Acad. Sci. Paris (à paraître 1984) · Zbl 0567.58008
[17] Ekeland, I., Teman, R.: Analyse convexe et problèmes variationnels, Dunod-Gauthier-Villars, 1974; English translation, ?Convex analysis and variational problems?. North-Holland-Elsevier, 1976
[18] Gromoll, D., Meyer, W.: On differentiable functions with isolated critical points. Topology8, 361-369 (1969) · Zbl 0212.28903 · doi:10.1016/0040-9383(69)90022-6
[19] Girardi, M., Matzeu, M.: Some results on solutions of minimal period to superquadratic Hamiltonian equations. Nonlinear Anal., Theory Methods Appl.7, 475-482 (1983) · Zbl 0512.70021 · doi:10.1016/0362-546X(83)90039-1
[20] van Groesen, E.: Existence of multiple normal mode trajectories on convex energy surfaces of even, classical Hamiltonian systems. J. Differ. Equations (To appear) · Zbl 0519.70026
[21] Hofer, H.: A geometric description of the neighbourhood of a critical point given by the mountain pass theorem. J. Lond. Math. Soc. (To appear) · Zbl 0573.58007
[22] Hofer, H.: The topological degree at a critical point of mountain pass type. Proc. Am. Math. Soc. Summer Institute on Nonlinear Functional Analysis (Berkely, 1983) (To appear) · Zbl 0608.58013
[23] Krasnoselskii, M.A.: Topological methods in the theory of nonlinear integral equations. English translation, Pergamon press, 1963
[24] Palais, R.: Ljusternik-Schnirelman theory on Banach manifolds. Topology5, 115-132 (1966) · Zbl 0143.35203 · doi:10.1016/0040-9383(66)90013-9
[25] Rabinowitz, P.: Periodic solutions of Hamiltonian systems Comm. Pure Appl. Math.31, 157-184 (1978) · Zbl 0369.70017 · doi:10.1002/cpa.3160310203
[26] Rockafellar, R.T.: Convex analysis. Princeton: University Press (1970) · Zbl 0193.18401
[27] Takens, F.: A note on sufficiency of jets. Invent. Math. 225-231 (1971) · Zbl 0231.58008
[28] Yakubovich, V., Starzhinskii, V.: Linear differential equations with periodic coefficients. New York: Halsted Press, Wiley
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.