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Calculus of variations in mean and convex Lagrangians, II

Published online by Cambridge University Press:  17 April 2009

Joël Blot
Affiliation:
Faculté des Sciences de LimogesDépartment de Mathématiques123 Av. Albert Thomas87060 Limoges CedexFrance
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Abstract

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We prove the Legendre Necessary Condition of the Calculus of Variations in Mean an arbitrary finite dimension. When the Lagrangian is convex, we establish that if the Euler-Lagrange equation possesses an almost periodic solution then it possesses periodic and constant solutions. We deduce from this fact various consequences on the structure of the set of almost periodic solutions.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Besicovitch, A.S., Almost periodic functions (Cambridge U. Press, Cambridge, 1932).Google Scholar
[2]Blot, J., ‘Une approche variationnelle des orbites quasi-périodiques des systémes hamiltoniens’, Ann. Sci. Math. Québec (to appear).Google Scholar
[3]Blot, J., ‘Calculus of Variations in Mean and Convex Lagrangians’, J. Math. Anal. Appl. 134 (1988), 312321.CrossRefGoogle Scholar
[4]Blot, J., ‘Calcul des variations en moyenne temporelle’, C.R. Acad. Sci. Paris Sér I Math t.306 (1988), 809811.Google Scholar
[5]Dunford, N. and Schwartz, J.T., Linear operators, Part I: General theory (Interscience Publ. Inc., New York, 1958).Google Scholar