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Minimax Theorems for Indefinite Functionals

  • Jean Mawhin
  • Michel Willem
Part of the Applied Mathematical Sciences book series (AMS, volume 74)

Abstract

The dual least action principle has provided sharp existence theorems for the periodic solutions of Hamiltonian systems when the Hamiltonian is convex in u. When it is not the case, the existence of critical points of saddle point type can be proved by using some minimax arguments. To motivate them, we can consider the following intuitive situation. If φ ∈ C 1(R 2,R), we can view φ(x, y) as the altitude of the point of the graph of φ having (x,y) as projection on R2. Assume that there exists points u 0R 2, u 1 ∈ R2 and a bounded open neighborhood Ω of u 0 such that u 1R 2 \ Ω and φ(u) > max(φ(u 0), φ(u 1)) whenever u ∈ ∂Ω (that is the case for example if u 0 and u 1 are two isolated local minimums of φ).

Keywords

Banach Space Periodic Solution Convergent Subsequence Critical Point Theory Minimax Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Jean Mawhin
    • 1
  • Michel Willem
    • 1
  1. 1.Institut de Mathematique Pure et AppliqueeLouvain-la-NeuveBelgium

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