Abstract
The mountain pass theorem for scalar functionals is a fundamental result
of the minimax methods in variational analysis. In this work we extend this theorem
to the class of C^1 functions f : R^n → R^m, where the image space is ordered by the
nonnegative orthant R^m_+. Under suitable geometrical assumptions, we prove the
existence of a critical point of f and we localize this point as a solution of a minimax
problem. We remark that the considered minimax problem consists of an inner
vector maximization problem and of an outer set-valued minimization problem. To
deal with the outer set-valued problem we use an ordering relation among subsets of
R^m introduced by Kuroiwa. In order to prove our result, we develop an Ekeland-type
principle for set-valued maps and we extensively use the notion of vector
pseudogradient.
Lingua originale | English |
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pagine (da-a) | 569-587 |
Numero di pagine | 19 |
Rivista | Set-Valued and Variational Analysis |
Volume | 19 |
DOI | |
Stato di pubblicazione | Pubblicato - 2011 |
Keywords
- critical points
- mountain pass theorem
- pseudogradient