A Mountain Pass-type Theorem for Vector-valued Functions

作者：Bednarczuk Ewa M; Miglierina Enrico; Molho Elena^*

来源：Set-Valued and Variational Analysis, 2011, 19(4): 569-587.

DOI：10.1007/s11228-011-0182-z

• Minimax
• Ekeland's principle
• Vector optimization
• Set-valued optimization

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.

• 出版日期2011-12