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.
Bednarczuk, E., Miglierina, E., Molho, E., A Mountain Pass-type Theorem for Vector-valued Functions, <<SET-VALUED AND VARIATIONAL ANALYSIS>>, 2011; 19 (4): 569-587. [doi:10.1007/s11228-011-0182-z] [http://dx.medra.org/10.1007/s11228-011-0182-z] [http://hdl.handle.net/10807/1425]
A Mountain Pass-type Theorem for Vector-valued Functions
Miglierina, Enrico;
2011
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.