A Mountain Pass-type Theorem for Vector-valued Functions |
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Authors: | Ewa M Bednarczuk Enrico Miglierina Elena Molho |
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Institution: | 1.Systems Research Institute,Polish Academy of Sciences,Warszawa,Poland;2.Cardinal Stefan Wyszynski University,Warszawa,Poland;3.Dipartimento di Economia,Università dell’Insubria,Varese,Italy;4.Dipartimento di Economia Politica e Metodi Quantitativi,Pavia,Italy |
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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 \(\mathcal{C}^{1}\) functions \(f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}\), where the image space is ordered by the nonnegative orthant \(\mathbb{R}_{+}^{m}\). 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 \(\mathbb{R}^{m}\) 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. |
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