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2018•Nonlinear functional analysis and applicationsRequires access

THE MINIMAX PROBLEM OF TWO VECTOR-VALUED FUNCTIONS

Chuang Liang Zhang

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Abstract

In this paper, we study the minimax problem in a partial ordered Hausdorff topological vector space. We obtain a vector form Ky Fan minimax inequality theorem which involves two vector-valued functions. The proof of this theorem don’t use separation theorems and nonlinear scalarization functions. Forthermore, we also use the fixed point theorem of set-valued mappings by Browder [1] to prove some results of minimax problem of two vector-valued functions, and the minimax theorem in [16] is extended to vector form.

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What this paper is about

In this paper, we study the minimax problem in a partial ordered Hausdorff topological vector space. We obtain a vector form Ky Fan minimax inequality theorem which involves two vector-valued functions. The proof of this theorem don’t use separation theorems and nonlinear scalarization functions. Forthermore, we also use the fixed point theorem of set-valued mappings by Browder [1] to prove some results of minimax problem of two vector-valued functions, and the minimax theorem in [16] is extended to vector form.

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Available abstract

In this paper, we study the minimax problem in a partial ordered Hausdorff topological vector space. We obtain a vector form Ky Fan minimax inequality theorem which involves two vector-valued functions. The proof of this theorem don’t use separation theorems and nonlinear scalarization functions. Forthermore, we also use the fixed point theorem of set-valued mappings by Browder [1] to prove some results of minimax problem of two vector-valued functions, and the minimax theorem in [16] is extended to vector form.

Key concepts: Mathematics, Minimax, Minimax theorem, Hausdorff space, Danskin's theorem, Fixed-point theorem, Vector space, Topological vector space

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