A generalized section theorem and a minimax inequality for a vector-valued mapping¹

Minimax theorems for set-valued maps without continuity assumptions

“…(ii) F is said to be (in the sense of [12 ,Definition 3.6]) type-(v) properly S−quasiconvex on X (see [9]), iff for any x 1 , x 2 ∈ X and λ ∈ [0, 1], either F (λx 1…”

“…iff, for each n and for every x 1 , x 2 , ..., x n ∈ X, [26]) Let X be a non-empty convex subset of a topological vector space E, let Y be a subset of a topological vector space Z and S a pointed closed convex cone in Z with its interior intS = ∅. A vector-valued map- 1]. This condition is equivalent with the following condition: there exists µ ∈ [0, 1] such that…”

A minimax theorem for vector-valued functions in lexicographic order

“…In recent years, based on the development of vector optimization, a great deal of research has been devoted to the study of the minimax theorem for vector-valued functions, such as [4][5][6][7][8][9][10][11], where the partial order is introduced by a closed convex pointed cone. In [7], Nieuwenhuis generalized the notions of minimax, maximin and saddle point from real-valued functions to vector-valued functions.…”

Minimax Problems of Uniformly Same-Order Set-Valued Mappings

“…Shi and Ling [22] proved a minimax theorem and a cone saddle point theorem for a class of vector-valued functions which includes separated functions as its proper subset. Chen [4] obtained a Ky Fan minimax inequality for a vector-valued function on H-spaces by using a generalized Fan's section theorem. Chang et al [3] proved a Ky Fan minimax inequality for a vector-valued function on W-spaces.…”