Minimax theorems for set-valued maps without continuity assumptions

Abstract

We introduce several classes of set-valued maps with generalized convexity. We obtain minimax theorems for set-valued maps which satisfy the introduced properties and are not continuous, by using a fixed point theorem for weakly naturally quasi-concave set-valued maps defined on a simplex in a topological vector space.Comment: 22 page

Applications of the KKM Property to Coincidence Theorems, Equilibrium Problems, Minimax Inequalities and Variational Relation Problems

“…In fact, we obtain a new general minimax inequality of the following type: inf x∈X sup y∈Y t(x, y) ≤ sup y∈Y infz∈Z q(y,z) infz∈Z sup x∈X p(x,z) . Another recent result, due to the author, regarding minimax inequalities for dicontinuous correspondences, is [27].…”

Convergence Theorems in Interval-Valued Riemann–Lebesgue Integrability

“…The theory of non-additive set functions and nonlinear integrals has become an important tool in many domains such as: potential theory, subjective evaluation, optimization, economics, decision making, data mining, artificial intelligence, accident rates estimations (e.g. [20,21,38,44,53,56,60,63,75,[79][80][81]84]). In the literature several methods of integration for (multi)functions based on extensions of the Riemann and Lebesgue integrals have been introduced and studied (see for example, [2-9, 13-15, 17, 18, 25, 29-34, 36, 37, 39-43, 54, 55, 58, 68]).…”

“…The study of non-additive set functions and nonlinear integrals has received a wide recognition because of its applications in many domains such as: potential theory, subjective evaluation, optimization, economics, decision-making, data mining, artificial intelligence, and accident rate estimations (e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). In the literature, there are several methods of integration for (multi)functions based on extensions of the Riemann and Lebesgue integrals.…”