S. Kobayashi (Department of Mathematics, Niigata University)

Fan-Takahashi minimax inequality in [1,4] is one of the important results in convex analysis as well as nonlinear analysis. In 2010, Kuwano, Tanaka and Yamada extend classical Fan-Takahashi minimax inequality into set-valued versions by using the following scalarizing functions for sets based on set-relations
$\leq^{(j)}_{C}$ $(j=1,\dots, 6)$ where $C$ is a convex cone in a vector space:
$$( {\rm I}^{(j)}_{k,V} \circ{F})(x):= \inf\left\{t\in{\mathbb{R}}\,\left|\,F(x)\leq^{(j)}_{C}(tk+V)\right.\right\}$$
$$ ( {\rm S}^{(j)}_{k,V} \circ{F})(x):= \sup\left\{t\in{\mathbb{R}}\,\left|\,(tk+V)\leq^{(j)}_{C}F(x)\right.\right\}$$
where $F$ is a set-valued map, $V\in 2^{Y}\setminus \{\emptyset \}$, direction $k\in {\mbox{\rm int}\mbox{$\,C$}}$ and
the set-relations $\leq^{(j)}_{C}$ ($j = 1, \ldots, 6$).
In 2012, Saito, Tanaka and Yamada ([3]) propose
a certain Ricceri's theorem ([2]) on Fan-Takahashi minimax inequality
for set-valued maps with respect to ``$\leq^{(5)}_{C}$.''
These scalarizing functions have some kinds of montonicity and convexity, and certain inherited properties from a parent set-valued map:if set-valued map $F$ has some kind of convexity, then ${\rm I}^{(j)}_{k,V} \circ{F}$ and ${\rm S}^{(j)}_{k,V} \circ{F}$ have also similar properies. In this talk, we take an overview of this kind of scalarization technique and show some applications to set-valued inequality.
Reference:
[1] K. Fan, A minimax inequality and its applications, Inequalities III, O. Shisha (ed.), Academic Press, New York, 1972, pp.103-113.
[2] B. Ricceri, Existence Theorems for Nonlinear Problems, Rend. Accad. Naz. Sci. XL, 11 (1987), 77-99.
[3] Y. Saito, T. Tanaka, and S. Yamada, On generalization of Ricceri's theorem for Fan-Takahashi minimax inequality into set-valued maps via scalarization, J. Nonlinear and Convex Anal. 16 (2015), 9-19.
[4] W. Takahashi, Nonlinear variational inequalities and xed point theorems, J. Math. Soc. Japan 28 (1976), 168-181.