There are many interesting theorems devoted to inequality for real-valued functions in the field of Nonlinear Analysis. Our main purpose is to study how to generalize some of them into cases for set-valued mappings. In , we can find some of generalizations to set-valued cases, which are based on the set-relations (see ) and certain sublinear scalarizing functions (see ) which are monotone with respect to each set-relation. They are set-valued versions of Fan-Takahashi’s inequality (see ), which is known as Ky Fan minimax inequality.
On the other hand, Ricceri provides interesting results on a nonlinear problem in , in which he gives a reasonable substitution for Fan-Takahashi’s inequality, that is, the same conclusion is derived under a slight alternative condition.
In this talk, we propose a set-valued version for Ricceri’s inequality by using the same methodology as the approach in  based on one of the set-relations.
 D. Kuroiwa, T. Tanaka, and T.X.D. Ha, On cone convexity of set-valued maps, Nonlinear Anal., 30 (1997), pp.1487-1496.
 I. Kuwano, T. Tanaka, and S. Yamada, Characterization of nonlinear scalarizing functions for set-valued maps, Nonlinear Analysis and Optimization, S. Akashi, W. Takahashi and T. Tanaka (eds.), Yokohama Publishers, Yokohama, 2009, pp.193-204.
 I. Kuwano, T. Tanaka, and S. Yamada, Unified scalarization for sets and set-valued Ky Fan minimax inequality, Journal of Nonlinear and Convex Anal., 11, 3 (2010), pp.513-525.
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