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 [3], we can find some of generalizations to set-valued cases, which are based on the set-relations (see [1]) and certain sublinear scalarizing functions (see [2]) which are monotone with respect to each set-relation. They are set-valued versions of Fan-Takahashi’s inequality (see [5]), which is known as Ky Fan minimax inequality.
On the other hand, Ricceri provides interesting results on a nonlinear problem in [4], 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 [3] based on one of the set-relations.
References
[1] D. Kuroiwa, T. Tanaka, and T.X.D. Ha, On cone convexity of set-valued maps, Nonlinear Anal., 30 (1997), pp.1487-1496.
[2] 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.
[3] 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.
[4] B. Ricceri, Existence theorems for nonlinear problems, Rend. Accad. Naz. Sci. XL, 11 (1987), pp.77-99.
[5] W. Takahashi, Nonlinear variational inequalities and fixed point theorems, J. Math. Soc. Japan, 28 (1976), pp.168-181.