Vector-valued set-valued variants of Ky Fan's inequality (Q2708288)

The authors establish a two-function result which is a slightly more general form of a two-function result of \textit{S. Simons} [Proc. Symp. Pure Math. 45(2), 377-392 (1986; MR 87h:49019)] and implies the classical Fan's inequality. By using the two-function result, the existence of a solution of two kinds of vector-valued Fan's type inequalities for multifunctions, known as generalized vector equilibrium problems, is proved. One of the result (Theorem 3.1) generalizes the main result of \textit{Q. H. Ansari} and \textit{J.-C. Yao} [Appl. Math. Lett. 12, No. 8, 53-56 (1999; Zbl 1014.49008)] in the sense that it contains Fan's inequality in its full generality (for lower-semicontinuous functions), while their result contains it only for continuous functions.