A minimax inequality and its applications to variational inequalities
Chi-Lin Yen · Pacific Journal of Mathematics · 1981
In this paper we get a slight generalization of a Ky Fan's result which concerns with a minimax inequality.We shall use this result to give a direct proof for the existence of solutions of the following two variational inequalities: (1) inf ^h{x)-h(y) for all xeX, weTy and (2) sup ^h(x)-h(y) for all xeX, w BTX where TczExE' is monotone, E is a reflexive Banach space with its dual E', X is a closed convex bounded subset of E, and h is a lower semicontinuous convex function from X into R.In this paper we get a slight generalization of a Ky Fan's result [4] which concerns with a minimax inequality.We shall use this result to give a direct proof for the existence of solutions of the following two variational inequalities: (1) inf (w, yx) ^ h(x)h(y) for all x e X , ιv e Ty and (2) sup (w, yx)