Vol. 97, No. 2, 1981

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A minimax inequality and its applications to variational inequalities

Chi-Lin Yen

Vol. 97 (1981), No. 2, 477–481
Abstract

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:

inf ⟨w, y− x⟩ ≦ h(x)− h(y) for all x ∈ X,
w∈Ty
(1)

and

sup ⟨w, y− x⟩ ≦ h(x)− h(y) for all x ∈ X,
w∈Tx
(2)

where T E ×Eis 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.

Mathematical Subject Classification 2000
Primary: 49A29, 49A29
Secondary: 47H05
Milestones
Received: 20 June 1980
Revised: 27 October 1980
Published: 1 December 1981
Authors
Chi-Lin Yen