考慮單調變分不等式問題：x ∈C, ‹Fx, y - x ›≥0, y∈C ，其中C 是實Hilbert 空間H
的非空閉凸子集，F 是C 到H 的單調算子。若F 是強單調且Lipschitzian，則變分不等
式問題等價於壓縮映像的固定點問題。因此，可以應用Banach 壓縮映像原理去解決問
題。
然而，在F 是反強單調的情況下，變分不等式問題不等價於壓縮映像的固定點問
題，而等價於非擴張映像的固定點問題。在這篇論文中我們提供一些對於非擴張映像的
疊代方法應用結果去解決變分不等式問題。我們將介紹Mann 演算法以及Halpern 演算
法且證明它們在某些特殊條件下分別會弱收斂和強收斂到變分不等式的解。

Consider the variational inequality (VI)
x* ∈C, ‹Fx*, x - x* ›≥0, x∈C (*)
where C is a nonempty closed convex subset of a real Hilbert space H and
F : C→ H is a monotone operator form C into H. It is known that if F is
strongly monotone and Lipschitzian, then VI (*) is equivalently turned into
a fixed point problem of a contraction; hence Banach's contraction principle
applies. However, in the case where F is inverse strongly monotone, VI (*)
is equivalently transformed into a fixed point problem of a nonexpansive
mapping. The purpose of this paper is to present some results which apply
iterative methods for nonexpansive mappings to solve VI (*). We introduce
Mann's algorithm and Halpern's algorithm and prove that the sequences
generated by these algorithms converge weakly and respectively, strongly to
a solution of VI (*), under appropriate conditions imposed on the parameter
sequences in the algorithms.

1 Introduction 1
2 Preliminaries 2
3 Variational Inequalities Governed by Inverse Strongly Monotone Operators 8
References 20

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