Existence and approximation of solutions of nonlinear variational inequalities.
Existence and approximation of solutions of nonlinear variational inequalities.
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DOI：
10.1073/pnas.56.4.1080
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发表时间：
1966-10
影响因子：
11.1
通讯作者：
F. Browder
中科院分区：
文献类型：
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作者：
F. Browder
在前面的注释中，我们表述并证明了一类问题解的基本存在性定理，我们称之为非线性变分不等式如前所述，这类特殊情况包括：单调算子方程，凸集上的变分问题，凸集上的单调算子不等式。这些问题的定义如下：我们给定一个自反的Banach空间X，一个从X到X*的域D(T)上的单调非线性算子T，以及一个从X到(-a， + ao]的下半连续凸函数f， f; + ao。设w是X*中的一个给定元素。我们求D（T)的元素usO，其中（two w, v -uo）） f(uo) -f(v), v e D(T)。（1)
In a preceding note,' we formulated and proved the basic existence theorems for solutions of a class of problems which we shall call nonlinear variational inequalities.2 As we remarked, this class includes as special cases: monotone operator equations, variational problems on convex sets, and monotone operator inequalities on convex sets. The definition of these problems runs as follows: We are given a reflexive Banach space X, a monotone nonlinear operator T from a domain D(T) in X to X*, and a lower semicontinuous convex function f from X to (-a, + ao ] with f ; + ao. Let w be a given element of X*. We ask after elements usO of D(T) for which: (Tuo w, v -uo) ) f(uo) -f(v), v e D(T). (1)