Weak convergence of a projection algorithm for variational inequalities in a Banach space
Weak convergence of a projection algorithm for variational inequalities in a Banach space
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DOI：
10.1016/j.jmaa.2007.07.019
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发表时间：
2008-03-01
影响因子：
1.3
通讯作者：
Takahashi, Wataru
中科院分区：
文献类型：
--
作者：
Iiduka, Hideaki;Takahashi, Wataru
设C是Banach空间E的非空闭凸集。本文在阿尔伯[Ya. I.阿尔伯，度量和广义投影算子在Banach空间：性质和应用，在：A.G. Kartsatos（Ed.），增生型和单调型非线性算子的理论和应用，在：纯粹应用数学讲义，第178卷，Dekker，纽约，1996年，pp. [15-50]中，我们引入了以下迭代方案来寻找Banach空间中逆强单调算子A的变分不等式问题的解：x（1）= x是C的元素，并且x（n+1）= Pi（C）J（-1）（Jx（n）-lambda（n）Ax（n）），其中n = 1。二，其中Pi（C）是从E到C的广义投影，J是从E到E-* 的对偶映射，{lambda（n）}是正真实的数列。然后我们证明了一个弱收敛定理（定理3.1）。最后，利用这个结果，我们考虑凸极小化问题，互补问题，以及寻找满足0 = Au的点U E E的问题. (c)2007年爱思唯尔公司All rights reserved.
Let C be a nonempty, closed convex subset of a Banach space E. In this paper, motivated by Alber [Ya.I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, in: Lecture Notes Pure Appl. Math., vol. 178, Dekker, New York, 1996, pp. 15-50], we introduce the following iterative scheme for finding a solution of the variational inequality problem for an inversestrongly-monotone operator A in a Banach space: x(1) = x is an element of C andx(n+1) = Pi(C)J(-1)(Jx(n)-lambda(n)Ax(n))for every n = 1. 2, ..., where Pi(C) is the generalized projection from E onto C, J is the duality mapping from E into E-* and {lambda(n)} is a sequence of positive real numbers. Then we show a weak convergence theorem (Theorem 3.1). Finally, using this result, we consider the convex minimization problem, the complementarity problem, and the problem of finding a point U E E satisfying 0 = Au. (c) 2007 Elsevier Inc. All rights reserved.