Hybrid Steepest-Descent Methods for Solving Variational Inequalities Governed by Boundedly Lipschitzian and Strongly Monotone Operators
© S. He and X.-L. Liang. 2010
Received: 30 September 2009
Accepted: 13 January 2010
Published: 24 January 2010
Let be a real Hilbert space and let be a boundedly Lipschitzian and strongly monotone operator. We design three hybrid steepest descent algorithms for solving variational inequality of finding a point such that , for all , where is the set of fixed points of a strict pseudocontraction, or the set of common fixed points of finite strict pseudocontractions. Strong convergence of the algorithms is proved.
This is known as the variational inequality problem (i.e., initially introduced and studied by Stampacchia  in 1964. In the recent years, variational inequality problems have been extended to study a large variety of problems arising in structural analysis, economics, optimization, operations research, and engineering sciences; see [1–6] and the references therein.
Yamada  proposed hybrid methods to solve , where is composed of fixed points of a nonexpansive mapping; that is, is of the form
He and Xu  proved that has a unique solution and iterative algorithms can be devised to approximate this solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . In the case where is the set of fixed points of a nonexpansive mapping, they invented a hybrid iterative algorithm to approximate the unique solution of and this extended the Yamada's results.
The main purpose of this paper is to continue our research in . We assume that is a boundedly Lipschitzian and strongly monotone operator as in , but is the set of fixed points of a strict pseudo-contraction , or the set of common fixed points of finite strict pseudo-contractions . For the two cases of , we will design the hybrid iterative algorithms for solving and prove their strong convergence, respectively. Relative definitions are stated as below.
Obviously, the nonexpansive mapping class is a proper subclass of the strict pseudo-contraction class and the Lipschitzian operator class is a proper subclass of the boundedly Lipschitzian operator class, respectively.
We will use the following notations:
We need some facts and tools which are listed as lemmas below.
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Lemma 2.5 (see ).
Lemma 2.8 (see ).
3. Further Extension of Hybrid Iterative Algorithm
Yamada got the following result.
Theorem 3.1 (see ).
He and Xu  proved that has a unique solution if is a boundedly Lipschitzian and strongly monotone operator and is a closed convex subset of . Using this result, they were able to relax the global Lipschitz condition on in Theorem 3.1 to the weaker bounded Lipschitz condition and invented a hybrid iterative algorithm to approximate the unique solution of . Their result extended the Yamada's above theorem.
We have the following result.
By (3.2) and Lemma 2.7, we have
4. Parallel Algorithm and Cyclic Algorithm
In this section, we discuss the parallel algorithm and the cyclic algorithm, respectively, for solving the variational inequality over the set of the common fixed points of finite strict pseudo-contractions.
Since is a nonempty closed convex subset of , (4.1) has a unique solution. Throughout this section, is an arbitrary fixed point, , is the Lipschitz constant of on , the fixed constant satisfies , and the sequence belongs to .
Using Lemma 2.8 and Thorem 3.2, the following conclusion can be deduced directly.
We get the following result
This research is supported by the Fundamental Research Funds for the Central Universities (GRANT:ZXH2009D021).
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