- Research Article
- Open Access
An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces
© Nguyen Buong and Nguyen Thi Quynh Anh. 2011
- Received: 17 December 2010
- Accepted: 7 March 2011
- Published: 15 March 2011
We introduce a new implicit iteration method for finding a solution for a variational inequality involving Lipschitz continuous and strongly monotone mapping over the set of common fixed points for a finite family of nonexpansive mappings on Hilbert spaces.
- Hilbert Space
- Variational Inequality
- Nonexpansive Mapping
- Inequality Problem
- Real Hilbert Space
Variational inequalities were initially studied by Kinderlehrer and Stampacchia in  and ever since have been widely investigated, since they cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [1–3]).
where denotes the metric projection from onto and is an arbitrarily fixed positive constant.
where , for integer , with the mod function taking values in the set . They proved the following result.
Let be a real Hilbert space and a nonempty closed convex subset of . Let be nonexpansive self-maps of such that , where . Let and be a sequence in such that . Then, the sequence defined implicitly by (1.5) converges weakly to a common fixed point of the mappings .
They proved the following result.
Let be a real Hilbert space and a mapping such that for some constants , is -Lipschitz continuous and -strongly monotone. Let be nonexpansive self-maps of such that . Let , and let and satisfying the conditions: and , for some . Then, the sequence defined by (1.7) converges weakly to a common fixed point of the mappings . The convergence is strong if and only if .
Recently, Ceng et al.  extended the above result to a finite family of asymptotically self-maps.
denotes the identity mapping of , and the parameters for all satisfy the following conditions: as and .
We formulate the following facts for the proof of our results.
Lemma 2.1 (see ).
(i) and for any fixed , (ii) , for all .
Put ; for any nonexpansive mapping of , we have the following lemma.
Lemma 2.2 (see ).
and for a fixed number , where .
Lemma 2.3 (Demiclosedness Principle ).
Assume that is a nonexpansive self-mapping of a closed convex subset of a Hibert space . If has a fixed point, then is demiclosed; that is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that .
Now, we are in a position to prove the following result.
Then, the net defined by (1.8)-(1.9) converges strongly to the unique element in (1.1).
So, is a contraction in . By Banach's Contraction Principle, there exists a unique element such that for all .
that implies the boundedness of . So, are the nets .
as for .
It means that as for .
and , it follows that . Similarly, we obtain that , for and as .
The uniqueness of in (1.1) guarantees that . Again, replacing in (2.8) by , we obtain the strong convergence for . This completes the proof.
It is well known  that a mapping by with a fixed for all is a nonexpansive mapping and . Using this fact, we can extend our result to the case , where is -strictly pseudocontractive as follows.
So, we have the following result.
Theorem 3.1 ..
where , for , are defined by (3.2) and , converges strongly to the unique element in (1.1).
It is known in  that where with and for -strictly pseudocontractions . Moreover, is -strictly pseudocontractive with . So, we also have the following result.
Theorem 3.2 ..
where , , and , converges strongly to the unique element in (1.1).
This work was supported by the Vietnamese National Foundation of Science and Technology Development.
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