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An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2011, Article number: 276859 (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.
Let be a nonempty closed and convex subset of a real Hilbert space with inner product and norm , and let be a nonlinear mapping. The variational inequality problem is formulated as finding a point such that
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]).
It is well known that if is an -Lipschitz continuous and -strongly monotone, that is, satisfies the following conditions:
where and are fixed positive numbers, then (1.1) has a unique solution. It is also known that (1.1) is equivalent to the fixed-point equation
where denotes the metric projection from onto and is an arbitrarily fixed positive constant.
Let be a finite family of nonexpansive self-mappings of . For finding an element , Xu and Ori introduced in  the following implicit iteration process. For and , the sequence is generated as follows:
The compact expression of the method is the form
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 .
Further, Zeng and Yao introduced in  the following implicit method. For an arbitrary initial point , the sequence is generated as follows:
The scheme is written in a compact form as
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.
Clearly, from we have that as . To obtain strong convergence without the condition , in this paper we propose the following implicit algorithm:
where are defined by
denotes the identity mapping of , and the parameters for all satisfy the following conditions: as and .
2. Main Result
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.
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 , such that
Then, the net defined by (1.8)-(1.9) converges strongly to the unique element in (1.1).
By using Lemma 2.2 with , that is, , we have that
So, is a contraction in . By Banach's Contraction Principle, there exists a unique element such that for all .
Next, we show that is bounded. Indeed, for a fixed point , we have that for , and hence
that implies the boundedness of . So, are the nets .
Further, for the sake of simplicity, we put and prove that
as for .
Let be an arbitrary sequence converging to zero as and . We have to prove that , where are defined by (2.5) with and . Let be a subsequence of such that
Let be a subsequence of such that
From (2.6) and Lemma 2.1, it implies that
By Lemma 2.1,
Without loss of generality, we can assume that for some . Then, we have
This together with (2.13) implies that
It means that as for .
Next, we show that as . In fact, in the case that we have . So, as . Further, since
and , we have that . Therefore, from
it follows that as . On the other hand, since
we obtain that as . Now, from
and , it follows that . Similarly, we obtain that , for and as .
Let be any sequence of converging weakly to as . Then, , for and also converges weakly to . By Lemma 2.3, we have and from (2.8), it follows that
Since , by replacing by in the last inequality, dividing by and taking in the just obtained inequality, we obtain
The uniqueness of in (1.1) guarantees that . Again, replacing in (2.8) by , we obtain the strong convergence for . This completes the proof.
Recall that a mapping is called a -strictly pseudocontractive if there exists a constant such that
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.
Let be fixed numbers. Then, with , a nonexpansive mapping, for , and hence
So, we have the following result.
Theorem 3.1 ..
Let be a real Hilbert space and a mapping such that for some constants , is -Lipschitz continuous and -strongly monotone. Let be -strictly pseudocontractive self-maps of such that . Let and let , such that
Then, the net defined by
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 ..
Let be a real Hilbert space and a mapping such that for some constants , is -Lipschitz continuous and -strongly monotone. Let be -strictly pseudocontractive self-maps of such that . Let , where , , and let , such that
Then, the net , defined by
where , , and , converges strongly to the unique element in (1.1).
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This work was supported by the Vietnamese National Foundation of Science and Technology Development.
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Buong, N., Quynh Anh, N. An Implicit Iteration Method for Variational Inequalities over the Set of Common Fixed Points for a Finite Family of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2011, 276859 (2011). https://doi.org/10.1155/2011/276859
- Hilbert Space
- Variational Inequality
- Nonexpansive Mapping
- Inequality Problem
- Real Hilbert Space