Abstract
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.
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1. Introduction
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 [1] 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 [4] 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.
Theorem 1.1.
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 [5] 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.
Theorem 1.2.
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. [6] 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 [7]).
(i) 
and for any fixed 
, (ii) 
, for all 
.
Put 
; for any nonexpansive mapping 
of 
, we have the following lemma.
Lemma 2.2 (see [8]).
and for a fixed number 
, where 
.
Lemma 2.3 (Demiclosedness Principle [9]).
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.
Theorem 2.4.
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).
Proof..
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
Therefore,
that implies the boundedness of 
. So, are the nets 
.
Put
Then,
Moreover,
Thus,
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
Hence,
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.
3. Application
Recall that a mapping 
is called a 
-strictly pseudocontractive if there exists a constant 
such that
It is well known [10] 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 [11] 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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Acknowledgment
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
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DOI: https://doi.org/10.1155/2011/276859