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A new explicit iteration method for variational inequalities on the set of common fixed points for a finite family of nonexpansive mappings
Journal of Inequalities and Applications volume 2013, Article number: 419 (2013)
In this paper, we introduce a new explicit iteration method based on the steepest descent method and Krasnoselskii-Mann type method for finding a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common fixed points for a finite family of nonexpansive mappings in a real Hilbert space.
MSC:41A65, 47H17, 47H20.
1 Introduction and preliminaries
Let C be a nonempty closed and convex subset of a real Hilbert space H with the inner product and the norm , and let be a nonlinear mapping. The variational inequality problem is to find a point such that
Variational inequalities were initially studied by Kinderlehrer and Stampacchia in , and since then have been widely investigated. They cover partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance (see [1–3]).
It is well known that if F is an L-Lipschitz continuous and η-strongly monotone, i.e., F satisfies the following conditions:
where L and η 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 C and μ is an arbitrarily positive constant.
The fixed point formulation (1.2) involves the metric projection . To overcome the complexity caused by , Yamada  introduced a hybrid steepest descent method for solving (1.1). His idea is stated as follows. Assume that , the set of common fixed points of a finite family of nonexpansive mappings on H with an integer .
Recall that is nonexpansive if
denotes the fixed point set of T. Yamada proposed the following algorithm in 
where , for integer , with the mod-function taking values in the set , and , and proved that the sequence in (1.3) converges strongly to under the following conditions:
Further, Zeng and Yao  proved the same result with (L3) replaced by
Theorem 1.1 
Let H be a real Hilbert space, and let be an L-Lipschitz continuous and η-strongly monotone mapping for some constants . Let be N nonexpansive self-maps of H such that , , and let conditions (L1), (L2), (L4) be satisfied. Assume in addition that
Then the sequence defined by (1.3) converges strongly to the unique element in (1.1).
It is not difficult to show that (L3) implies (L4) if exists. However, in general, conditions (L3) and (L4) are not comparable, i.e., neither one of them implies the other (see  for details).
Recently, Zeng et al.  proposed the following iterative scheme:
and proved the following result.
Theorem 1.2 
Let H be a real Hilbert space, and let be an L-Lipschitz continuous and η-strongly monotone mapping for some constants . Let be N nonexpansive self-maps of H such that
and let . Assume that the following conditions hold:
, where ;
for some ;
Assume in addition that (1.4) holds. If
then the sequence defined by (1.5) converges strongly to the unique element in (1.1).
They also showed that conditions (L1), (L2) and (L4) are sufficient for to be bounded and
So, (1.6) is satisfied. They did not give another sufficient condition different from (L1), (L2) and (L4).
Let , where A is a self-adjoint bounded linear mapping such that A is strongly positive, i.e.,
and u is some fixed element in H. Xu  introduced the following iteration process:
where I is the identity mapping of H, and proved the following result.
Theorem 1.3 
Let conditions (L1), (L2) and (L3) or (L4) be satisfied. Assume in addition that (1.4) holds. Then the sequence generated by algorithm (1.7) converges strongly to the unique solution of (1.1) with .
Very recently, Liu and Cui  showed that the condition
is sufficient for (1.4) if .
In this paper, we introduce a new algorithm based on a combination of the steepest descent method for variational inequalities with the Krasnoselskii-Mann method for fixed point problems to solve (1.1) with , where is a nonexpansive mapping on H for each i.
Given a starting point , the iteration is defined by
and the sequences of parameters and satisfy the following conditions:
In Section 2, we prove the strong convergence theorem for (1.9)-(1.10) without conditions (L3), (L4) and (1.8). An application to the case that is a -strictly pseudocontractive mapping is given in Section 3.
2 Main results
We need the following lemmas for the proof of our main result.
Lemma 2.1 
, , and for any fixed .
From , we have the following lemma.
Lemma 2.2 
, and for a fixed number , , where and for .
Assume that T is a nonexpansive self-map of a closed convex subset K of a real Hilbert space H. If T has a fixed point, then is demiclosed; that is, whenever is a sequence in K weakly converging to some and the sequence strongly converges to some y, it follows that .
Lemma 2.4 
Let and be bounded sequences in a Banach space E such that
for , where is in such that
Lemma 2.5 
Let be a sequence of nonnegative real numbers satisfying the condition
where and are sequences of real numbers such that
Now, we are in a position to prove the following main result.
Theorem 2.6 Let H be a real Hilbert space, and let be an L-Lipschitz continuous and η-strongly monotone mapping for some constants . Let be N nonexpansive self-maps of H such that
Then the sequence defined by (1.9)-(1.10) converges strongly to the unique element in (1.1).
Proof First, we prove that is bounded. By Lemma 2.2, we have, for any , from (1.9) that
Put . Then . So, if , then . This conclusion has a place for with . Indeed,
Therefore, the sequence is bounded. So, the sequences , , and () are also bounded. Without loss of generality, we assume that they are bounded by a positive constant .
Let . Then we have from (1.9) that
On the other hand,
So, we obtain that
Since and , , we have
By Lemma 2.4, as , i.e., .
Now, we prove that for . First, we prove . Let be a subsequence of such that
and let be a subsequence of such that
Next, by Lemma 2.1 we have
On the other hand, by Lemma 2.1 we get
Without loss of generality, assume that , for and some . Then we obtain that
which with and (2.1) implies that as . So, as .
Similarly, we obtain that , …, as .
Further, we prove that as for . First, note that as because and , and because , for . Now, from
and , it follows that for .
Further, we have
Indeed, let be a subsequence of that converges weakly to such that
Then . So, by Lemma 2.3, . Therefore, from (1.1) it implies (2.2).
Finally, we estimate the value
On the other hand, since
Using Lemma 2.5 with
and (2.2), we have that . This completes the proof. □
Recall that a mapping is called γ-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 each . So, we have the following result.
Theorem 3.1 Let H be a real Hilbert space, and let be an L-Lipschitzian and η-strongly monotone mapping for some constants . Let be N -strictly pseudocontractive self-maps of H such that
Let , . Assume that satisfy (1.10). Then the sequence defined by (1.9) with replaced by of (3.1) converges strongly to the unique element of (1.1).
4 Numerical example
Consider the following optimization problem: find an element
where , , Euclid space, and , defined by
Clearly, the above problem possesses a unique solution and F, the Fréchet derivative of φ, is 1-Lipschitz continuous and -strongly monotone. Starting with the point , and , we obtained the result in Table 1.
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This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042138).
The authors declare that they have no competing interests.
The main idea of this paper was proposed by JKK. JKK and NB prepared the manuscript initially and performed all the steps of proof in this research. All authors read and approved the final manuscript.
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Kim, J.K., Buong, N. A new explicit iteration method for variational inequalities on the set of common fixed points for a finite family of nonexpansive mappings. J Inequal Appl 2013, 419 (2013). https://doi.org/10.1186/1029-242X-2013-419
- common fixed points
- hybrid steepest descent method
- nonexpansive mappings
- monotone mappings