- Open Access
A new iteration method for variational inequalities on the set of common fixed points for a finite family of quasi-pseudocontractions in Hilbert spaces
© Zhou and Wang; licensee Springer. 2014
Received: 14 November 2013
Accepted: 8 May 2014
Published: 30 May 2014
In this paper, we propose a new iteration method based on the hybrid steepest descent method and Ishikawa-type method for seeking 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 Lipschitz continuous and quasi-pseudocontractive mappings in a real Hilbert space.
MSC: 41A65, 47H17, 47J20.
1 Introduction and preliminaries
for all .
Variational inequalities were initially investigated by Kinderlehrer and Stampacchia in , and have been widely studied by many authors ever since, due to the fact that they cover as diverse disciplines as partial differential equations, optimization, optimal control, mathematical programming, mechanics and finance (see [1–3]).
When F is a k-Lipschitz continuous and η-strongly monotone mapping, as , the sequence generated by (1.4) converges strongly to a unique solution of (1.2).
where , taking values in , and is a sequence of real numbers in , and proved that, under the following conditions:
(L3) , and
the sequence generated by (1.5) converges strongly to a unique solution of (1.2). The algorithms and convergence results of Yamada in  have been improved and extended to a finite or an infinite family of nonexpansive mappings; see, for example, Xu and Kim , Zeng , Liu and Cai , and Iemoto and Takahashi . However, all such improvements and extensions are confined to a finite or an infinite family of nonexpansive mappings.
In this paper, we propose a new iterative algorithm based on a combination of the projected gradient method for variational inequalities with the Ishikawa-type method for fixed point problems to solve (1.2) with , where is a finite family of -Lipschitz continuous and quasi-pseudocontractive mappings on Ω, where Ω is a nonempty closed and convex subset of H, while is a k-Lipschitz continuous and η-strongly monotone mapping.
where , while μ is a fixed constant satisfying .
By virtue of new analysis techniques, we prove that the sequence generated by (1.6) converges strongly to a unique solution of (1.2) with .
In order to reach our goal, we need the following conceptions and facts.
for all . When , T is said to be pseudocontractive.
for all but .
for all but , respectively.
We note that if T is κ-strictly pseudocontractive, then it is Lipschitz continuous and pseudocontractive; if T is a pseudocontraction with a fixed point, then T is a quasi-pseudocontraction; however, the converse may be not true.
that is, for any point , if and only if and .
Let be a metric projection from H on a nonempty closed convex subset E of H. Then the following conclusions hold true:
for all ;
for all and ;
- (iii)for all and () such that , the following equality holds:
Lemma 1.3 
for all .
Lemma 1.4 
Let E be a nonempty closed convex subset of a real Hilbert space H and be L-Lipschitz continuous and quasi-pseudocontractive. Then is a nonempty closed convex subset of E, and therefore is well defined for each .
Lemma 1.5 
Let E be a nonempty closed convex subset of a real Hilbert space H and be a demicontinuous pseudocontraction from E into itself. Then is a closed convex subset of E and is demiclosed at zero.
Lemma 1.6 
for all sufficiently large numbers .
Lemma 1.7 
where and satisfy the following conditions: , , and . Then as .
2 Main results
Theorem 2.1 Let Ω be a nonempty, closed and convex subset of a real Hilbert space H. Let be -Lipschitz continuous and quasi-pseudocontractive with Lipschitz constants , respectively. Let be a k-Lipschitz continuous and η-strongly monotone mapping. Assume that and are demiclosed at zero for . Let be defined by (1.6). Then converges strongly to a unique solution of (1.2), where .
Indeed, in view of Lemma 1.3, we know that is a contraction, and hence is also a contraction on Ω. Then we use the Banach contraction mapping principle to deduce (2.1).
for and all .
for and all .
for all and all .
for all , and therefore is bounded; consequently, , and are all bounded.
We next show that ().
for and all , where and are fixed positive constants.
for and all .
Now we consider two possible cases.
In this case, we have exists.
where . Now Lemma 1.7 can be used to deduce as .
By using a reasoning similar to case 1, we can obtain that , and hence by (2.17), i.e., as , which derives as ; consequently, as , since for sufficiently large . This completes the proof. □
Remark 2.1 When , in (1.6) can be dropped.
Corollary 2.1 Let Ω be a nonempty, closed and convex subset of a real Hilbert space H. Let be N -Lipschitz continuous and strongly pseudocontractive with Lipschitz constants , respectively. Let F, ℱ and be the same as in Theorem 2.1. Then converges strongly to a unique solution of (1.2), where .
Proof By virtue of Lemma 1.5, we know that are closed convex for and hence is nonempty, closed and convex. Lemma 1.5 also ensures that are demiclosed at zero for and hence the conclusion of Corollary 2.1 follows exactly from Theorem 2.1. □
Corollary 2.2 Let Ω be a nonempty, closed and convex subset of a real Hilbert space H. Let be N strict pseudocontractions, respectively. Let F, ℱ and be the same as in Theorem 2.1. Then converges strongly to a unique solution of (1.2), where .
Proof Since every strictly pseudocontractive mapping is Lipschitz continuous and pseudocontractive, we have the desired conclusion. □
Corollary 2.3 Let Ω be a nonempty, closed and convex subset of a real Hilbert space H. Let be N nonexpansive mappings, respectively. Let F, ℱ and be the same as in Theorem 2.1. Then converges strongly to a unique solution of (1.2), where .
Proof Since any nonexpansive mapping is 1-Lipschitz continuous and pseudocontractive, we have the desired conclusion by Corollary 2.2. □
Remark 2.2 When , we have the following strong convergence theorem.
Corollary 2.4 Let Ω be a nonempty, closed and convex subset of a real Hilbert space H. Let be N -Lipschitz continuous and quasi-pseudocontractive with Lipschitz constants , respectively. Let ℱ and be the same as in Theorem 2.1. Let be defined by (1.6) with . Then converges strongly to the minimum-norm fixed point of the family .
3 Numerical example
Example 3.1 
The results of the algorithm in 
The results of algorithm ( 1.6 )
Example 3.2 
Then , is 5-Lipschitz continuous and pseudocontractive and is 10-Lipschitz continuous and pseudocontractive. We find the point with the minimum-norm. To do so, set .
Values of with initial values and
This research was supported by the National Natural Science Foundation of China (11071053).
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