- Open Access
A new explicit iterative algorithm for solving a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings
© Zhang and Yang; licensee Springer. 2014
- Received: 6 November 2013
- Accepted: 19 February 2014
- Published: 7 March 2014
In this paper, we introduce a new explicit iterative algorithm for finding a solution for a class of variational inequalities over the common fixed points set of a finite family of nonexpansive mappings in Hilbert spaces. Under suitable assumptions, we prove that the sequence generated by the iterative algorithm converges strongly to the unique solution of the variational inequality. Our result improves and extends the corresponding results announced by many others. At the end of the paper, we extend our result to the more broad family of λ-strictly pseudo-contractive mappings.
- nonexpansive mapping
- strong convergence
- variational inequalities
- common fixed points
where C is a nonempty closed convex subset of H, u is a given point in H and A is a strongly positive bounded linear operator on H.
where u is some point of H and is a sequence in . He proved that the sequence converges strongly to the unique solution of the minimization problem (1.1) with .
where h is a potential function for γf (i.e., for ).
In fact, there are many nonexpansive mappings which do not satisfy (1.8).
From [, Lemma 3.1], we know that .
where and h is a potential function for γf (i.e., ).
Under certain appropriate conditions, without (1.8), they proved that defined by (1.12) converges strongly to the unique solution of (1.10) which is also the optimality condition for (1.11).
They proved that the sequence defined by (1.15) converges strongly to the unique solution of the variational inequality (1.14) in a faster rate of convergence.
Motivated and inspired by the results of Zhou et al., in this paper, we consider a new iterative algorithm to solve the class of variational inequalities (1.6). The iterative algorithm improves and extends the results of Yao et al., and the corresponding results announced by many others. At the end of this paper, we extend our iterative algorithm to the more broad family of λ-strictly pseudo-contractive mappings.
Throughout this paper, we write and to indicate that converges weakly to x and converges strongly to x, respectively.
for all .
In order to prove our results, we collect some necessary conceptions and lemmas in this section.
where is the identity mapping and is nonexpansive. More precisely, when (2.1) holds, we say that T is α-averaged.
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then both and are α-averaged, where .
- (ii)If the mappings are averaged and have a common fixed point, then
In particular, if , we have .
Lemma 2.2 ()
for every .
Lemma 2.3 ()
Assume V is a contraction on a Hilbert space H with coefficient , and is an L-Lipschitzian continuous and η-strongly monotone operator with , . Then, for , is strongly monotone with coefficient .
Lemma 2.4 ()
Let H be a Hilbert space, C a closed convex subset of H, and a nonexpansive mapping with . If is a sequence in C weakly converging to and converges strongly to , then . In particular, if , then .
Lemma 2.5 ()
Lemma 2.6 ()
Lemma 2.7 ()
Assume S is a λ-strictly pseudo-contractive mapping on a Hilbert space H. Define a mapping T by for all and . Then T is a nonexpansive mapping such that .
Now we state and prove our main results in this paper.
Equivalently, we have .
Proof Since our methods easily deduce the general case, we prove Theorem 3.1 for .
where D is an arbitrary bounded subset of H.
Consequently, according to the conditions (i) and (ii), (3.14), and Lemma 2.6, we conclude that as . This completes the proof. □
where . By virtue of Lemma 2.7, we know that is a family of nonexpansive mappings. Thus we extend Theorem 3.1 to the family of -strictly pseudo-contractions.
This research is supported by the Fundamental Science Research Funds for the Central Universities (Program No. 3122013k004).
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