Strong convergence theorems for quasi-nonexpansive mappings and maximal monotone operators in Hilbert spaces
© Wu et al.; licensee Springer. 2014
Received: 11 February 2014
Accepted: 30 July 2014
Published: 21 August 2014
We present the strong convergence theorem for the iterative scheme for finding a common element of the fixed-point set of a quasi-nonexpansive mapping and the zero set of the sums of maximal monotone operators in Hilbert spaces. Our results extend and improve the recent results of Takahashi et al. (J. Optim. Theory Appl. 147:27-41, 2010) and Takahashi and Takahashi (Nonlinear Anal. 69:1025-1033, 2008).
MSC:47H05, 47H09, 47J25.
Keywordsquasi-nonexpansive mapping maximal monotone operator variational inequality
Let H be a real Hilbert space and let K be a nonempty closed convex subset of H. Let be a mapping. We denote by the fixed-point set of T, that is, . A mapping is nonexpansive if for all . Approximation methods for fixed points of nonexpansive mappings have attracted considerable attention (see [1–5]). A mapping is quasi-nonexpansive if and for all and . It is well known that the fixed-point set of a quasi-nonexpansive mapping is closed and convex (see [6, 7]). There are some quasi-nonexpansive mappings which are not nonexpansive (see [8–10]). For example, the level set of a continuous convex function is characterized as the fixed-point set of a nonlinear mapping called the subgradient projection, which is not nonexpansive but quasi-nonexpansive. Quasi-nonexpansive mappings have been discussed in the recent literature (see [9–11]).
We say that a mapping is demiclosed at zero if for any sequence which converges weakly to x, the strong convergence of the sequence to zero implies . It is well known that is demiclosed whenever T is nonexpansive. In fact, this property is satisfied for more general mappings (see [12, 13]).
where is a nonexpansive mapping.
Motivated by the above results, especially by Chuang et al.  and Takahashi et al. , we obtain the strong convergence theorem for the iterative scheme for finding a common element of the fixed-point set of a quasi-nonexpansive mapping and the zero set of the sums of maximal monotone operators in Hilbert spaces. Our results extend and improve the recent results of  and .
The rest of this paper is organized as follows. Section 2 contains some important facts and tools. In Section 3, we introduce a new iterative scheme for finding a common element of the fixed-point set of a quasi-nonexpansive mapping and the zero set of the sums of maximal monotone operators, and we prove strong convergence theorem in Hilbert spaces.
The subdifferential of is a maximal monotone operator since is a proper lower semicontinuous convex function on H. The resolvent of for r is (see ).
The solution set of (2.3) is denoted by . Some methods have been proposed to study the variational inequality problem (see [25–28] and the references therein). It is easy to see that , where A is an inverse strongly monotone mapping of K into H (for more details, see ).
We collect some useful lemmas.
Lemma 2.1 
In particular, if , then is a nonexpansive mapping.
Lemma 2.2 
Lemma 2.3 
for all and .
The following lemma is an immediate consequence of the inner product on H.
Lemma 2.4 For all , the inequality holds.
Lemma 2.5 
3 Strong convergence theorems
In this section, a new iterative scheme for finding a common element of the fixed-point set of a quasi-nonexpansive mapping and the zero set of the sums of maximal monotone operators is presented.
Suppose the following conditions are satisfied:
(c1) and ;
Then the sequence converges strongly to .
Proof Observe that the set Ω is closed and convex since , and are closed and convex.
which shows that is bounded. So are and .
Set , where . We divide the rest proof into two cases.
In view of the boundedness of , without loss of generality, we assume that . Now we show that . According to the fact that is contained in K and K is a closed convex set, one has .
Note that the expressions (3.8) and (3.9) yield and . By the fact that is demiclosed at zero, the expression (3.10) implies .
Since is nonexpansive, the demiclosedness for a nonexpansive mapping implies that , that is, .
Using a similar argument, we get . In fact, we have obtained .
The inequality (3.11) is obtained.
It follows from (3.11) and Lemma 2.5 that converges strongly to .
The proof is completed. □
The following result is a direct consequence of Theorem 3.1.
If conditions (c1)-(c4) are satisfied, then the sequence converges strongly to the element .
Proof Letting and in Theorem 3.1, the desired result follows. □
Let us consider the variational inequality problem. Recall that the subdifferential of is a maximal monotone operator and , where A is an inverse strongly monotone mapping. We obtain the following result.
If conditions (c1)-(c4) are satisfied, then the sequence converges strongly to the element .
Proof Corollary 3.2 easily yields the desired result. □
The nonexpansive mapping is extended to the quasi-nonexpansive mapping.
The constant vector u is replaced by the variables with .
The condition is removed.
The authors are grateful to referees and editors for their valuable comments and suggestions.
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