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Strong convergence theorems for quasi-nonexpansive mappings and maximal monotone operators in Hilbert spaces
Journal of Inequalities and Applications volume 2014, Article number: 318 (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.
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]).
Let B be a mapping from H into . The effective domain of B is denoted by , namely, . The graph of B is
A multi-valued mapping B is said to be monotone if
A monotone operator B is said to be maximal if its graph is not properly contained in the graph of any other monotone operator. For a maximal monotone operator B on H and , we define a single-valued operator , which is called the resolvent of B for r. It is well known that is firmly nonexpansive, that is,
A basic problem for maximal monotone operator B is to
The classical method for solving problem (1.1) is the proximal point algorithm which was first introduced by Martinet . Rockafellar  obtained the weak convergence of the proximal point algorithm for maximal monotone operators. Güler  constructed a proximal point iteration that converges weakly but not strongly. Some researchers have devoted their work to modifications of the proximal point algorithm in order to obtain the strong convergence theorem (see [17, 18]). For a positive constant α, a mapping is said to be α-inverse strongly monotone if
We write for the zero set of , that is, , where the mapping is inverse strongly monotone and B is maximal monotone. It is well known that for all (see ). Takahashi et al.  presented the following iterative sequence. Let , and let be a sequence generated by
Under appropriate conditions they proved that the sequence converges strongly to a point . Lin and Takahashi  introduced an iterative sequence that converges strongly to an element of , where F is another maximal monotone operator. Takahashi et al.  established an iterative scheme for finding a point of as follows. Let and let be a sequence generated by
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.
Throughout this paper, let H be a real Hilbert space with inner product and norm , and let K be a nonempty closed convex subset of H. Let ℕ be the set of positive integers. We denote the strong convergence and the weak convergence of to x by and , respectively. For any , there exists a unique point such that
is called the metric projection of H onto K. Note that is a nonexpansive mapping. For and , we have
Let f be a proper lower semicontinuous convex function of H into . The subdifferential ∂f of f is defined as
for all . Rockafellar  claimed that ∂f is a maximal monotone operator. Let be the indicator function of K, i.e.,
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 ).
Let be a nonlinear mapping. The variational inequality problem is to find such that
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 
Let be an α-inverse strongly monotone mapping. For all and , we have
In particular, if , then is a nonexpansive mapping.
Lemma 2.2 
Let be a sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence of such that
Define the sequence of integers as follows:
where such that . Then, for all , the following hold:
Lemma 2.3 
Let B be a maximal monotone operator on H. Then the following holds:
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 
Let be a sequence of nonnegative real numbers satisfying , where
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.
Theorem 3.1 Let K be a nonempty closed convex subset of a real Hilbert space H. Let and be α-inverse strongly monotone and γ-inverse strongly monotone, respectively. Suppose that B and D are maximal monotone operators on H such that the domains of B and D are contained in K and that is a quasi-nonexpansive mapping such that is demiclosed at zero. Assume that . Let and be sequences in and let be a sequence in K. Let be a sequence generated by
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.
By Lemma 2.1, for any , we have
It follows that
The sequence is bounded due to condition (c2). Hence there exists a positive number L such that . By a simple inductive process, we have
which shows that is bounded. So are and .
Thus we get
This implies that
Set , where . We divide the rest proof into two cases.
Case 1. Suppose that for all . In this case, the limit exists and then . We obtain
It follows that
Therefore we get
In a similar way, we have
Letting , we have
from which one deduces that
Using (3.1), we see that
It follows from (3.5) that
This implies that
By (3.6), a similar argument shows that
combining (3.3), (3.8) and (3.9) gives
Next we prove that
To show this inequality, we choose a subsequence of such that
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 .
We prove that . Due to (c4), there is a subsequence of such that . Without loss of generality, we assume that . Observe that
Since is nonexpansive, the demiclosedness for a nonexpansive mapping implies that , that is, .
Using a similar argument, we get . In fact, we have obtained .
By (3.12) and (2.1), we have
The inequality (3.11) is obtained.
Finally, we prove that . With the help of Lemma 2.4, we obtain
It follows from (3.11) and Lemma 2.5 that converges strongly to .
Case 2. Suppose that there exists a subsequence of such that
We define by
Lemma 2.2 shows that . Therefore we have from (3.2)
As in the proof of Case 1, we obtain
It follows that
which implies that
Thus we get
It follows from (3.15) and (3.18) that
Lemma 2.2 implies that
The proof is completed. □
The following result is a direct consequence of Theorem 3.1.
Corollary 3.2 Let K be a nonempty closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone operator of K into H and let B be a maximal monotone operator on H such that the domain of B is contained in K. Let be a quasi-nonexpansive mapping such that is demiclosed at zero. Assume that . Let and be sequences in and let be a sequence in K. Let be a sequence generated by
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.
Corollary 3.3 Let K be a nonempty closed convex subset of a real Hilbert space H. Let A be an α-inverse strongly monotone operator of K into H and let be a quasi-nonexpansive mapping such that is demiclosed at zero. Assume that . Let and be sequences in and let be a sequence in K. Let be a sequence generated by
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.
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The authors are grateful to referees and editors for their valuable comments and suggestions.
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Wu, Hc., Cheng, Cz. & Qu, Dn. Strong convergence theorems for quasi-nonexpansive mappings and maximal monotone operators in Hilbert spaces. J Inequal Appl 2014, 318 (2014). https://doi.org/10.1186/1029-242X-2014-318
- quasi-nonexpansive mapping
- maximal monotone operator
- variational inequality