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.
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.
- Cai Y, Tang Y, Liu L: Iterative algorithms for minimum-norm fixed point of nonexpansive mapping in Hilbert space. Fixed Point Theory Appl. 2012. Article ID 49, 2012: Article ID 49Google Scholar
- Cui YL, Liu X: Notes on Browder’s and Halpern’s methods for nonexpansive mappings. Fixed Point Theory 2009, 10: 89-98.MathSciNetMATHGoogle Scholar
- Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73: 957-961. 10.1090/S0002-9904-1967-11864-0View ArticleMathSciNetMATHGoogle Scholar
- Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506-510. 10.1090/S0002-9939-1953-0054846-3View ArticleMathSciNetMATHGoogle Scholar
- Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch. Math. 1992, 58: 486-491. 10.1007/BF01190119MathSciNetView ArticleMATHGoogle Scholar
- Itoh S, Takahashi W: The common fixed point theory of single-valued mappings and multivalued mappings. Pac. J. Math. 1978, 79: 493-508. 10.2140/pjm.1978.79.493MathSciNetView ArticleMATHGoogle Scholar
- Kocourek P, Takahashi W, Yao JC: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan. J. Math. 2010, 14: 2497-2511.MathSciNetMATHGoogle Scholar
- Combettes PL, Pesquet JC: Image restoration subject to a total variation constraint. IEEE Trans. Image Process. 2004, 13: 1213-1222. 10.1109/TIP.2004.832922View ArticleGoogle Scholar
- Kim GE: Weak and strong convergence theorems of quasi-nonexpansive mappings in a Hilbert space. J. Optim. Theory Appl. 2012, 152: 727-738. 10.1007/s10957-011-9924-1MathSciNetView ArticleMATHGoogle Scholar
- Yamada I, Ogura N: Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings. Numer. Funct. Anal. Optim. 2004, 25: 619-655.MathSciNetView ArticleMATHGoogle Scholar
- Chuang CS, Lin LJ, Takahashi W: Halpern’s type iterations with perturbations in Hilbert spaces: equilibrium solutions and fixed points. J. Glob. Optim. 2013, 56: 1591-1601. 10.1007/s10898-012-9911-6MathSciNetView ArticleMATHGoogle Scholar
- Lin LJ, Chuang CS, Yu ZT: Fixed point theorems for some new nonlinear mappings in Hilbert spaces. Fixed Point Theory Appl. 2011. Article ID 51, 2011: Article ID 51Google Scholar
- Marino G, Xu HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336-346. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleMATHGoogle Scholar
- Martinet B: Regularisation dinequations variationnelles par approximations successives. Rev. Fr. Inf. Rech. Oper. Ser. 1970, 4: 154-158.MathSciNetMATHGoogle Scholar
- Rockafellar RT: Monotone operators and proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877-898. 10.1137/0314056MathSciNetView ArticleMATHGoogle Scholar
- Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 1991, 29: 403-419. 10.1137/0329022MathSciNetView ArticleMATHGoogle Scholar
- Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory 2000, 106: 226-240. 10.1006/jath.2000.3493MathSciNetView ArticleMATHGoogle Scholar
- Xu HK: A regularization method for the proximal point algorithm. J. Glob. Optim. 2006, 36: 115-125. 10.1007/s10898-006-9002-7View ArticleMathSciNetMATHGoogle Scholar
- Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J. Nonlinear Convex Anal. 2007, 8: 471-489.MathSciNetMATHGoogle Scholar
- Takahashi W, Wong NC, Yao JC: Two generalized strong convergence theorems of Halpern’s type in Hilbert spaces and applications. Taiwan. J. Math. 2012, 16: 1151-1172.MathSciNetMATHGoogle Scholar
- Lin LJ, Takahashi W: A general iterative method for hierarchical variational inequality problems in Hilbert spaces and applications. Positivity 2012, 16: 429-453. 10.1007/s11117-012-0161-0MathSciNetView ArticleGoogle Scholar
- Takahashi S, Takahashi W, Toyoda M: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 2010, 147: 27-41. 10.1007/s10957-010-9713-2MathSciNetView ArticleMATHGoogle Scholar
- Takahashi S, Takahashi W: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 2008, 69: 1025-1033. 10.1016/j.na.2008.02.042MathSciNetView ArticleMATHGoogle Scholar
- Rockafellar RT: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 1970, 33: 209-216. 10.2140/pjm.1970.33.209MathSciNetView ArticleMATHGoogle Scholar
- Chang SS, Lee HWJ, Chan CK: Generalized system for relaxed cocoercive variational inequalities in Hilbert spaces. Appl. Math. Lett. 2007, 20: 329-334. 10.1016/j.aml.2006.04.017MathSciNetView ArticleMATHGoogle Scholar
- Chang SS, Lee HWJ, Chan CK, Liu JA: A new method for solving a system of generalized nonlinear variational inequalities in Banach spaces. Appl. Math. Comput. 2011, 217: 6830-6837. 10.1016/j.amc.2011.01.021MathSciNetView ArticleMATHGoogle Scholar
- Wangkeeree R, Kamraksa U: A general iterative method for solving the variational inequality problem and fixed point problem of an infinite family of nonexpansive mappings in Hilbert spaces. Fixed Point Theory Appl. 2009. Article ID 369215, 2009: Article ID 369215Google Scholar
- Yao YH, Liou YC, Yao JC: An extragradient method for fixed point problems and variational inequality problems. J. Inequal. Appl. 2007. Article ID 38752, 2007: Article ID 38752Google Scholar
- Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings. Nonlinear Anal. 2005, 61: 341-350. 10.1016/j.na.2003.07.023MathSciNetView ArticleMATHGoogle Scholar
- Mainge PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16: 899-912. 10.1007/s11228-008-0102-zMathSciNetView ArticleMATHGoogle Scholar
- Xu HK: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 2002, 65: 109-113. 10.1017/S0004972700020116View ArticleMathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.