Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings
© Qing and Shang; licensee Springer 2013
Received: 31 December 2012
Accepted: 21 February 2013
Published: 25 March 2013
In this paper, we investigate the problem of finding some common element in the set of common fixed points of an infinite family of nonexpansive mappings and in the set of solutions of variational inequalities based on an extragradient-like iterative algorithm. Strong convergence of the purposed iterative algorithm is obtained.
MSC:47H05, 47H09, 47J25, 90C33.
Iterative algorithms have been playing an important role in the approximation solvability, especially of nonlinear variational inequalities as well as of nonlinear equations in several fields such as mechanics, traffic, economics, information, medicine, and many others. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonlinear mappings; see, for example, [1–11]. The Mann iterative algorithm is an efficient method to study the class of nonexpansive mappings. Indeed, Picard cannot converge even that the fixed point set of nonexpansive mappings is nonempty.
It is known that Mann iterative algorithm only has weak convergence for nonexpansive mappings in infinite-dimensional Hilbert spaces; see  for more details and the references therein. In many disciplines, including economics , image recovery , quantum physics [15–20], and control theory , problems arise in infinite dimension spaces. To improve the weak convergence of the Mann iterative algorithm, many authors considered using contractions to approximate nonexpansive mappings; for more details, see  and  and the references therein.
In this paper, we focus on the problem of finding some common element in the set of common fixed points of an infinite family of nonexpansive mappings and in the set of solutions of variational inequalities based on an extragradient-like iterative algorithm. Some deduced sub-results and applications are obtained.
Throughout this paper, we assume that H is a real Hilbert space whose inner product and norm are denoted by and , respectively. Let K be a nonempty, closed, and convex subset of H. Let be the metric projection from H onto K.
For such a case, B is also said to be μ-inverse-strongly monotone.
In this paper, we use to denote the fixed point set of the mapping T.
For such a case, f is also said to be an α-contraction.
Then S is maximal monotone and iff ; see  for more details.
where is a monotone mapping. It is known that is a solution to (2.1) iff u is a fixed point of the mapping , where is a constant and I stands for the identity mapping. In this paper, we use to denote the solution set of the variational inequality (2.1).
strongly converges to the unique solution of the minimization problem (2.2) provided that the sequence satisfies certain restriction.
where h is a potential function for γf, that is, for .
where is a nonexpansive mapping, is a μ-inverse-strongly monotone mapping, is a sequence in , and is a sequence in . They showed that the sequence generated in (2.3) weakly converges to some point .
where is a nonexpansive mapping, is a μ-inverse-strongly monotone mapping, is a sequence in , and is a sequence in . They proved that the sequence strongly converges to some point .
where are real numbers such that , is an infinite family of mappings of K into itself. Nonexpansivity of each ensures the nonexpansivity of .
Regarding , we have the following lemmas which are important to prove our main results.
Lemma 2.1 
Let K be a nonempty, closed, and convex subset of a strictly convex Banach space E. Let be nonexpansive mappings of K into itself such that is nonempty, and let be real numbers such that for any . Then, for every and , the limit exists.
Such a mapping W is called W-mapping generated by and .
Throughout this paper, we will assume that for each .
Lemma 2.2 
Let K be a nonempty, closed, and convex subset of a strictly convex Banach space E. Let be nonexpansive mappings of K into itself such that is nonempty, and let be real numbers such that for each . Then .
In this paper, motivated by the above results, we investigate the problem of approximating a common element in the solution set of variational inequalities and in the common fixed point set of a family of nonexpansive mappings based on an extragradient-like iterative algorithm. Strong convergence theorems of common elements are established in the framework of Hilbert spaces.
In order to prove our main results, we also need the following lemmas.
Lemma 2.4 
Assume A is a strongly positive linear bounded self-adjoint operator on a Hilbert space H with the coefficient and . Then .
Lemma 2.5 
Equivalently, we have .
Lemma 2.6 
Lemma 2.7 
Let K be a nonempty closed convex subset of a Hilbert space H, be a family of infinitely nonexpansive mappings with , be a real sequence such that for each . If C is any bounded subset of K, then .
3 Main results
, , and ;
, where .
Then the sequence strongly converges to , where .
This shows that is nonexpansive, so is . Noticing the condition (a), we may assume, with no loss of generality, that for each . It follows from Lemma 2.4 that .
Apply Lemma 2.6 to (3.23) to conclude that as . This completes the proof. □
If , the zero mapping, then Theorem 3.1 is reduced to the following.
, , and ;
, where .
Then the sequence strongly converges to , where .
Remark 3.3 Corollary 3.2 includes the corresponding results in Iiduka and Takahashi  as a special case.
As an application of our main results, we consider another class of important nonlinear operators: strict pseudocontractions.
It is easy to see that the class of κ-strict pseudocontractions strictly includes the class of nonexpansive mappings as a special case.
Putting , where is a κ-strict pseudocontraction, we know that B is -inverse-strongly monotone; see  and the references therein.
, , and ;
, where .
Then the sequence strongly converges to , where .
Proof Put and . Then is -inverse-strongly monotone and is -inverse-strongly monotone, respectively. We have , , and . The desired conclusion can be immediately obtained from Theorem 3.1. □
This research was supported by the Natural Science Foundation of Hebei Province (A2010001943), the Science Foundation of Shijiazhuang Science and Technology Bureau (121130971) and the Science Foundation of Beijing Jiaotong University (2011YJS075).
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