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Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings
Fixed Point Theory and Applications volume 2013, Article number: 67 (2013)
Abstract
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.
1 Introduction
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 [12] for more details and the references therein. In many disciplines, including economics [13], image recovery [14], quantum physics [15–20], and control theory [21], 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 [22] and [23] 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.
2 Preliminaries
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.
Recall that a mapping is said to be inverse-strongly monotone iff there exists a positive real number μ such that
For such a case, B is also said to be μ-inverse-strongly monotone.
Recall that a mapping is said to be nonexpansive iff
In this paper, we use to denote the fixed point set of the mapping T.
Recall that a mapping is said to be a contraction iff there exists a coefficient such that
For such a case, f is also said to be an α-contraction.
Recall that a linear bounded operator is strongly positive iff there exists a constant such that
Recall that a set-valued mapping is said to be monotone iff and imply
A monotone mapping is maximal iff the graph of of S is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping S is maximal iff for , for every implies . Let be a monotone mapping and be the normal cone to K at , i.e., , and define
Then S is maximal monotone and iff ; see [24] for more details.
Recall that the classical variational inequality is to find a such that
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).
Iterative algorithms for nonexpansive mappings have recently been applied to solve convex minimization problems. A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping T on a real Hilbert space H,
where A is a linear bounded self-adjoint operator on H and u is a given point in H. In [25], it is proved that the sequence defined by the iterative algorithm
strongly converges to the unique solution of the minimization problem (2.2) provided that the sequence satisfies certain restriction.
Recently, Marino and Xu [26] reconsidered the problem by viscosity approximation method. They investigated the following iterative algorithm:
where A is a linear bounded self-adjoint operator on H, is a nonexpansive mapping, and is a contraction. They proved that the sequence generated in the above iterative process converges strongly to the unique solution of the following variational inequality:
which is the optimality condition for the minimization problem
where h is a potential function for γf, that is, for .
Recently, the problem of finding a common element in the fixed point set of a nonexpansive mapping and in the solution set of a variational inequality has been considered by many authors; see, for example, [27–40] and the references therein. In 2003, Takahashi and Toyoda [35] considered the following iterative algorithm:
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 .
Iiduka and Takahashi [36] reconsidered the common element problem via the following iterative algorithm:
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 .
In this paper, we will consider an infinite family of nonexpansive mappings. More precisely, we consider the mapping defined by
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 [41]
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.
Using Lemma 2.1, one can define the mapping W as follows:
Such a mapping W is called W-mapping generated by and .
Throughout this paper, we will assume that for each .
Lemma 2.2 [41]
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 2.3 In a real Hilbert space H, the following inequality holds:
Lemma 2.4 [26]
Assume A is a strongly positive linear bounded self-adjoint operator on a Hilbert space H with the coefficient and . Then .
Lemma 2.5 [26]
Let H be a Hilbert space. Let A be a strongly positive linear bounded self-adjoint operator with the coefficient . Assume that . Let T be a nonexpansive mapping with a fixed point of the contraction . Then converges strongly as to a fixed point of T, which solves the variational inequality
Equivalently, we have .
Lemma 2.6 [42]
Assume that is a sequence of nonnegative real numbers such that
where is a sequence in and is a sequence such that
-
(a)
;
-
(b)
or .
Then .
Lemma 2.7 [39]
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
Theorem 3.1 Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Let be -inverse-strongly monotone mappings for each , and be an α-contraction. Let be a strongly positive linear bounded self-adjoint operator with the coefficient . Let be a sequence generated in the following extragradient-like iterative algorithm:
where is the metric projection from H onto K, is a mapping defined by (2.5), is a real number sequence in , and , are two positive real number sequences. Assume that , and the following restrictions are satisfied:
-
(a)
, , and ;
-
(b)
, ;
-
(c)
, where .
Then the sequence strongly converges to , where .
Proof First, we show that and are nonexpansive. Indeed, we see from the restriction (c) that
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 .
Next, we show that the sequence is bounded. Letting , we see that
It follows that
By simple induction, we have
which yields that the sequence is bounded, so is . Notice that
Putting , we have
Substituting (3.2) into (3.3), we arrive at
where is an appropriate constant such that
Notice that
Since and are nonexpansive, we have from (2.6) that
where is an appropriate constant such that for each . Substituting (3.4) and (3.6) into (3.5), we arrive at
where is an appropriate constant such that
From the restrictions (a) and (b), we obtain from Lemma 2.6 that
Notice that
It follows from the restriction (a) that
Notice that
In a similar way, we find that
On the other hand, we have
Substituting (3.11) into (3.12) gives
It follows from the restriction (c) that
In view of the restriction (a), we obtain from (3.8) that
From (3.12), we also have
Combining (3.10) with (3.14), we arrive at
which implies from the restriction (c) that
In view of the restriction (a), we obtain from (3.8) that
On the other hand, we see from the firm expansivity of that
which yields
In the same way, we can obtain that
Substituting (3.16) into (3.14) yields
It follows that
In view of (3.7) and (3.15), we see from the restriction (a) that
Similarly, we can obtain that
Observe that
It follows from (3.7), (3.9), (3.18) and (3.19) that
From Lemma 2.7, we have as . This in turn implies that
Next, we show that
To show it, we choose a subsequence of such that
As is bounded, we have that a subsequence of converges weakly to p. We may assume, without loss of generality, that . From (3.18) and (3.19), we also have and , respectively. Notice that . Indeed, let us first show that . Put
Then S is maximal monotone. Let . Since and , we have
On the other hand, we have from that
and hence
It follows that
which implies that . We have and hence . In a similar way, we can show . Next, let us show . Since Hilbert spaces satisfy Opial’s condition, we see from (3.21) that
which derives a contradiction. Thus, we have . From Lemma 2.5, we see that there exists a unique such that . It follows that
That is, (3.22) holds. It follows from Lemma 2.3 that
It follows that
where is an appropriate constant such that . Put and . That is,
In view of the restrictions (a) and (b), we see from (3.22) 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.
Corollary 3.2 Let K be a nonempty, closed, and convex subset of a real Hilbert space H. Let be -inverse-strongly monotone mappings and be an α-contraction. Let be a strongly positive linear bounded self-adjoint operator with the coefficient . Assume that . Let be a sequence generated in the following iterative algorithm:
where is the metric projection from H onto K, is a mapping defined by (2.5), is a real number sequence in , and is a positive real number sequence. Assume that and the following restrictions are satisfied:
-
(a)
, , and ;
-
(b)
;
-
(c)
, where .
Then the sequence strongly converges to , where .
Remark 3.3 Corollary 3.2 includes the corresponding results in Iiduka and Takahashi [36] as a special case.
As an application of our main results, we consider another class of important nonlinear operators: strict pseudocontractions.
Recall that a mapping is said to be a κ-strict pseudocontraction if there exists a constant such that
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 [43] and the references therein.
Corollary 3.4 Let H be a real Hilbert space and K be a nonempty closed convex subset of H. Let be -inverse-strongly monotone mappings for each and be an α-contraction. Let be a strongly positive linear bounded self-adjoint operator with the coefficient . Assume that . Let be a sequence generated in the following iterative process:
where is the metric projection from H onto K, is a mapping defined by (2.5), is a real number sequence in , and , are two positive real number sequences. Assume that and the following restrictions are satisfied:
-
(a)
, , and ;
-
(b)
, ;
-
(c)
, 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. □
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Acknowledgements
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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Qing, Y., Shang, M. Convergence of an extragradient-like iterative algorithm for monotone mappings and nonexpansive mappings. Fixed Point Theory Appl 2013, 67 (2013). https://doi.org/10.1186/1687-1812-2013-67
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DOI: https://doi.org/10.1186/1687-1812-2013-67