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Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces
Fixed Point Theory and Applications volume 2010, Article number: 907275 (2010)
Abstract
Using strongly accretive and strictly pseudocontractive mapping, we introduce a general iterative method for finding a common fixed point of a semigroup of nonexpansive mappings in a Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded realvalued functions of the semigroup. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality.
1. Introduction
Let be a real Hilbert space. A mapping of into itself is called nonexpansive if , for all . By , we denote the set of fixed points of (i.e., ).
Mann [1] introduced an iteration procedure for approximation of fixed points of a nonexpansive mapping on a Hilbert space as follows. Let and
where is a sequence in . See also [2].
On the other hand, Moudafi [3] introduced the viscosity approximation method for fixed point of nonexpansive mappings (see [4] for further developments in both Hilbert and Banach spaces). Let be a contraction on a Hilbert space (i.e., for all and ). Starting with an arbitrary initial , define a sequence recursively by
where is sequence in . It is proved in [3, 4] that, under appropriate condition imposed on , the sequence generated by (1.2) converges strongly to the unique solution in of the variational inequality:
Assume that is strongly positive, that is, there is a constant with the property
In [4] (see also [5]), it is proved that the sequence defined by the iterative method below, with the initial guess chosen arbitrarily,
converges strongly to the unique solution of the minimization problem
provided that the sequence satisfies certain conditions. Marino and Xu [6] combined the iterative (1.5) with the viscosity approximation method (1.2) and considered the following general iterative methods:
where . They proved that if is a sequence in satisfying the following conditions:
either or ,
then, the sequence generated by (1.7) converges strongly, as , to the unique solution of the variational inequality:
which is the optimality condition for minimization problem
where is a potential function for (i.e., , for all ).
Let be the topological dual of a Banach space . The value of at will be denoted by or . With each , we associate the set
Using the HahnBanach theorem, it is immediately clear that for each . The multivalued mapping from into is said to be the (normalized) duality mapping. A Banach space is said to be smooth if the duality mapping is single valued. As it is well known, the duality mapping is the identity when is a Hilbert space; see [7].
Let and be two positive real numbers such that . Recall that a mapping with domain and range in is called strongly accretive if, for each , there exists such that
Recall also that a mapping is called strictly pseudocontractive if, for each , there exists such that
It is easy to see that (1.12) can be rewritten as
see [8].
In this paper, motivated and inspired by Atsushiba and Takahashi [9], Lau et al. [10], Marino and Xu [6] and Xu [4, 11], we introduce the iterative below, with the initial guess chosen arbitrarily,
where is strongly accretive and strictly pseudocontractive with , is a contraction on a Hilbert space with coefficient , is a positive real number such that , and is a nonexpansive semigroup on such that the set of common fixed point of is nonempty, is a subspace of such that and the mapping is an element of for each , and is a sequence of means on . Our purpose in this paper is to introduce this general iterative algorithm for approximating a common fixed points of semigroups of nonexpansive mappings which solves some variational inequality. We will prove that if is left regular and is a sequence in satisfying the conditions and , then converges strongly to , which solves the variational inequality:
Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of nonexpansive mappings are also presented. It is worth mentioning that we obtain our result without assuming condition .
2. Preliminaries
Let be a semigroup and let be the space of all bounded realvalued functions defined on with supremum norm. For and , we define elements and in by
Let be a subspace of containing , and let be its dual. An element in is said to be a mean on if . We often write instead of for and . Let be left invariant (resp., right invariant), that is, (resp., ) for each . A mean on is said to be left invariant (right invariant) if (resp. ) for each and . is said to be left (resp., right) amenable if has a left (resp., right) invariant mean. is amenable if is both left and right amenable. As it is well known, is amenable when is a commutative semigroup; see [12]. A net of means on is said to be left regular if
for each , where is the adjoint operator of .
Let be a nonempty closed and convex subset of a reflexive Banach space . A family of mapping from into itself is said to be a nonexpansive semigroup on if is nonexpansive and for each . We denote by the set of common fixed points of , that is,
The open ball of radius centered at is denoted by . For subset of , by , we denote the closed convex hull of . Weak convergence is denoted by , and strong convergence is denoted by .
Let f be a function of semigroup into a reflexive Banach space such that the weak closure of is weakly compact, and let be a subspace of containing all functions with . Then, for any , there exists a unique element in such that
for all . Moreover, if is a mean on then
One can write by
Lemma 2.2 (see [13]).
Let be a closed convex subset of a Hilbert space , a semigroup from into such that , the mapping an element of for each and , and a mean on . If one writes instead of , then the following holds.
(i) is nonexpansive mapping from into .
(ii) for each .
(iii) for each .
(iv)If is left invariant, then is a nonexpansive retraction from onto .
Let be a nonempty subset of a normed space , and let . An element is said to be the best approximation to if
where . The number is called the distance from to or the error in approximating by . The (possibly empty) set of all best approximation from to is denoted by
This defines a mapping from into and is called metric (the nearest point) projection onto .
Lemma 2.3 (see [7]).
Let be a nonempty convex subset of a smooth Banach space and let and . Then, the following is equivalent.
(i) is the best approximation to .
(ii) is a solution of the variational inequality
Let be a nonempty subset of a Banach space and a mapping. Then is said to be demiclosed at if, for any sequence in , the following implication holds:
Lemma 2.4 (see [14]).
Let be a nonempty closed convex subset of a Hilbert space and suppose that is nonexpansive. Then, the mapping is demiclosed at zero.
The following lemma is well known.
Lemma 2.5.
Let be a real Hilbert space. Then, for all
(i)
(ii)
Lemma 2.6 (see [11]).
Let be a sequence of nonnegative real numbers such that
where and are sequences of real numbers satisfying the following conditions:
(i)
(ii)either or
Then,
The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.
Lemma 2.7.
Let be a real smooth Banach space and a mapping.
(i)If is strongly accretive and strictly pseudocontractive with , then, is contractive with constant .
(ii)If is strongly accretive and strictly pseudocontractive with , then, for any fixed number , is contractive with constant .
Proof.

(i)
From (1.11) and (1.13), we obtain
(2.11)Because , we have
(2.12)and, therefore, is contractive with constant.

(ii)
Because is contractive with constant, for each fixed number , we have
(2.13)
This shows that is contractive with constant .
Throughout this paper, will denote a strongly accretive and strictly pseudocontractive mapping with , and is a contraction with coefficient on a Hilbert space . We will also always use to mean a number in .
3. Strong Convergence Theorem
The following is our main result.
Theorem 3.1.
Let be a nonexpansive semigroup on a real Hilbert space such that . Let be a left invariant subspace of such that , and the function is an element of for each . Let be a left regular sequence of means on , and let be a sequence in such that and . Let and be generated by the iteration algorithm (1.14). Then, converges strongly, as , to , which is a unique solution of the variational inequality (1.15). Equivalently, one has
Proof.
First, we claim that is bounded. Let ; by Lemmas 2.2 and 2.7 we have
By induction,
Therefore, is bounded and so is .
Set . We remark that is invariant bounded closed convex set and . Now we claim that
Let . By [15, Theorem ], there exists such that
Also by [15, Corollary ], there exists a natural number such that
for all and . Let . Since is strongly left regular, there exists such that for and . Then we have
By Lemma 2.2 we have
It follows from (3.5), (3.6), (3.7), and (3.8) that
for all and . Therefore,
Since is arbitrary, we get (3.4). In this stage, we will show that
Let and . Then, there exists , which satisfies (3.5). Take
From and (3.4) there exists such that and , for all . By Lemma 2.7, we have
for all . Therefore, we have
for all . This shows that
Since is arbitrary, we get (3.11).
Let . Then is a contraction of into itself. In fact, we see that
and hence is a contraction due to
Therefore, by Banach contraction principal, has a unique fixed point . Then using Lemma 2.3, is the unique solution of the variational inequality
We show that
Indeed, we can choose a subsequence of such that
Because is bounded, we may assume that . In terms of Lemma 2.4 and (3.11), we conclude that . Therefore,
Finally, we prove that as . By Lemmas 2.5 and 2.7 we have
On the other hand
Since and are bounded, we can take a constant such that
So from the above, we reach the following:
Substituting (3.24) in (3.21), we obtain
It follows that
where
Since is bounded and , by (3.18), we get
Consequently, applying Lemma 2.6, to (3.26), we conclude that .
Corollary 3.2.
Let , , , and be as in Theorem 3.1. Suppose that a strongly positive bounded linear operator on with coefficient and . Let be defined by the iterative algorithm
Then, converges strongly, as , to , which is a unique solution of the variational inequality
Proof.
Because is strongly positive bounded linear operator on with coefficient , we have
Therefore, is strongly accretive. On the other hand,
Since is strongly positive if and only if is strongly positive, we may assume, with no loss of generality, that , so that
This shows that is strictly pseudocontractive. Now apply Theorem 3.1 to conclude the result.
Corollary 3.3.
Let , , and be as in Theorem 3.1. Suppose and define a sequence by the iterative algorithm
Then, converges strongly, as , to a , which is a unique solution of the variational inequality
Proof.
It is sufficient to take and in Theorem 3.1.
4. Some Application
Corollary 4.1.
Let and be nonexpansive mappings on a Hilbert space with such that . Let be a sequence in satisfying conditions and . Let , and define a sequence by the iterative algorithm:
Then, converges strongly, as , to which solves the variational inequality:
Proof.
Let for each . Then is a semigroup of nonexpansive mappings on . Now, for each and , we define Then, is regular sequence of means [16]. Next, for each and , we have
Therefore, applying Theorem 3.1, the result follows.
Corollary 4.2.
Let be a strongly continuous semigroup of nonexpansive mappings on a Hilbert space such that . Let be a sequence in satisfying conditions and . Let and . Let be a sequence defined by the iterative algorithm:
where is an increasing sequence in such that and . Then, converges strongly, as , to , which solves the variational inequality
Proof.
For , we define for each , where denotes the space of all realvalued bounded continuous functions on with supremum norm. Then, is regular sequence of means [16]. Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.
Corollary 4.3.
Let be a strongly continuous semigroup of nonexpansive mappings on a Hilbert space such that . Let be a sequence in satisfying conditions and . Let and . Let be a sequence defined by the iterative algorithm
where is an decreasing sequence in such that . Then converges strongly, as , to , which solves the variational inequality
Proof.
For , we define for each . Then is regular sequence of means [16]. Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.
Corollary 4.4.
Let be a nonexpansive mapping on a Hilbert space such that . Let be a sequence in satisfying conditions and and let be a strongly regular matrix. Let and . Let be a sequence defined by the iterative algorithm
Then, converges strongly, as , to which solves the variational inequality
Proof.
For each , we define
for each . Since is a strongly regular matrix, for each , we have , as ; see [17]. Then, it is easy to see that is regular sequence of means. Furthermore, for each , we have Now, apply Theorem 3.1 to conclude the result.
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Acknowledgments
The authors thank the referee(s) for the helpful comments, which improved the presentation of this paper. This paper is dedicated to Professor Anthony To Ming Lau. This paper is based on final report of the research project of the Ph.D. thesis which is done with financial support of research office of the University of Tabriz.
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Piri, H., Vaezi, H. Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces. Fixed Point Theory Appl 2010, 907275 (2010). https://doi.org/10.1155/2010/907275
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DOI: https://doi.org/10.1155/2010/907275
Keywords
 Hilbert Space
 Banach Space
 Variational Inequality
 Iterative Algorithm
 Nonexpansive Mapping