Strong Convergence of a Generalized Iterative Method for Semigroups of Nonexpansive Mappings in Hilbert Spaces
© H. Piri and H. Vaezi. 2010
Received: 20 April 2010
Accepted: 18 June 2010
Published: 8 July 2010
Using -strongly accretive and -strictly pseudocontractive mapping, we introduce a general iterative method for finding a common fixed point of a semigroup of non-expansive mappings in a Hilbert space, with respect to a sequence of left regular means defined on an appropriate space of bounded real-valued functions of the semigroup. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality.
Let be a real Hilbert space. A mapping of into itself is called non-expansive if , for all . By , we denote the set of fixed points of (i.e., ).
where is a sequence in . See also .
where . They proved that if is a sequence in satisfying the following conditions:
either or ,
where is a potential function for (i.e., , for all ).
Using the Hahn-Banach 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 .
Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of non-expansive mappings are also presented. It is worth mentioning that we obtain our result without assuming condition .
for each , where is the adjoint operator of .
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 .
One can write by
Lemma 2.2 (see ).
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 non-expansive mapping from into .
(ii) for each .
(iii) for each .
(iv)If is left invariant, then is a non-expansive retraction from onto .
This defines a mapping from into and is called metric (the nearest point) projection onto .
Lemma 2.3 (see ).
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 .
Lemma 2.4 (see ).
Let be a nonempty closed convex subset of a Hilbert space and suppose that is non-expansive. Then, the mapping is demiclosed at zero.
The following lemma is well known.
Let be a real Hilbert space. Then, for all
Lemma 2.6 (see ).
where and are sequences of real numbers satisfying the following conditions:
The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.
Let be a real smooth Banach space and a mapping.
(i)If is -strongly accretive and -strictly pseudo-contractive with , then, is contractive with constant .
(ii)If is -strongly accretive and -strictly pseudo-contractive with , then, for any fixed number , is contractive with constant .
- (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 pseudo-contractive 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.
Therefore, is bounded and so is .
Since is arbitrary, we get (3.11).
and hence is a contraction due to
Consequently, applying Lemma 2.6, to (3.26), we conclude that .
This shows that is -strictly pseudo-contractive. Now apply Theorem 3.1 to conclude the result.
It is sufficient to take and in Theorem 3.1.
4. Some Application
Therefore, applying Theorem 3.1, the result follows.
For , we define for each , where denotes the space of all real-valued bounded continuous functions on with supremum norm. Then, is regular sequence of means . Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.
For , we define for each . Then is regular sequence of means . Furthermore, for each , we have . Now, apply Theorem 3.1 to conclude the result.
for each . Since is a strongly regular matrix, for each , we have , as ; see . 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.
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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