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.
where is a sequence in . See also .
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 .
Lemma 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
The following lemma is well known.
Lemma 2.6 (see ).
The following lemma will be frequently used throughout the paper. For the sake of completeness, we include its proof.
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.
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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