Strong convergence of a relaxed three-step iterative algorithm for countable families of pseudocontractions
© Deng; licensee Springer. 2013
Received: 29 April 2013
Accepted: 29 July 2013
Published: 14 August 2013
An up-to-date method for the approximation of common fixed points of countable families of nonlinear operators is introduced, by which a relaxed three-step iterative algorithm is developed for the class of pseudocontractive mappings, and a strong convergence theorem is established in the framework of Hilbert spaces. Since there is no need to impose uniformity assumption on the involved Lipschitzian and closed mappings, the results improve and extend those announced by Cheng et al. (Fixed Point Theory Appl. 2013:100, 2013) and other authors with the related interest.
MSC:47H05, 47H09, 47H10.
holds for any and for all .
Not only from its being an important generalization of nonexpansive mappings, but also from the firm connection with the important class of nonlinear monotone mappings stems the interest in the class of pseudocontractions. We observe that A is monotone if and only if is pseudocontractive, and hence a zero of A, that is, , is just a fixed point of T. It is well known (see, e.g., ) that if A is monotone, then the solutions to the equation correspond to the equilibrium points of some evolution systems. Considerable efforts have then been devoted to developing iterative techniques for approximating fixed points of pseudocontractive mappings (see, for example, [2–4] and the references contained therein).
A countable family of mapping is called uniformly closed if as , and imply .
Inspired and motivated by the studies mentioned above, in this paper, we introduced an up-to-date method for the approximation of common fixed points of countable families of nonlinear operators, by which a relaxed three-step iterative algorithm is developed for the class of pseudocontractive mappings, and a strong convergence theorem is established in the framework of Hilbert spaces. No compactness assumption is imposed either on the involved mappings or on the set C. The results are more applicable than those of other authors with the related interest.
In what follows, we shall make use of the following lemmas.
Lemma 2.1 
Lemma 2.2 
If and , then exists.
Lemma 2.3 
where denotes the maximal integer that is not larger than x.
3 Main results
Recall that an operator T on a Hilbert space is closed if and as , then .
Then converges strongly to an .
So, by Lemma 2.2, we conclude that exists.
Note that, from (3.16), as . It immediately follows from (3.18) and the closedness of that for each , and hence . This completes the proof. □
We now give an example, to which the results of Cheng et al.  cannot be applied.
while is obviously not a member of F.
Remark 3.3 By using a specific way of choosing the indexes of the involved mappings and parameters, we propose an up-to-date iterative approach to approximating common fixed points of countable families of pseudocontractive mappings. The results extend previous results, announced by the authors with the related research interest.
The author is very grateful to the referees for their useful suggestions, by which the contents of this article has been improved. This work is supported by the National Natural Science Foundation of China (Grant No. 11061037).
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