A modified Picard-Mann hybrid iterative algorithm for common fixed points of countable families of nonexpansive mappings
© Deng; licensee Springer. 2014
Received: 25 April 2013
Accepted: 25 February 2014
Published: 6 March 2014
An up-to-date method is used for approximating common fixed points of countable families of nonlinear mappings. A modified Picard-Mann hybrid iterative algorithm is introduced with the help of our method for the class of nonexpansive mappings. Strong convergence and weak convergence theorems are established in the framework of uniformly convex Banach spaces. Our results extend the corresponding ones announced by Khan (Fixed Point Theory Appl. 2013:69, 2013, doi:10.1186/1687-1812-2013-69) to the case of countable families of nonexpansive mappings.
where is a real sequence in . He showed that the new process converges faster than all of Picard, Mann and Ishikawa iterative processes in the sense of Berinde  for contractions. He also proved strong convergence and weak convergence theorems with the help of his process for the class of nonexpansive mappings in general Banach spaces and apply it to obtain a result in uniformly convex Banach spaces.
Inspired and motivated by the studies mentioned above, in this paper, we use an up-to-date method for the approximation of common fixed points of countable families of nonlinear operators. We introduce a modified Picard-Mann hybrid iterative algorithm with the help of our method for the class of nonexpansive mappings. We prove strong convergence and weak convergence theorems in the framework of Banach spaces. Our results extend the corresponding ones for one map in .
In the sequel, we use to denote the set of fixed points of a mapping T.
exists for each . In this case, the norm of E is said to be Gâteaux differentiable. The space E is said to have uniformly Gâteaux differentiable norm if for each , the limit (2.3) is attained uniformly for . The norm of E is said to be Fréchet differentiable if for each , the limit (2.3) is attained uniformly for . The norm of E is said to be uniformly Fréchet differentiable (and E is said to be uniformly smooth) if the limit (2.3) is attained uniformly for .
Note The readers can find all the definitions and concepts mentioned above in .
for all with , where denotes that converges weakly to x.
A mapping T with domain and range in E is said to be demi-closed at p if whenever is a sequence in such that converges weakly to and converges strongly to p, then .
If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E.
If E is reflexive and strictly convex, then is norm-weak-continuous.
If E is a smooth, strictly convex and reflexive Banach space, then the normalized duality mapping is single valued, one-to-one and onto.
A Banach space E is uniformly smooth if and only if is uniformly convex.
Each uniformly convex Banach space E has the Kadec-Klee property, i.e., for any sequence , if and , then as .
We need the following lemmas for our main results.
Lemma 2.2 
imply that , where is a constant.
Lemma 2.3 
Let E be a real uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let be a nonexpansive mapping. Then is demi-closed at zero.
Lemma 2.4 
where denotes the maximal integer that is not larger than x.
3 Main results
exists, where ;
This shows that is decreasing and hence exists.
(2) This conclusion can easily be shown by taking the infimum in (3.3) for all .
which implies that .
Note that as . It then follows from (3.8) and (3.10) that (3.11) holds obviously. This completes the proof. □
Remark 3.2 The key point of the proof of Lemma 3.1 lies in the use of a special way of choosing the indices of involved mappings, which makes the generalization of finite families of nonlinear mappings to infinite ones possible. Moreover, with the help of our method, some known results on the common fixed points of countable families of nonexpansive mappings have been improved. We now give an example to show why our work, compared with that of others, is an improvement.
In 2011, for the approximation of common fixed points of a countable family of nonexpansive mappings , Zhang et al.  introduced in his iterative algorithm a mapping T defined by a convex linear combination of , i.e., , () with . However, it is easy to see that the accurate computation of at each step of the iteration process is not easily attainable, which will leads to gradually increasing errors. By using a special way of choosing the indices of involved mappings, Deng  recently improved the corresponding results announced by Zhang et al. . Since the strong convergence theorems for solving some variational inequality problems and hierarchical fixed point problems are obtained without the aid of the convex linear combination of a countable family of nonexpansive mappings, our results are more applicable than those of other authors with related research interest.
Theorem 3.3 Let E be a real uniformly convex Banach space and K a nonempty closed convex subset of E. Let be a sequence of nonexpansive mappings from K to itself. Suppose that is a sequence defined by (3.1). If and there exist and a nondecreasing function with and for all such that for all , then converges strongly to some common fixed point of .
which implies by the definition of the function f.
Now we show that is a Cauchy sequence. Since , then for any , there exists a positive integer N such that for all . On the other hand, there exists a such that , because and F is closed.
This implies that is a Cauchy sequence, and hence there exists an such that as . Then yields . Further, it follows from the closedness of F that . This completes the proof. □
Theorem 3.4 Let E be a real uniformly convex Banach space satisfying Opial’s condition and K a nonempty closed convex subset of E. Let be a sequence of nonexpansive mappings from K to itself. Suppose that is a sequence defined by (3.1). If , then converges weakly to some common fixed point of .
Proof For any , by Lemma 3.1, we know that exists. We now prove that has a unique weakly subsequential limit in F. First of all, Lemmas 2.3 and 3.1 guarantee that each weakly subsequential limit of is a common fixed point of . Secondly, Opial’s condition guarantees that the weakly subsequential limit of is unique. Consequently, converges weakly to a common fixed point of . This completes the proof. □
Remark 3.5 The results presented in this paper extend those of Khan , whose research areas are limited to the situation of a single nonexpansive mapping.
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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