Abstract and Applied Analysis

Volume 2012 (2012), Article ID 626547, 10 pages

http://dx.doi.org/10.1155/2012/626547

## Strong Convergence of Parallel Iterative Algorithm with Mean Errors for Two Finite Families of Ćirić Quasi-Contractive Operators

Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University, Zhejiang, Hangzhou 310036, China

Received 23 June 2012; Revised 19 August 2012; Accepted 21 August 2012

Academic Editor: Yonghong Yao

Copyright © 2012 Feng Gu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The purpose of this paper is to establish a strong convergence of a new parallel iterative algorithm with mean errors to a common fixed point for two finite families of Ćirić quasi-contractive operators in normed spaces. The results presented in this paper generalize and improve the corresponding results of Berinde, Gu, Rafiq, Rhoades, and Zamfirescu.

#### 1. Introduction and Preliminaries

Let be a metric space. A mapping is said to be -*contraction*, if , where .

The mapping is said to be Kannan mapping [1], if there exists such that .

A mapping is said to be Chatterjea mapping [2], if there exists such that .

Combining these three definitions, Zamfirescu [3] proved the following important result.

Theorem Z (see [3]). * Let be a complete metric space and a mapping for which there exist the real numbers , and satisfying , such that for each pair , at least one of the following conditions holds:* *,* *,* *.**Then has a unique fixed point and the Picard iteration defined by
**
converges to for any arbitrary but fixed . *

*Remark 1.1. * An operator satisfying the contractive conditions in the above theorem is called -operator.

*Remark 1.2. * The conditions can be written in the following equivalent form:
for all . Thus, a class of mappings satisfying the contractive conditions is a subclass of mappings satisfying the following condition:
. The class of mappings satisfying is introduced and investigated by iri [4] in 1971.

*Remark 1.3. *A mapping satisfying is commonly called iri generalized contraction.

In 2000, Berinde [5] introduced a new class of operators on a normed space satisfying
for any , and .

Note that (1.3) is equivalent to for any , and .

Berinde [5] proved that this class is wider than the class of Zamfiresu operators and used the Mann [6] iteration process to approximate fixed points of this class of operators in a normed space given in the form of following theorem.

Theorem B (see [5]). *Let be a nonempty closed convex subset of a normed space . Let be an operator satisfying (1.3) and . For given , let be generated by the algorithm
**
where be a real sequence in [0, 1]. If , then converges strongly to the unique fixed point of .*

In 2006, Rafiq [7] considered a class of mappings satisfying the following condition: . This class of mappings is a subclass of mappings satisfying the following condition: . The class of mappings satisfying was introduced and investigated by iri [8] in 1974 and a mapping satisfying is commonly called iri quasi-contraction.

Rafiq [7] proved the following result.

Theorem R (see [7]). *Let be a nonempty closed convex subset of a normed space . Let be an operator satisfying the condition . For given , let be generated by the algorithm
**
where , and be three real sequences in [0, 1] satisfying for all , is a bounded sequences in . If and , then converges strongly to the unique fixed point of . *

In 2007, Gu [9] proved the following theorem.

Theorem G (see [9]). *Let be a nonempty closed convex subset of a normed space . Let be operators satisfying the condition with (the set of common fixed points of ). Let , , and be three real sequences in [0, 1] satisfying for all , a bounded sequences in satisfying the following conditions:*(i)*;*(ii)* or . ** Suppose further that is any given point and is generated by the algorithm
**
where . Then converges strongly to a common fixed point of . *

*Remark 1.4. *It should be pointed out that Theorem G extends Theorem R from a iri quasi-contractive operator to a finite family of iri quasi-contractive operators.

Inspired and motivated by the facts said above, we introduced a new two-step parallel iterative algorithm with mean errors for two finite family of operators and as follows: where are two finite sequences of positive number such that and , , , and are four real sequences in [0, 1] satisfying and for all , and are two bounded sequences in and is a given point.

Especially, if , are two sequences in [0, 1] satisfying for all , satisfying , is a bounded sequence in and is a given point in , then the sequence defined by is called the one-step parallel iterative algorithm with mean errors for a finite family of operators .

The purpose of this paper is to study the convergence of two-steps parallel iterative algorithm with mean errors defined by (1.8) to a common fixed point for two finite family of iri quasi-contractive operators in normed spaces. The results presented in this paper generalized and extend the corresponding results of Berinde [5], Gu [9], Rafiq [7], Rhoades [10], and Zamfirescu [3]. Even in the case of or for all or are also new.

In order to prove the main results of this paper, we need the following Lemma.

Lemma 1.5 (see [11]). *Suppose that , , and are three nonnegative real sequences satisfying the following condition:
**
where is some nonnegative integer, , , and . Then . *

#### 2. Main Results

We are now in a position to prove our main results in this paper.

Theorem 2.1. *Let be a nonempty closed convex subset of a normed space . Let be operators satisfying the condition and be operators satisfying the condition with , where and are the set of fixed points of and in , respectively. Let , , , and be four real sequences in [0, 1] satisfying and for all , two finite sequences of positive number such that and , and two bounded sequences in satisfying the following conditions:*(i)*;*(ii)*;*(iii)* or . ** Suppose further that is any given point and is an iteration sequence with mane errors defined by (1.8), then converges strongly to a common fixed point of and . *

* Proof. * Since is iri operator satisfying the condition , hence there exists () such that
For each fixed . Denote , then and
hold for each fixed . If from (2.2) we have
then
Hence
which yields (using the fact that )
Also, from (2.2), if
holds, then

(a) , which implies and hence, as ,
or

(b) , which implies
Thus, if (2.7) holds, then from (2.8) and (2.9) we have
Denote
Then we have and . Combining (2.2),(2.6), and (2.10) we get
holds for all and .

On the other hand, since is iri operator satisfying the condition , similarly, we can prove
for all and , where and .

Let ; using (1.8) we have
where . Now for and , (2.12) gives
Substituting (2.15) into (2.14), we obtain that
Again it follows from (1.8) that
Now for and , (2.13) gives
Combining (2.17) and (2.18) we get
Substituting (2.19) into (2.16), we obtain that
where
or
From the conditions (i)–(iii) it is easy to see that , , , and . Thus using (2.20) and Lemma 1.5 we have , and so . This completes the proof of Theorem 2.1.

Theorem 2.2. *Let be a nonempty closed convex subset of a normed space . Let be operators satisfying the condition (2.12) and let be operators satisfying the condition (2.13) with , where and are the set of fixed points of and in , respectively. Let , , , and be four real sequences in [0, 1] satisfying and for all , two finite sequences of positive number such that , and , and two bounded sequences in satisfying the following conditions:*(i)*;*(ii)*;*(iii)* or . ** Suppose further that is any given point and is an iteration sequence defined by (1.8), then converges strongly to a common fixed point of and . *

Theorem 2.3. * Let be a nonempty closed convex subset of a normed space . Let be operators satisfying the condition with (the set of common fixed points of ). Let and be two real sequences in [0, 1] satisfying for all , a finite sequence of positive number such that , and a bounded sequence in satisfying the following conditions:*(i)*;*(ii)* or . ** Suppose further that is any given point and is an iteration sequence with mane errors defined by (1.9), then converges strongly to a common fixed point of . *

Theorem 2.4. * Let be a nonempty closed convex subset of a normed space . Let be operators satisfying the condition (2.12) with (the set of common fixed points of ). Let and be two real sequences in [0, 1] satisfying for all , a finite sequence of positive number such that , and a bounded sequence in satisfying the following conditions:*(i)*;*(ii)* or . ** Suppose further that is any given point and is an iteration sequence defined by (1.9), then converges strongly to a common fixed point of . *

Corollary 2.5 (see [7]). *Let be a nonempty closed convex subset of a normed space . Let be an operators satisfying the condition . Let , , and be three real sequences in [0, 1] satisfying for all and a bounded sequences in satisfying the following conditions:*(i)*;*(ii)* or . ** Suppose further that is any given point and is an explicit iteration sequence as follows:
**
then converges strongly to the unique fixed point of . *

*Proof. * By iri [8], we know that has a unique fixed point in . Taking in Theorem 2.3, then the conclusion of Corollary 2.5 can be obtained from Theorem 2.3 immediately. This completes the proof of Corollary 2.5.

*Remark 2.6. *Theorems 2.2–2.4 and Corollary 2.5 improve and extend the corresponding results of Berinde [5], Gu [9], Rafiq [7], Rhoades [10], and Zamfirescu [3].

#### Acknowledgments

The present study was supported by the National Natural Science Foundation of China (11071169, 11271105) and the Natural Science Foundation of Zhejiang Province (Y6110287, Y12A010095).

#### References

- R. Kannan, “Some results on fixed points,”
*Bulletin of the Calcutta Mathematical Society*, vol. 60, pp. 71–76, 1968. View at Google Scholar · View at Zentralblatt MATH - S. K. Chatterjea, “Fixed-point theorems,”
*Comptes Rendus de l'Académie Bulgare des Sciences*, vol. 25, no. 6, pp. 727–730, 1972. View at Google Scholar · View at Zentralblatt MATH - T. Zamfirescu, “Fix point theorems in metric spaces,”
*Archiv der Mathematik*, vol. 23, no. 1, pp. 292–298, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. B. Ćirić, “Generalized contractions and fixed point theorems,”
*Publications de l'Institut Mathématique*, vol. 12, no. 26, pp. 19–26, 1971. View at Google Scholar · View at Zentralblatt MATH - V. Berinde,
*Iterative Approximation of Fixed Points*, Efemeride, Baia Mare, Romania, 2000. View at Zentralblatt MATH - W. R. Mann, “Mean value methods in iteration,”
*Proceedings of the American Mathematical Society*, vol. 4, pp. 506–510, 1953. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Rafiq, “Fixed points of Ćirić quasi-contractive operators in normed spaces,”
*Mathematical Communications*, vol. 11, no. 2, pp. 115–120, 2006. View at Google Scholar · View at Zentralblatt MATH - L. B. Ćirić, “A generalization of Banach’s contraction principle,”
*Proceedings of the American Mathematical Society*, vol. 45, no. 2, pp. 267–273, 1974. View at Google Scholar - F. Gu, “Strong convergence of an explicit iterative process with mean errors for a finite family of Ćirić quasi-contractive operators in normed spaces,”
*Mathematical Communications*, vol. 12, no. 1, pp. 75–82, 2007. View at Google Scholar · View at Zentralblatt MATH - B. E. Rhoades, “Fixed point iterations using infinite matrices,”
*Transactions of the American Mathematical Society*, vol. 196, pp. 161–176, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. S. Chang, Y. J. Cho, and H. Y. Zhou,
*Iterative Methods for Nonlinear Operator Equations in Banach Spaces*, Nova Science, New York, NY, USA, 2002. View at Zentralblatt MATH