- Research Article
- Open Access
Comparison of the Rate of Convergence among Picard, Mann, Ishikawa, and Noor Iterations Applied to Quasicontractive Maps
© B. E. Rhoades and Zhiqun Xue. 2010
- Received: 12 October 2010
- Accepted: 14 December 2010
- Published: 21 December 2010
We provide sufficient conditions for Picard iteration to converge faster than Krasnoselskij, Mann, Ishikawa, or Noor iteration for quasicontractive operators. We also compare the rates of convergence between Krasnoselskij and Mann iterations for Zamfirescu operators.
- Banach Space
- Iteration Method
- Contractive Condition
- Principal Result
- Iteration Scheme
Let be a complete metric space, and let be a self-map of . If has a unique fixed point, which can be obtained as the limit of the sequence , where any point of , then is called a Picard operator (see, e.g., ), and the iteration defined by is called Picard iteration.
for each , where is a real number satisfying .
Not every map which has a unique fixed point enjoys the Picard property. For example, let with the absolute value metric, defined by . Then, has a unique fixed point at , but if one chooses as a starting point for any , then successive function iterations generate the bounded divergent sequence .
In this paper, we will consider the following four iterations.
where for , and .
Three of these iteration schemes have also been used to obtain fixed points for some Picard maps. Consequently, it is reasonable to try to determine which process converges the fastest.
In this paper, we will discuss this question for the above quasicontractions and for Zamfirescu operators. For this, we will need the following result, which is a special case of the Theorem in .
Let be any nonempty closed convex subset of a Banach space , and let be a quasicontractive self-map of . Let be the Ishikawa iteration process defined by (1.5), where each and . then converges strongly to the fixed point of .
To avoid trivialities, we shall always assume that , where denotes the fixed point of the map .
Let be a Banach space, a closed convex subset of , and a quasicontractive self-map of , then, for , Picard iteration converges faster than Krasnoselskij iteration.
where is the fixed point of .
By setting each and each , it follows from Theorem 1.1 that converges to .
as , since .
Let , and be as in Theorem 2.1. And let for all .
(A)If the constant , then Picard iteration converges faster than Mann iteration.
(B)If the constant , then Picard iteration converges faster than Ishikawa iteration.
(C)If the constant , then Picard iteration converges faster than Noor iteration.
We have the following cases
Case A (Mann Iteration).
as , since for each .
Case B (Ishikawa Iteration).
as , since for each .
Case C (Noor Iteration).
First we must show that converges to . The proof will follow along the lines of that of Theorem 1.1.
then is bounded.
a contradiction, since .
Similarly, , , , , and for any .
Hence, , that is, . By induction on , we obtain , a contradiction.
which implies that , and by induction on , we get .
From Cases 2 and 3, , and for some , that is, . If , we obtain that . Therefore; , other cases, omitting.
it leads to . Again by induction on , we have . Similarly, if or, , we also get ; other cases, omitting.
these imply that . By Case 3, we obtain that .
which implies that . Using induction on , we have .
In view of the above cases, so we have shown that . It remains to show that is bounded.
where , then .
Similarly, if , or we again get . Hence, is bounded, that is, is bounded.
Let , and be as in Theorem 2.1, and that , then , as defined by (1.6), converges strongly to the unique fixed point of .
, since .
and is Cauchy sequence. Since is closed, there exists such that . Also, .
Since , it follows that , and is a fixed point of . But the fixed point is unique. Therefore, .
as , since for .
It is not possible to compare the rates of convergence between the Krasnoselskij, Mann, and Noor iterations for quasicontractive maps. However, if one considers Zamfirescu maps, then some comparisons can be made.
A selfmap is called a Zamfirescu operator if there exist real numbers satisfying such that, for each at least one of the following conditions is true:
for some .
In the following results, we shall use the representation (3.1).
Let , and be as in Theorem 2.1, a Zamfirescu selfmap of , then if with the constant for each , Krasnoselskij iteration converges faster than Mann, Ishikawa, or Noor iteration.
Since Zamfirescu maps are special cases of quasicontractive maps, from Theorem 1.1 , and converge to the unique fixed point of , which we will call .
as , since .
The proofs for Ishikawa and Noor iterations are similar.
Let , and be as in Theorem 3.1, then if with the constant for any , Mann iteration converges faster than Krasnoselskij iteration.
as , since .
It is not possible to compare the rates of convergence for Mann, Ishikawa, and Noor iterations, even for Zamfirescu maps.
Krasnoselskij and Mann iterations were developed to obtain fixed point iteration methods which converge for some operators, such as nonexpansive ones, for which Picard iteration fails. Ishikawa iteration was invented to obtain a convergent fixed point iteration procedure for continuous pseudocontractive maps, for which Mann iteration failed. To date, there is no example of any operator that requires Noor iteration; that is, no example of an operator for which Noor iteration converges, but for which neither Mann nor Ishikawa converges.
The authors would like to thank the reviewers for valuable suggestions, and the National Natural Science Foundation of China Grant 10872136 for the financial support.
- Rus IA: Weakly Picard mappings. Commentationes Mathematicae Universitatis Carolinae 1993,34(4):769–773.MathSciNetMATHGoogle Scholar
- \'Cirić LjB: A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society 1974, 45: 267–273.MathSciNetMATHGoogle Scholar
- Rhoades BE: A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society 1977, 226: 257–290.MathSciNetView ArticleMATHGoogle Scholar
- Xu HK: A note on the Ishikawa iteration scheme. Journal of Mathematical Analysis and Applications 1992,167(2):582–587. 10.1016/0022-247X(92)90225-3MathSciNetView ArticleMATHGoogle Scholar
- Berinde V: Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. Fixed Point Theory and Applications 2004, (2):97–105.Google Scholar
- Rafiq A: Fixed points of Cirić quasi-contractive operators in normed spaces. Mathematical Communications 2006,11(2):115–120.MathSciNetMATHGoogle Scholar
- Qing Y, Rhoades BE: Letter to the editor: comments on the rate of convergence between Mann and Ishikawa iterations applied to Zamfirescu operators. Fixed Point Theory and Applications 2008, -3.Google Scholar
- Babu GVR, Vara Prasad KNVV: Mann iteration converges faster than Ishikawa iteration for the class of Zamfirescu operators. Fixed Point Theory and Applications 2006, -6.Google Scholar
This article is published under license to BioMed Central Ltd. 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.