Open Access

Comparison of the Rate of Convergence among Picard, Mann, Ishikawa, and Noor Iterations Applied to Quasicontractive Maps

Fixed Point Theory and Applications20102010:169062

https://doi.org/10.1155/2010/169062

Received: 12 October 2010

Accepted: 14 December 2010

Published: 21 December 2010

Abstract

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.

1. Introduction

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., [1]), and the iteration defined by is called Picard iteration.

One of the most general contractive conditions for which a map is a Picard operator is that of Ćirić [2] (see also [3]). A self-map is called quasicontractive if it satisfies
(1.1)

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 .

To obtain fixed points for some maps for which Picard iteration fails, a number of fixed point iteration procedures have been developed. Let be a Banach space, the corresponding quasicontractive mapping is defined by
(1.2)

In this paper, we will consider the following four iterations.

Krasnoselskij:
(1.3)

where .

Mann:
(1.4)

where for , and .

Ishikawa:
(1.5)

where .

Noor:
(1.6)

where .

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 [4].

Theorem 1.1.

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 .

2. Results for Quasicontractive Operators

To avoid trivialities, we shall always assume that , where denotes the fixed point of the map .

Let be two convergent sequences with the same limit , then is said to converge faster than (see, e.g., [5]) if
(2.1)

Theorem 2.1.

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.

Proof.

From Theorem 1 of [2] and (1.2),
(2.2)

where is the fixed point of .

From (1.3), with ,
(2.3)

By setting each and each , it follows from Theorem 1.1 that converges to .

Therefore,
(2.4)

as , since .

Theorem 2.2.

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.

Proof.

We have the following cases

Case A (Mann Iteration).

Using Theorem 1.1 with each , converges to . Using (1.4),
(2.5)
Therefore,
(2.6)

as , since for each .

Case B (Ishikawa Iteration).

From Theorem 1.1, converges to . Using (1.5),
(2.7)
Hence,
(2.8)

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.

Lemma 2.3.

Define
(2.9)

then is bounded.

Proof.

Case 1.

Suppose that for some , then, from (1.2) and the definition of ,
(2.10)

a contradiction, since .

Similarly, , , , , and for any .

Case 2.

Suppose that , without loss of generality we let . Then, from (1.6),
(2.11)

Hence, , that is, . By induction on , we obtain , a contradiction.

Case 3.

Suppose that for some . If , then we have, using (1.6),
(2.12)

which implies that , and by induction on , we get .

Case 4.

Suppose that or for some , then
(2.13)

From Cases 2 and 3, , and for some , that is, . If , we obtain that . Therefore; , other cases, omitting.

Case 5.

Suppose that or for some , then if ,
(2.14)

it leads to . Again by induction on , we have . Similarly, if or, , we also get ; other cases, omitting.

Case 6.

Suppose that or for some , then, using Case 1,
(2.15)
or
(2.16)

these imply that . By Case 3, we obtain that .

Case 7.

Suppose that or for some , then if , using Case 2,
(2.17)

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.

Indeed, suppose that for some , then, using Case 1,
(2.18)

where , then .

Similarly, if , or we again get . Hence, is bounded, that is, is bounded.

Lemma 2.4.

Let , and be as in Theorem 2.1, and that , then , as defined by (1.6), converges strongly to the unique fixed point of .

Proof.

From Ćirić [2], has a unique fixed point . For each , define
(2.19)
Then, using the same proof as that of Lemma 2.3, it can be shown that
(2.20)
Using (1.2) and (1.6),
(2.21)

, since .

For any with ,
(2.22)

and is Cauchy sequence. Since is closed, there exists such that . Also, .

Using (1.2),
(2.23)

Since , it follows that , and is a fixed point of . But the fixed point is unique. Therefore, .

Returning to the proof of Case C, from (1.6),
(2.24)
So,
(2.25)

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.

3. Zamfirescu Maps

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:

(1) ,

(2) ,

(3) .

In [6] it was shown that the above set of conditions is equivalent to
(3.1)

for some .

In the following results, we shall use the representation (3.1).

Theorem 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.

Proof.

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 .

Using (1.2),
(3.2)
Using (3.1),
(3.3)
Therefore,
(3.4)
and
(3.5)
Thus,
(3.6)

as , since .

The proofs for Ishikawa and Noor iterations are similar.

Theorem 3.2.

Let , and be as in Theorem 3.1, then if with the constant for any , Mann iteration converges faster than Krasnoselskij iteration.

Proof.

Using (1.4) and (3.1),
(3.7)
And again using (1.3), (3.1), we have
(3.8)
Thus,
(3.9)

as , since .

It is not possible to compare the rates of convergence for Mann, Ishikawa, and Noor iterations, even for Zamfirescu maps.

Remark 3.3.

It has been noted in [7] that the principal result in [8] is incorrect.

Remark 3.4.

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.

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
Department of Mathematics, Indiana University
(2)
Department of Mathematics and Physics, Shijiazhuang Railway University

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Copyright

© B. E. Rhoades and Zhiqun Xue. 2010

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.