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.
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.
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.
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 .
2. Results for Quasicontractive Operators
We have the following cases
Case A (Mann Iteration).
Case B (Ishikawa Iteration).
Case C (Noor Iteration).
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
In the following results, we shall use the representation (3.1).
The proofs for Ishikawa and Noor iterations are similar.
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.
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