Comments on the Rate of Convergence between Mann and Ishikawa Iterations Applied to Zamfirescu Operators
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Letter to the Editor Comments on the Rate of Convergence between Mann and Ishikawa Iterations Applied to Zamfirescu Operators Yuan Qing1 and B. E. Rhoades2 1 2
Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, Zhejiang, China Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA
Correspondence should be addressed to Yuan Qing, [email protected] Received 16 November 2007; Revised 8 February 2008; Accepted 13 March 2008 In the work of Babu and Vara Prasad 2006, the claim is made that Mann iteration converges faster than Ishikawa iteration when applied to Zamfirescu operators. We provide an example to demonstrate that this claim is false. Copyright q 2008 Y. Qing and B. E. Rhoades. 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.
We begin with some definitions. Definition 1. Suppose that {an } and {bn } are two real convergent sequences with limits a and b, respectively. Then {an } is said to converge faster than {bn } if an − a 0. 1 lim bn − b Definition 2. Let X, d be a complete metric space, and T : X→X a map for which there exist real numbers, a, b, and c satisfying 0 < a < 1, 0 < b, c < 1/2 such that for each pair x, y ∈ X, at least one of the following is true: 1 dT x, T y ≤ adx, y; 2 dT x, T y ≤ bdx, T x dy, T y; 3 dT x, T y ≤ cdx, T y dy, T x. Definition 3. Let E denote an arbitrary Banach space, T , a self-map of E. The sequence {xn } defined by x0 ∈ E, xn1 1 − αn xn αn T xn , n 0, 1, 2, . . . , 2 where 0 ≤ an < 1 for n 1, 2, . . . , is called Mann iteration, and will be denoted by Mx0 , αn , T .
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Fixed Point Theory and Applications The sequence {yn } defined by y0 ∈ E, yn1 1 − αn yn αn T zn , zn 1 − βn yn βn T yn , n 0, 1, 2, . . . ,
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where 0 ≤ αn , βn ≤ 1 for n 1, 2, . . . , is commonly called Ishikawa iteration, and will be denoted by Iy0 , αn , βn , T . The following appears in 1, Theorem 2.1. Theorem 4. Let E be an arbitrary Banach space, K a closed convex subset of E, T a Zamfirescu operator, 0 ≤ αn , βn ≤ 1, and αn ∞. Then Mann iteration Mx0 , αn , T converges faster than Ishikawa iteration Iy0 , αn , βn , T to the fixed point x∗ of T , provided that x0 y0 ∈ K. Let T be a nondecreasing continuous self-map of 0, 1 with p a fixed point of T . It was shown in 2, Theorem 7, that |yn1 − p| ≤ |xn1 − p| for each n 1, 2, . . .. Therefore, the condition xn1 − p 0 lim yn1 − p
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is impossible for any Zamfirescu operator on 0, 1. The error is caused by the inconsistent in 1, Definiton 1.3 see also 3. In fact, we will give an example satisfying the condition of 1, Theorem 2.1 such that the Ishikawa iteration converges faster than the Mann iteration. Example 5. Suppose 1 T : 0, 1 −→ 0, 1 : x, 2 αn βn 0,
n 1, 2, 3, . . . , 15;
4 αn
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