tailieunhanh - Báo cáo hóa học: " Letter to the Editor Comments on the Rate of Convergence between Mann and Ishikawa Iterations Applied to "

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Letter to the Editor Comments on the Rate of Convergence between Mann and Ishikawa Iterations Applied to | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 387504 3 pages doi 2008 387504 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 Department of Mathematics Hangzhou Normal University Hangzhou 310036 Zhejiang China 2 Department of Mathematics Indiana University Bloomington IN 47405-7106 USA Correspondence should be addressed to Yuan Qing yuanqingbuaa@ 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 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 im b b 0 1 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 G X at least one of the following is true 1 d Tx Ty ad x yfi 2 d Tx Ty b d x Tx d y Ty 3 d Tx Ty c d x Ty d y Tx . Definition 3. Let E denote an arbitrary Banach space T a self-map of E. The sequence xn defined by x0 G E xn 1 1 - an xn anTxn n 0 1 2 . 2 where 0 an 1 for n 1 2 . is called Mann iteration and will be denoted by M x0 an T . 2 Fixed Point Theory and Applications The sequence yn defined by y0 E yn 1 _ 1 an yn anTzn Zn 1 - fify ộnTyn n 0 1 2 . 3 where 0 an pn 1 for n 1 2 . is commonly called Ishikawa iteration and will be denoted by I y0 ữn ộn T . .