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Báo cáo hóa học: " Review Article Remarks of Equivalence among Picard, Mann, and Ishikawa Iterations in Normed Spaces Xue Zhiqun"

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: Review Article Remarks of Equivalence among Picard, Mann, and Ishikawa Iterations in Normed Spaces Xue Zhiqun | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 61434 5 pages doi 10.1155 2007 61434 Review Article Remarks of Equivalence among Picard Mann and Ishikawa Iterations in Normed Spaces Xue Zhiqun Received 1 April 2007 Revised 16 April 2007 Accepted 21 June 2007 Recommended by J. R. L. Webb We show that the convergence of Picard iteration is equivalent to the convergence of Mann iteration schemes for various Zamfirescu operators. Our result extends of Soltuz 2005 . Copyright 2007 Xue Zhiqun. 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. 1. Introduction Let E be a real normed space D a nonempty convex subset of E and T a self-map of D let p0 u0 x0 e D. The Picard iteration is defined by 1.1 pn 1 Tpn n 0. The Mann iteration is defined by un 1 Q - afi un anTun n 0. 1.2 The Ishikawa iteration is defined by yn Ơ - xn 1 Ơ - bn xn bnTxn n 0 afix anTyn n 0 1.3 where an bn are sequences of positive numbers in 0 1 . Obviously for an 1 the Mann iteration 1.2 reduces to the Picard iteration and for bn 0 the Ishikawa iteration 1.3 reduces to the Mann iteration 1.2 . Definition 1.1 1 Definition 1 . Let T D - D be a map for which there exist real numbers a b c satisfying 0 a 1 0 b 1 2 0 c 1 2. Then T is called a Zamfirescu operator 2 Fixed Point Theory and Applications if for each pair x y in D T satisfies at least one of the following conditions given in 1 - 3 1 Tx - Ty a x - yH 2 Tx - Ty b x- Tx IIy - Tyll 3 Tx - Ty c x- Ty IIy - Tx . It is easy to show that every Zamfirescu operator T satisfies the inequality Tx- Ty 8 x-yll 28 x- Tx 1.4 for all x y e D where 8 max a b 1 - b c 1 - c with 0 8 1 See Soltuz 1 . Recently Soltuz 1 had studied that the equivalence of convergence for Picard Mann and Ishikawa iterations and proved the following results. Theorem 1.2 1 Theorem 1 . Let X .