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: Research Article Data Dependence for Ishikawa Iteration When Dealing with Contractive-Like Operators | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 242916 7 pages doi 2008 242916 Research Article Data Dependence for Ishikawa Iteration When Dealing with Contractive-Like Operators . M. oltuz1 2 and Teodor Grosan3 1 Departamento de Matematicas Universidad de los Andes Carrera 1 No. 18A-10 Bogota Colombia 2 The Institute of Numerical Analysis . Box 68-1 Cluj-Napoca Romania 3 Department of Applied Mathematics Babes-Bolyai University Cluj-Napoca Romania Correspondence should be addressed to Teodor Grosan tgrosan@ Received 13 February 2008 Accepted 27 May 2008 Recommended by Hichem Ben-El-Mechaiekh We prove a convergence result and a data dependence for Ishikawa iteration when applied to contraction-like operators. An example is given in which instead of computing the fixed point of an operator we approximate the operator with a contractive-like one. For which it is possible to compute the fixed point and therefore to approximate the fixed point of the initial operator. Copyright 2008 S. M. Soltuz and T. Grosan. 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 X be a real Banach space let B c X be a nonempty convex closed and bounded set. Let T S B B be two maps. For a given x0 u0 G B we consider the Ishikawa iteration see 1 for T and S xn 1 1 an xn anTy n yn 1 - fin xn finTx n un 1 _ 1 an un anSvn vn 1 fin un finSun where an c 0 1 ộn c 0 1 and TO lim an limpn 0 Van TO. n -x. nxx n 1 Set pn 0 Vn G N to obtain the Mann iteration see 2 . The map T is called Kannan mappings see 3 if there exists b G 0 1 2 such that for all x y G B Tx - Ty b Ilx - Txll by - Tyll . 2 Fixed Point Theory and Applications Similar mappings are Chatterjea mappings see 4 for which there exists c G 0 1 2 such that for all x y G B WTx - .
