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Báo cáo hóa học: " Research Article The Equivalence between T-Stabilities of The Krasnoselskij and The Mann Iterations"

Tham khảo luận văn - đề án 'báo cáo hóa học: " research article the equivalence between t-stabilities of the krasnoselskij and the mann iterations"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007 Article ID 60732 7 pages doi 10.1155 2007 60732 Research Article The Equivalence between T-Stabilities of The Krasnoselskij and The Mann Iterations Stefan M. Soltuz Received 20 June 2007 Accepted 14 September 2007 Recommended by Hichem Ben-El-Mechaiekh We prove the equivalence between the T-stabilities of the Krasnoselskij and the Mann iterations a consequence is the equivalence with the T-stability of the Picard-Banach iteration. Copyright 2007 Stefan M. Soltuz. 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 normed space and T a selfmap of X. Let x0 be a point of X and assume that xn 1 f T xn is an iteration procedure involving T which yields a sequence xn of points fromX. Suppose xn converges to a fixed point x of T. Let ịfin be an arbitrary sequence in X and set en 1 - f T for all n e N. Definition 1.1 1 . If iimn. . en 0 iimn. . n p then the iteration procedure xn 1 f T xn is said to be T-stable with respect to T. Remark 1.2 1 . In practice such a sequence fin could arise in the following way. Let x0 be a point in X. Set xn 1 f T xn . Let 0 x0. Now x1 f T x0 . Because of rounding or discretization in the function T a new value approximately equal to x1 might be obtained instead of the true value of f T x0 . Then to approximate x2 the value f T J is computed to yield fi2 an approximation of f T J. This computation is continued to obtain fin an approximate sequence of xn . Let X be a normed space D a nonempty convex subset of X and T a selfmap of D let p0 e0 e D. The Mann iteration see 2 is defined by en 1 1 - afi en anTen 1.1 2 Fixed Point Theory and Applications where an c 0 1 . The Ishikawa iteration is defined see 3 by Xn 1 1 - a xn anTyn yn 1 fn xn fnTxn 1.2 where an c 0 1 fn c 0 1 . The .