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Báo cáo hóa học: "Review Article T -Stability Approach to Variational Iteration Method for Solving Integral Equations"

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 T -Stability Approach to Variational Iteration Method for Solving Integral Equations | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 393245 9 pages doi 10.1155 2009 393245 Review Article T-Stability Approach to Variational Iteration Method for Solving Integral Equations R. Saadati 1 S. M. Vaezpour 1 and B. E. Rhoades2 1 Department of Mathematics and Computer Science Amirkabir University ofTechnology 424 Hafez Avenue Tehran 15914 Iran 2 Department of Mathematics Indiana University Bloomington IN 47405-7106 USA Correspondence should be addressed to B. E. Rhoades rhoades@indiana.edu Received 16 February 2009 Accepted 26 August 2009 Recommended by Nan-jing Huang We consider T-stability definition according to Y. Qing and B. E. Rhoades 2008 and we show that the variational iteration method for solving integral equations is T -stable. Finally we present some text examples to illustrate our result. Copyright 2009 R. Saadati et al. 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 and Preliminaries Let X II II be a Banach space and T a self-map of X. Let xn 1 f T xn be some iteration procedure. Suppose that F T the fixed point set of T is nonempty and that xn converges to a point q e F T . Let yn Q X and define en ịịyn 1 - f T yn . If lim en 0 implies that lim yn q then the iteration procedure xn 1 f T xn is said to be T-stable. Without loss of generality we may assume that yn is bounded for if yn is not bounded then it cannot possibly converge. If these conditions hold for xn 1 Txn that is Picard s iteration then we will say that Picard s iteration is T-stable. Theorem 1.1 see 1 . Let X II II be a Banach space and T a self-map of X satisfying IITx - Ty L x - Tx a x - y II 1.1 for all x y e X where L 0 0 a 1. Suppose that T has a fixed point p. Then T is Picard T -stable. Various kinds of analytical methods and numerical methods 2-10 were .