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- Title
Necessary and Sufficient Condition for Mann Iteration Converges to a Fixed Point of Lipschitzian Mappings.
- Authors
Xiang, Chang-He; Zhang, Jiang-Hua; Zhe Chen
- Abstract
Suppose that E is a real normed linear space, C is a nonempty convex subset of E, T : C ? C is a Lipschitzian mapping, and x0 £ C is a fixed point of T. For given x0 C, suppose that the sequence {xn} C is the Mann iterative sequence defined by x(sub)n(/sub)C C [0,1] £xn(1-α(sub)n(/sub) (anTxn, n = 0, where {an} is a sequence in [(0), 1(G), B Σn-C=0 a2Σn=0a2=8,Σn=0an8 n0 anTxn [C8).We prove that the sequence {xn} strongly converges to x if and only if there exists a strictly increasing function F:(0,8)E ? (F0,8)E with F(0) (0 such that lim supn?8infj)xn-xJxn-x(E{HTxn-x, j(Dxn-x E-xn-x 2xn-xo)E} = 0.
- Subjects
ITERATIVE methods (Mathematics); STOCHASTIC convergence; VECTOR spaces; CONVEX sets; MATHEMATICAL mappings; FIXED point theory; MATHEMATICAL sequences
- Publication
Journal of Applied Mathematics, 2012, p1
- ISSN
1110-757X
- Publication type
Article
- DOI
10.1155/2012/327878