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Fixed point iteration for pseudocontractive maps

Authors:	C E Chidume Chika Moore

Institution:	The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy ; The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

Abstract:	Let be a compact convex subset of a real Hilbert space, ; $T:K\rightarrow K$ a continuous pseudocontractive map. Let $\{a_{n}\}, \{b_{n}\}, \{c_{n}\}, \{a_{n}^{'}\}, \{b_{n}^{'}\}$ and $\{c_{n}^{'}\}$ be real sequences in 0,1] satisfying appropriate conditions. For arbitrary $x_{1}\in K,$ define the sequence $\{x_{n}\}_{n=1}^{\infty}$ iteratively by $x_{n+1} = a_{n}x_{n} + b_{n}Ty_{n} + c_{n}u_{n}; y_{n} = a_{n}^{'}x_{n} + b_{n}^{'}Tx_{n} + c_{n}^{'}v_{n}, n\geq 1,$ where $\{u_{n}\}, \{v_{n}\}$ are arbitrary sequences in . Then, $\{x_{n}\}_{n=1}^{\infty}$ converges strongly to a fixed point of . A related result deals with the convergence of $\{x_{n}\}_{n=1}^{\infty}$ to a fixed point of when is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

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