Fixed point iteration for pseudocontractive maps

Chidume, C.; Moore, Chika

doi:10.1090/S0002-9939-99-05050-9

Fixed point iteration for pseudocontractive maps
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by C. E. Chidume and Chika Moore PDF

Proc. Amer. Math. Soc. 127 (1999), 1163-1170 Request permission

Abstract:

Let $K$ be a compact convex subset of a real Hilbert space, $H$; $T:K\rightarrow K$ 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 $K$. Then, $\{x_{n}\}_{n=1}^{\infty }$ converges strongly to a fixed point of $T$. A related result deals with the convergence of $\{x_{n}\}_{n=1}^{\infty }$ to a fixed point of $T$ when $T$ is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.

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