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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Fixed point iteration for pseudocontractive maps


Authors: C. E. Chidume and Chika Moore
Journal: Proc. Amer. Math. Soc. 127 (1999), 1163-1170
MSC (1991): Primary 47H05, 47H06, 47H10, 47H15
MathSciNet review: 1625729
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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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Additional Information

C. E. Chidume
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email: chidume@ictp.trieste.it

Chika Moore
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05050-9
PII: S 0002-9939(99)05050-9
Received by editor(s): August 1, 1997
Communicated by: David R. Larson
Article copyright: © Copyright 1999 American Mathematical Society