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FIBONACCI–MANN ITERATION FOR MONOTONE ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Published online by Cambridge University Press:  13 March 2017

M. R. ALFURAIDAN
Affiliation:
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia email [email protected]
M. A. KHAMSI*
Affiliation:
Department of Mathematical Sciences, The University of Texas at El Paso, El Paso, TX 79968, USA email [email protected]
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Abstract

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We extend the results of Schu [‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl.158 (1991), 407–413] to monotone asymptotically nonexpansive mappings by means of the Fibonacci–Mann iteration process

$$\begin{eqnarray}x_{n+1}=t_{n}T^{f(n)}(x_{n})+(1-t_{n})x_{n},\quad n\in \mathbb{N},\end{eqnarray}$$
where $T$ is a monotone asymptotically nonexpansive self-mapping defined on a closed bounded and nonempty convex subset of a uniformly convex Banach space and $\{f(n)\}$ is the Fibonacci integer sequence. We obtain a weak convergence result in $L_{p}([0,1])$, with $1<p<+\infty$, using a property similar to the weak Opial condition satisfied by monotone sequences.

Type
Research Article
Copyright
© 2017 Australian Mathematical Publishing Association Inc. 

Footnotes

The authors acknowledge the support provided by the deanship of scientific research at King Fahd University of Petroleum and Minerals in funding this work through project no. IN141040.

References

Aksoy, A. and Khamsi, M. A., Nonstandard Methods in Fixed Point Theory (Springer, New York, 1990).Google Scholar
Alfuraidan, M. R. and Khamsi, M. A., ‘A fixed point theorem for monotone asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc., to appear.Google Scholar
Beauzamy, B., Introduction to Banach Spaces and Their Geometry (North-Holland, Amsterdam, 1985).Google Scholar
Carl, S. and Heikkilä, S., Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory (Springer, New York, 2011).CrossRefGoogle Scholar
Goebel, K. and Kirk, W. A., ‘A fixed point theorem for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 35 (1972), 171174.Google Scholar
Jachymski, J., ‘The contraction principle for mappings on a metric space with a graph’, Proc. Amer. Math. Soc. 136 (2008), 13591373.Google Scholar
Khamsi, M. A. and Kirk, W. A., An Introduction to Metric Spaces and Fixed Point Theory (John Wiley, New York, 2001).Google Scholar
Opial, Z., ‘Weak convergence of the sequence of successive approximations for nonexpansive mappings’, Bull. Amer. Math. Soc. 73 (1967), 591597.Google Scholar
Ran, A. C. M. and Reurings, M. C. B., ‘A fixed point theorem in partially ordered sets and some applications to matrix equations’, Proc. Amer. Math. Soc. 132(5) (2004), 14351443.Google Scholar
Schu, J., ‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl. 158 (1991), 407413.Google Scholar