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FIBONACCI–MANN ITERATION FOR MONOTONE ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

  • M. R. ALFURAIDAN (a1) and M. A. KHAMSI (a2)
Abstract

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 , using a property similar to the weak Opial condition satisfied by monotone sequences.

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mohamed@utep.edu
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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.

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References
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[1] Aksoy A. and Khamsi M. A., Nonstandard Methods in Fixed Point Theory (Springer, New York, 1990).
[2] Alfuraidan M. R. and Khamsi M. A., ‘A fixed point theorem for monotone asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc., to appear.
[3] Beauzamy B., Introduction to Banach Spaces and Their Geometry (North-Holland, Amsterdam, 1985).
[4] 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).
[5] Goebel K. and Kirk W. A., ‘A fixed point theorem for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 35 (1972), 171174.
[6] Jachymski J., ‘The contraction principle for mappings on a metric space with a graph’, Proc. Amer. Math. Soc. 136 (2008), 13591373.
[7] Khamsi M. A. and Kirk W. A., An Introduction to Metric Spaces and Fixed Point Theory (John Wiley, New York, 2001).
[8] Opial Z., ‘Weak convergence of the sequence of successive approximations for nonexpansive mappings’, Bull. Amer. Math. Soc. 73 (1967), 591597.
[9] 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.
[10] Schu J., ‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl. 158 (1991), 407413.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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