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

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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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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

A. Aksoy and M. A. Khamsi , Nonstandard Methods in Fixed Point Theory (Springer, New York, 1990).

S. Carl and S. Heikkilä , Fixed Point Theory in Ordered Sets and Applications: From Differential and Integral Equations to Game Theory (Springer, New York, 2011).

K. Goebel and W. A. Kirk , ‘A fixed point theorem for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 35 (1972), 171174.

J. Jachymski , ‘The contraction principle for mappings on a metric space with a graph’, Proc. Amer. Math. Soc. 136 (2008), 13591373.

M. A. Khamsi and W. A. Kirk , An Introduction to Metric Spaces and Fixed Point Theory (John Wiley, New York, 2001).

Z. Opial , ‘Weak convergence of the sequence of successive approximations for nonexpansive mappings’, Bull. Amer. Math. Soc. 73 (1967), 591597.

A. C. M. Ran and M. C. B. Reurings , ‘A fixed point theorem in partially ordered sets and some applications to matrix equations’, Proc. Amer. Math. Soc. 132(5) (2004), 14351443.

J. Schu , ‘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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