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FIBONACCI–MANN ITERATION FOR MONOTONE ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

Part of: Normed linear spaces and Banach spaces; Banach lattices Nonlinear operators and their properties

Published online by Cambridge University Press:  13 March 2017

M. R. ALFURAIDAN  and
M. A. KHAMSI
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]
*
✉[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.

Keywords

asymptotically nonexpansive mappingFibonacci sequencefixed pointMann iteration processmonotone Lipschitzian mappingOpial conditionuniformly convex Banach space

MSC classification

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc.
Secondary: 46B20: Geometry and structure of normed linear spaces 47H10: Fixed-point theorems
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 96 , Issue 2 , October 2017 , pp. 307 - 316
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

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