## A fixed point theorem for monotone asymptotically nonexpansive mappings

HTML articles powered by AMS MathViewer

- by Monther Rashed Alfuraidan and Mohamed Amine Khamsi PDF
- Proc. Amer. Math. Soc.
**146**(2018), 2451-2456 Request permission

## Abstract:

Let $C$ be a nonempty, bounded, closed, and convex subset of a Banach space $X$ and $T: C \rightarrow C$ be a monotone asymptotically nonexpansive mapping. In this paper, we investigate the existence of fixed points of $T$. In particular, we establish an analogue to the original Goebel and Kirk’s fixed point theorem for asymptotically nonexpansive mappings.## References

- Mostafa Bachar and Mohamed Amine Khamsi,
*Fixed points of monotone mappings and application to integral equations*, Fixed Point Theory Appl. , posted on (2015), 2015:110, 7. MR**3368216**, DOI 10.1186/s13663-015-0362-x - S. Banach,
*Sur les opérations dans les ensembles abstraits et leurs applications*, Fund. Math. 3(1922), 133-181. - B. A. Bin Dehaish and M. A. Khamsi,
*Browder and Göhde fixed point theorem for monotone nonexpansive Mappings*, Fixed Point Theory and Applications 2016:20 DOI: 10.1186/s13663-016-0505-8 - Felix E. Browder,
*Nonexpansive nonlinear operators in a Banach space*, Proc. Nat. Acad. Sci. U.S.A.**54**(1965), 1041–1044. MR**187120**, DOI 10.1073/pnas.54.4.1041 - Siegfried Carl and Seppo Heikkilä,
*Fixed point theory in ordered sets and applications*, Springer, New York, 2011. From differential and integral equations to game theory. MR**2760654**, DOI 10.1007/978-1-4419-7585-0 - Michael Edelstein,
*The construction of an asymptotic center with a fixed-point property*, Bull. Amer. Math. Soc.**78**(1972), 206–208. MR**291917**, DOI 10.1090/S0002-9904-1972-12918-5 - K. Goebel and W. A. Kirk,
*A fixed point theorem for asymptotically nonexpansive mappings*, Proc. Amer. Math. Soc.**35**(1972), 171–174. MR**298500**, DOI 10.1090/S0002-9939-1972-0298500-3 - Dietrich Göhde,
*Zum Prinzip der kontraktiven Abbildung*, Math. Nachr.**30**(1965), 251–258 (German). MR**190718**, DOI 10.1002/mana.19650300312 - Jacek Jachymski,
*The contraction principle for mappings on a metric space with a graph*, Proc. Amer. Math. Soc.**136**(2008), no. 4, 1359–1373. MR**2367109**, DOI 10.1090/S0002-9939-07-09110-1 - W. A. Kirk,
*A fixed point theorem for mappings which do not increase distances*, Amer. Math. Monthly**72**(1965), 1004–1006. MR**189009**, DOI 10.2307/2313345 - Joram Lindenstrauss and Lior Tzafriri,
*Classical Banach spaces. II*, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 97, Springer-Verlag, Berlin-New York, 1979. Function spaces. MR**540367** - Juan J. Nieto and Rosana Rodríguez-López,
*Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations*, Order**22**(2005), no. 3, 223–239 (2006). MR**2212687**, DOI 10.1007/s11083-005-9018-5 - Zdzisław Opial,
*Weak convergence of the sequence of successive approximations for nonexpansive mappings*, Bull. Amer. Math. Soc.**73**(1967), 591–597. MR**211301**, DOI 10.1090/S0002-9904-1967-11761-0 - André C. M. Ran and Martine C. B. Reurings,
*A fixed point theorem in partially ordered sets and some applications to matrix equations*, Proc. Amer. Math. Soc.**132**(2004), no. 5, 1435–1443. MR**2053350**, DOI 10.1090/S0002-9939-03-07220-4 - Mihai Turinici,
*Fixed points for monotone iteratively local contractions*, Demonstratio Math.**19**(1986), no. 1, 171–180. MR**871128**

## Additional Information

**Monther Rashed Alfuraidan**- Affiliation: Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
- MR Author ID: 795781
- ORCID: 0000-0002-3641-290X
- Email: monther@kfupm.edu.sa
**Mohamed Amine Khamsi**- Affiliation: Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas 79968
- MR Author ID: 100900
- ORCID: 0000-0001-6787-7032
- Email: mohamed@utep.edu
- Received by editor(s): June 20, 2016
- Received by editor(s) in revised form: July 15, 2016
- Published electronically: February 28, 2018
- Communicated by: Mourad Ismail
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 2451-2456 - MSC (2010): Primary 46B20, 45D05; Secondary 47E10, 34A12
- DOI: https://doi.org/10.1090/proc/13385
- MathSciNet review: 3778148