The Equivalence between the Mann and Ishikawa Iterations for Generalized Contraction Mappings in a Cone

L. Jones Tarcius Doss, T. Esakkiappan
DOI: 10.4236/am.2011.211192                

Abstract

In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping in cone subset of a real Banach space is discussed.

Keywords

Mann Iteration, Ishikawa Iterations, Generalized Contraction, Cone

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	W. R. Mann, “Mean Value Methods in Iteration,” Proceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3
[2]	S. Ishikawa, “Fixed Points by a New Iteration Method,” Proceedings of the American Mathematical Society, Vol. 44, No. 1, 1974, pp. 147-150. doi:10.1090/S0002-9939-1974-0336469-5
[3]	L.-G. Huang and X. Zhang, “Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings,” Journal of Mathematical Analysis and Applications, Vol. 332, No. 2, 2007, pp.1468-1476. doi:10.1016/j.jmaa.2005.03.087
[4]	X. Weng, “Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping,” Proceedings of the American Mathematical Society, Vol. 113, No. 3, 1991, pp. 727-731. doi:10.1090/S0002-9939-1991-1086345-8
[5]	B. E. Rhoades, “Convergence of an Isikawa-Type Iteration Scheme for a Generalized Contraction,” Journal of Mathematical Analysis and Applications, Vol. 185, No. 2, 1994, pp. 350-355. doi:10.1006/jmaa.1994.1253
[6]	B. E. Rhoades and S. M. Soltuz, “The Eqivalence between Mann and Ishikawa Iterations Dealing with Generalized Contractions,” International Journal of Mathematics and Mathematical Sciences, Vol. 2006, 2006, pp. 1-5.