Abstract
We prove that Mann and Ishikawa iterations are equivalent models dealing with \(\psi\)-uniformly pseudocontractive or d-weakly contractive maps without bounded range
Authors
Stefan M. Soltuz
Tiberiu Popoviciu Institute of Numerical Analysis
B.E. Rhoades
Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, U.S.A
Keywords
\(\psi\)-uniformly pseudocontractive maps; d-weakly contractivemap;~ Mannand Ishikawa iterations
Paper coordinates
B.E. Rhoades, Ş.M. Şoltuz, The equivalence of Mann and Ishikawa iterations dealing with uniformly pseudocontractive maps without bounded range, Tamkang J. Math. 37 (3) (2006).
About this paper
Journal
Tamkang Journal of Mathematics
Publisher Name
Tamkang University Tamsui, Taiwan, R.O.C.
Print ISSN
0049-2930
Online ISSN
2073-9826
google scholar link
[1] Ya. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operatorequations in Banach spaces,Panamer. Math. J.4(1994), 39-54.
[2] Ya. I. Alber and S. Guerre-Delabriere,Principle of weakly contractive maps in Hilbertspaces, Operator Theory98(1997), 7-22.
[3] Ya. I. Alber and S. Guerre-Delabriere,On the projection methods for fixed point problems,Analysis21(2001), 17-39.
[4] M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977, page65.
[5] C. E. Chidume, H. Zegeye and S. J. Aneke,Approximation of fixed points of weakly con-tractive nonself maps in Banach spaces, J. Math. Anal. Appl.270(2002), 189-199.
[6] C. E. Chidume and H. Zegeye,Approximation methods for nonlinear operator equations,Proc. Amer. Math. Soc.131(2003), 2467-2478.
[7] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, 1985, page 115.
[8] S. Ishikawa,Fixed points by a new iteration method, Proc. Amer. Math. Soc.44(1974),147-150.
[9] W. R. Mann,Mean value in iteration, Proc. Amer. Math. Soc.4(1953), 506-510.
[10] C. Morales and J. S. Jung,Convergence of paths for pseudocontractive mappings in Banachspaces,Proc. Amer. Math. Soc.128(2000), 3411-3419.
[11] B. E. Rhoades and S. M. S ̧oltuz,On the equivalence of Mann and Ishikawa iterationmethods,Int. J. Math. Math. Sci.2003(7), 451-459.
[12] B. E. Rhoades and S. M. S ̧oltuz,The equivalence of Mann and Ishikawa iteration forψ–uniformly pseudocontractive orψ–uniformly accretive maps,Int. J. Math. Math. Sci.46(2004), 2443-2452.