Abstract
The purpose of this paper is to introduce a general iterative algorithm by viscosity method to approximate a common point of a finite family of m-accretive mappings in a reflexive Banach space which has a weakly continuous duality mapping. We obtain strong convergence theorems under some mild conditions imposed on parameters.
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References
R.E. Bruck, Nonexpansive projections on subsets of Banach spaces, Pacific J. Math., 47 (1973), 341–355.
F.E. Browder, Fixed points theorems for noncompact mappings in Hilbert spaces, Proc. Natl. Acad. Sci. USA, 53 (1965), 1272–1276.
F.E. Browder, Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Arch. Ration. Mech. Anal., 24 (1967), 82–90.
F.E. Browder, Covergence theorems for sequences of nonlinear operators in Banach space, Math. Z., 100 (1967) 201–225.
T.C. Lim, H.K. Xu, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal., 22 (1994), 1345–1355.
F.E. Browder, Covergence theorems for sequences of nonlinear operators in Banach space, Math. Z., 100 (1967), 201–225.
R. Chen, Z. Zhu, Viscosity approximation fixed points for nonexpansive and m-accretive operators, doi:10.1155/FPTA/2006/81325.
S.S. Chang, Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 323 (2006), 1402–1416.
T.D. Benavides, G.L. Acedo, H.K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr., 248–249 (2003), 62–71.
T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations, Nonlinear Appl., 61 (2005), 51–60.
A. Moudafi, Viscosity approximation methods for fixed points problems, J. Math. Anal. Appl., 241 (2000), 46–55.
X. Qin, Y. Su, Approximation of zero point of accretive operator in Banach spaces, J. Math. Anal. Appl., 329 (2007), 415–424.
S. Reich, Asymptotic behavior of contractions in banach spaces, J. Math. Anal. Appl., 44 (1973), 57–70.
T. Suzuki, Strong convergence theorems infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl. (2005), 103–123.
Y. Song, R. Chen, H. Zhou, Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces, Nonlinear Anal., 66 (2007), 1016–1024.
S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506–515.
W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama (2000).
H.K. Xu, Strong convergence of an iterative method for nonexpansive and accretive opertors, J. Math. Anal. Appl., 314 (2006), 631–643.
H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279–291.
H.K. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., 66 (2002), 240–256.
H. Zegeyea, N. Shahzad, Strong convergence theorems for a common zero of a finite family of m-accretive mappings, Nonlinear Anal., 66 (2007), 1161–1169.
H. Yao, M.A. Noor, On viscosity iterative methods for variational inequalities, J. Math. Anal. Appl., 325 (2007), 776–787.
Y. Yao, A general iterative method for a finite family of nonexpansive mappings, Nonlinear Anal., 66 (2007), 2676–2687.
Y. Yao, R. Chen, H. Zhou, Iterative algorithms to fixed point of nonexpansive mapping, Acta Math. Sinica, Chinese Series, 50 (2007), 139–144.
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Cho, Y.J., Qin, X. Viscosity approximation methods for a family of m-accretive mappings in reflexive Banach spaces. Positivity 12, 483–494 (2008). https://doi.org/10.1007/s11117-007-2181-8
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DOI: https://doi.org/10.1007/s11117-007-2181-8