Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces. (English) Zbl 1111.47057
By using viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, the author obtains sufficient and necessary conditions for the iterative sequence \( x_{n+1} = \alpha _{n+1}f(x_{n}) + (1 - \alpha _{n+1})T_{n+1}x_{n}\) to converge strongly to a common fixed point of the family. His statements extend and improve some recent results.
Reviewer: Edward Prempeh (Kumasi)
MSC:
47J25 | Iterative procedures involving nonlinear operators |
47H10 | Fixed-point theorems |
47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
47H05 | Monotone operators and generalizations |
Keywords:
finite family of nonexpansive maps; common fixed points; viscosity approximation method; strong convergence; uniformly smooth Banach spaceReferences:
[1] | Bauschke, H. H., The approximation of fixed points of compositions of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 202, 150-159 (1996) · Zbl 0956.47024 |
[2] | Browder, F. E., Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal., 24, 82-90 (1967) · Zbl 0148.13601 |
[3] | Chang, S. S., Some problems and results in the study of nonlinear analysis, Nonlinear Anal., 30, 7, 4197-4208 (1997) · Zbl 0901.47036 |
[4] | Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040 |
[5] | Goebel, K.; Kirk, W. A., Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., vol. 28 (1990), Cambridge Univ. Press · Zbl 0708.47031 |
[6] | Goebel, K.; Reich, S., Uniform Convexity, Nonexpansive Mappings and Hyperbolic Geometry (1984), Dekker · Zbl 0537.46001 |
[7] | Halpern, B., Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101 |
[8] | Jung, J. S., Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 302, 509-520 (2005) · Zbl 1062.47069 |
[9] | Lions, P. L., Approximation de points fixes de contractions’, C. R. Acad. Sci. Paris Sér. A, 284, 1357-1359 (1977) · Zbl 0349.47046 |
[10] | Moudafi, A., Viscosity approximation methods for fixed point problems, J. Math. Anal. Appl., 241, 46-55 (2000) · Zbl 0957.47039 |
[11] | Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75, 128-292 (1980) · Zbl 0437.47047 |
[12] | Wang, X., Fixed point iteration for local strictly pseudo-contractive mappings, Proc. Amer. Math. Soc., 113, 727-731 (1991) · Zbl 0734.47042 |
[13] | Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036 |
[14] | Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060 |
[15] | Xu, H. K., Remark on an iterative method for nonexpansive mappings, Comm. Appl. Nonlinear Anal., 10, 67-75 (2003) · Zbl 1035.47035 |
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