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Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces. (English) Zbl 1101.47053

Let \(E\) be a reflexive Banach space with a uniformly Gâteaux differentiable norm, \(C\) a closed convex subset of \(E\), \(f: C \to C\) a contraction, and \(T_1, \dots, T_N\) nonexpansive mappings from \(C\) into \(C\). Suppose that the set \(F\) of common fixed points of \(T_1, \dots, T_N\) is nonempty. The authors study the iterative process \(x_{n+1}=\lambda_{n+1} f(x_n)+ (1-\lambda_{n+1}) T_{n+1} x_n\), where \(x_0 \in C\), \(\{ \lambda_n \} \in (0,1)\) and \(T_n:= T_{n \text{ {mod}} N}\). Under appropriate conditions on \(\{ \lambda_n \}\), they prove that \(\{ x_n\}\) converges strongly to an element of \(F\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
49M05 Numerical methods based on necessary conditions
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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[1] Barbu, V.; Precupanu, Th., Convexity and Optimization in Banach Spaces (1978), Editura Academiei R. S. R.: Editura Academiei R. S. R. Bucharest · Zbl 0379.49010
[2] Bauschke, H. H., The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl., 202, 150-159 (1996) · Zbl 0956.47024
[3] Browder, F. E., Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal., 24, 82-90 (1967) · Zbl 0148.13601
[4] Cho, Y. J.; Kang, S. M.; Zhou, H. Y., Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal., 12, 2, 27-34 (2005) · Zbl 1088.47053
[5] Goebel, K.; Kirk, W. A., Topics in metric fixed point theory, (Cambridge Studies in Advanced Mathematics, vol. 28 (1990), Cambridge University Press: Cambridge University Press Cambridge, UK) · Zbl 0708.47031
[6] Goebel, K.; Reich, S., Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings (1984), Marcel Dekker: Marcel Dekker New York, Basel · Zbl 0537.46001
[7] Ha, K. S.; Jung, J. S., Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl., 147, 330-339 (1990) · Zbl 0712.47045
[8] Halpern, B., Fixed points of nonexpansive maps, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101
[9] 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
[10] Jung, J. S.; Cho, Y. J.; Agarwal, R. P., Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl., 2005, 2, 125-135 (2005) · Zbl 1109.47056
[11] Jung, J. S.; Kim, T. H., Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc., 34, 1, 93-102 (1997)
[12] Jung, J. S.; Morales, C., The Mann process for perturbed \(m\)-accretive operators in Banach spaces, Nonlinear Anal., 46, 231-243 (2001) · Zbl 0997.47042
[13] Lions, P. L., Approximation de points fixes de contractions, C. R. Acad. Sci. Sér. A-B, Paris, 284, 1357-1359 (1977) · Zbl 0349.47046
[14] Liu, L. S., Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl., 194, 114-125 (1995) · Zbl 0872.47031
[15] Moudafi, A., Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241, 46-55 (2000) · Zbl 0957.47039
[16] O’Hara, J. G.; Pillay, P.; Xu, H. K., Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal., 54, 1417-1426 (2003) · Zbl 1052.47049
[17] Reich, S., Product formulas, nonlinear semigroup and accretive operators, J. Funct. Anal., 36, 147-168 (1980) · Zbl 0437.47048
[18] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75, 287-292 (1980) · Zbl 0437.47047
[19] Shioji, N.; Takahashi, W., Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125, 12, 3641-3645 (1997) · Zbl 0888.47034
[20] Takahashi, W.; Ueda, Y., On Reich’s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl., 104, 546-553 (1984) · Zbl 0599.47084
[21] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 59, 486-491 (1992) · Zbl 0797.47036
[22] Xu, H. K., Iterative algorithms for nonlinear operators, J. London Math. Soc., 66, 2, 240-256 (2002) · Zbl 1013.47032
[23] Xu, H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 3, 659-678 (2003) · Zbl 1043.90063
[24] Xu, H. K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298, 279-291 (2004) · Zbl 1061.47060
[25] H.Y. Zhou, L. Wei, Y.J. Cho, Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press.; H.Y. Zhou, L. Wei, Y.J. Cho, Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press. · Zbl 1100.65049
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