## Approximation of the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings.(English)Zbl 1423.47055

Summary: We introduce an iterative process which converges strongly to the common minimum-norm fixed point of a finite family of asymptotically nonexpansive mappings. As a consequence, convergence result to a common minimum-norm fixed point of a finite family of nonexpansive mappings is proved.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:

### References:

 [1] Censor, Y; Elfving, T, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8, 221-239, (1994) · Zbl 0828.65065 [2] Byrne, C, Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl, 18, 441-453, (2002) · Zbl 0996.65048 [3] Censor, Y; Bortfeld, T; Martin, B; Trofimov, A, A unified approach for inversion problem in intensity-modulated radiation therapy, Phys. Med. Biol, 51, 2353-2365, (2006) [4] Goebel, K; Kirk, WA, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. Math. Soc, 35, 171-174, (1972) · Zbl 0256.47045 [5] Browder, FE, Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal, 24, 82-90, (1967) · Zbl 0148.13601 [6] Halpern, B, Fixed points of nonexpansive maps, Bull. Am. Math. Soc, 73, 957-961, (1967) · Zbl 0177.19101 [7] Wittmann, R, Approximation of fixed point of nonexpansive mappings, Arch. Math, 58, 486-491, (1992) · Zbl 0797.47036 [8] Shimizu, T; Takahashi, W, Strong convergence theorems for asymptotically nonexpansive semigroups in Hilbert spaces, Nonlinear Anal, 34, 87-99, (1998) · Zbl 0935.47039 [9] Bruck, RE; Kuczumow, T; Reich, S, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math, 65, 169-179, (1993) · Zbl 0849.47030 [10] Lim, TC; Xu, HK, Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal, 22, 1345-1355, (1994) · Zbl 0812.47058 [11] Morales, CH; Jung, JS, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Am. Math. Soc, 128, 3411-3419, (2000) · Zbl 0970.47039 [12] Reich, S, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl, 75, 287-292, (1980) · Zbl 0437.47047 [13] Schu, J, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl, 158, 407-413, (1991) · Zbl 0734.47036 [14] Schu, J, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Aust. Math. Soc, 43, 153-159, (1991) · Zbl 0709.47051 [15] Shioji, N; Takahashi, W, A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Arch. Math, 72, 354-359, (1999) · Zbl 0940.47045 [16] Shioji, N; Takahashi, W, Strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces, J. Approx. Theory, 97, 53-64, (1999) · Zbl 0932.47042 [17] 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 [18] Tan, KK; Xu, HK, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Am. Math. Soc, 122, 733-739, (1994) · Zbl 0820.47071 [19] Yang, X; Liou, Y-C; Yao, Y, Finding minimum norm fixed point of nonexpansive mappings and applications, No. 2011, (2011) · Zbl 1216.47102 [20] Yao, Y; Xu, H-K, Iterative methods for finding minimum-norm fixed points of nonexpansive mappings with applications, Optimization, 60, 645-658, (2011) · Zbl 1368.47093 [21] Zegeye, H, A hybrid iteration method for equilibrium, variational inequality problems and common fixed point problems in Banach spaces, Nonlinear Anal, 72, 2136-2146, (2010) · Zbl 1225.47121 [22] Takahashi W: Nonlinear Functional Analysis-Fixed Point Theory and Applications. Yokohama Publishers, Yokohama; 2000. · Zbl 0997.47002 [23] Chang, SS; Cho, YJ; Zhou, H, Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean Math. Soc, 38, 1245-1260, (2001) · Zbl 1020.47059 [24] Ohara, JG; Pillay, P; Xu, HK, Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Anal, 64, 2022-2042, (2006) · Zbl 1139.47056 [25] Maingé, PE, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal, 16, 899-912, (2008) · Zbl 1156.90426
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.