# zbMATH — the first resource for mathematics

The common minimal-norm fixed point of a finite family of nonexpansive mappings. (English) Zbl 1214.47050
Let $$C$$ be a closed convex subset of a real Hilbert space $$H$$ and $$\{T_1,\dots,T_N\}$$ be a finite family of nonexpansive self-mappings on $$C$$ such that the set of common fixed points
$F:=\bigcap_{i=1}^N \text{Fix}(T_i)\neq \emptyset.$ Under some appropriate conditions, the authors prove that there exists a unique minimum-norm element $$x^†\in F$$, that is,
$\|x^†\|=\min \big\{\|x\|:x\in F\big\}.$ They give both cyclic and parallel iteration methods to find this minimal-norm element.
Reviewer: Long Wei (Jiangxi)

##### MSC:
 47H10 Fixed-point theorems 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47J25 Iterative procedures involving nonlinear operators
Full Text:
##### References:
 [1] 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 [2] Cui, Y.L.; Liu, X., Notes on browder’s and halpern’s methods for nonexpansive mappings, Fixed point theory, 10, 1, 89-98, (2009) · Zbl 1190.47068 [3] Halpern, B., Fixed points of nonexpanding maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101 [4] Lions, P.L., Approximation de points fixes de contractions, C. R. acad. sci., Paris Sér. A-B, 284, 1357-1359, (1977) · Zbl 0349.47046 [5] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 1, 287-292, (1980) · Zbl 0437.47047 [6] Reich, S., Approximating fixed points of nonexpansive mappings, Panamer. math. J., 4, 2, 23-28, (1994) · Zbl 0856.47032 [7] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036 [8] Xu, H.K., Iterative algorithms for nonlinear operators, J. lond. math. soc., 66, 240-256, (2002) · Zbl 1013.47032 [9] Xu, H.K., Another control condition in an iterative method for nonexpansive mappings, Bull. aust. math. soc., 65, 109-113, (2002) · Zbl 1030.47036 [10] Lopez, G.; Xu, H.K., Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear anal., 10, 1, 67-75, (2003) [11] Marino, G.; Xu, H.K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, Nonlinear anal., 10, 1, 67-75, (2003) [12] Marino, G.; Xu, H.K., A general iterative method for nonexpansive mappings in Hilbert spaces, J. math. anal. appl., 318, 43-52, (2006) · Zbl 1095.47038 [13] 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 [14] O’Hara, J.G.; Pillay, P.; Xu, H.K., Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear anal., 64, 2022-2042, (2006) · Zbl 1139.47056 [15] Xu, H.K., Remarks on an iterative method for nonexpansive mappings, Comm. appl. nonlinear anal., 10, 1, 67-75, (2003) · Zbl 1035.47035 [16] Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. math. anal. appl., 298, 279-291, (2004) · Zbl 1061.47060 [17] Xu, H.K., Strong convergence of an iterative method for nonexpansive and accretive operators, J. math. anal. appl., 314, 631-643, (2006) · Zbl 1086.47060 [18] Xu, H.K., A regularization method for the proximal point algorithm, J. global optim., 36, 115-125, (2006) · Zbl 1131.90062 [19] Xu, H.K., A variable krasnoselskii – mann algorithm and the multiple-set split feasibility problem, Inverse problems, 22, 2021-2034, (2006) · Zbl 1126.47057 [20] Geobel, K.; Kirk, W.A., () [21] Geobel, K.; Reich, S., Uniform convexity, nonexpansive mappings, and hyperbolic geometry, (1984), Dekker [22] Opial, Z., Weak convergence of the sequence of successive approximations of nonexpansive mappings, Bull. amer. math. soc., 73, 595-597, (1967) · Zbl 0179.19902 [23] Bauschke, H.H., The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. math. anal. appl., 202, 1, 150-159, (1996) · Zbl 0956.47024 [24] Suzuki, T., Some notes on bauschke’s condition, Nonlinear anal., 67, 2224-2231, (2007) · Zbl 1133.47049
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.