×

zbMATH — the first resource for mathematics

Iterative methods for strict pseudo-contractions in Hilbert spaces. (English) Zbl 1133.47050
This article deals with two iterative algorithms of finding a common fixed points for \(N\) strict pseudo-contractions \(\{T_i\}_{i=1}^N\) defined on a closed convex subset \(C\) of a real Hilbert space \(H\) (an operator \(T: C \to C\) is a strict pseudo-contraction, if there exists a constant \(0 \leq k < 1\) such that \(\| Tx - Ty\| ^2 \leq \| x - y\| ^2 + k\| (I - T)x - (I - T)y\| ^2\)). The first algorithm, called parallel, is defined by the formula \[ x_{n+1} = \alpha_nx_n + (1 - \alpha_n) \sum_{i=1}^N \lambda_i^{(n)} T_ix_n, \;x_0 \in C,\;\lambda_i^{(n)} > 0, \;\lambda_1^{(n)} + \cdots + \lambda_N^{(n)} = 1;\tag{1} \] the second one, called cyclic, by the formula
\[ x_{n+1} = \alpha_nx_n + (1 - \alpha) T_{[n]}x_n, \quad x_0 \in C, \quad T_{[n]} = T_i, \;i = n(\text{ mod} \, N), \;1 \leq i \leq N.\tag{2} \] The main results describe (provided that \(F = \bigcap_{i=1}^N \text{Fix} (T_i) \neq \emptyset\)) conditions on the control sequence \(\{\alpha_n\}\) so that the approximations \(x_n\) converge weakly to a common fixed point of \(\{T_i\}_{i=1}^N\). At the end of the article, some modifications of algorithms (1) and (2) are proposed; it is proved that approximations \(x_n\) for these modified algorithms converge strongly to \(P_Fx_0\), where \(P_F\) is the nearest point projection from \(H\) onto \(F\).

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65J15 Numerical solutions to equations with nonlinear operators
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bauschke, 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
[2] Browder, F.E.; Petryshyn, W.V., Construction of fixed points of nonlinear mappings in Hilbert spaces, J. math. anal. appl., 20, 197-228, (1967) · Zbl 0153.45701
[3] Byrne, C., A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse problems, 20, 103-120, (2004) · Zbl 1051.65067
[4] Genel, A.; Lindenstrauss, J., An example concerning fixed points, Israel J. math., 22, 81-86, (1975) · Zbl 0314.47031
[5] Goebel, K.; Kirk, W.A., ()
[6] Goebel, K.; Reich, S., Uniform convexity, hyperbolic geometry, and nonexpansive mappings, (1984), Marcel Dekker · Zbl 0537.46001
[7] Güler, O., On the convergence of the proximal point algorithm for convex optimization, SIAM J. control optim., 29, 403-419, (1991) · Zbl 0737.90047
[8] Halpern, B., Fixed points of nonexpanding maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[9] Ishikawa, S., Fixed points by a new iteration method, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036
[10] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2003) · Zbl 1101.90083
[11] Kim, T.H.; Xu, H.K., Strong convergence of modified Mann iterations, Nonlinear anal., 61, 51-60, (2005) · Zbl 1091.47055
[12] Kim, T.H.; Xu, H.K., Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear anal., 64, 1140-1152, (2006) · Zbl 1090.47059
[13] Lions, P.L., Approximation de points fixes de contractions, C. R. acad. sci. Sèr. A-B Paris, 284, 1357-1359, (1977) · Zbl 0349.47046
[14] Mann, W.R., Mean value methods in iteration, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[15] Marino, G.; Xu, H.K., Convergence of generalized proximal point algorithms, Comm. pure appl. anal., 3, 791-808, (2004) · Zbl 1095.90115
[16] G. Marino, H.K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. (2006), in press (doi:10.1016/j.jmaa.2006.06.055). Available online 27 July 2006
[17] Matinez-Yanes, C.; Xu, H.K., Strong convergence of the CQ method for fixed point processes, Nonlinear anal., 64, 2400-2411, (2006) · Zbl 1105.47060
[18] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[19] 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
[20] 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
[21] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 67, 274-276, (1979) · Zbl 0423.47026
[22] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047
[23] Scherzer, O., Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems, J. math. anal. appl., 194, 911-933, (1991) · Zbl 0842.65036
[24] Shioji, N.; Takahashi, W., Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. amer. math. soc., 125, 3641-3645, (1997) · Zbl 0888.47034
[25] Solodov, M.V.; Svaiter, B.F., Forcing strong convergence of proximal point iterations in a Hilbert space, Math. program. ser. A, 87, 189-202, (2000) · Zbl 0971.90062
[26] Tan, K.K.; Xu, H.K., Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. math. anal. appl., 178, 2, 301-308, (1993) · Zbl 0895.47048
[27] Tan, K.K.; Xu, H.K., Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. amer. math. soc., 122, 733-739, (1994) · Zbl 0820.47071
[28] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036
[29] Xu, H.K., Iterative algorithms for nonlinear operators, J. London math. soc., 66, 240-256, (2002) · Zbl 1013.47032
[30] Xu, H.K., Remarks on an iterative method for nonexpansive mappings, Comm. appl. nonlinear anal., 10, 1, 67-75, (2003) · Zbl 1035.47035
[31] Xu, H.K., Strong convergence of an iterative method for nonexpansive mappings and accretive operators, J. math. anal. appl., 314, 631-643, (2006) · Zbl 1086.47060
[32] Xu, H.K., Strong convergence of approximating fixed point sequences for nonexpansive mappings, Bull. austral. math. soc., 74, 143-151, (2006) · Zbl 1126.47056
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.