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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
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[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., (Topics in Metric Fixed Point Theory. Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics, vol. 28 (1990), Cambridge University Press) · Zbl 0708.47031
[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; 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
[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
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