×

Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. (English) Zbl 1052.47049

In this paper, the authors establish the result that the iteration scheme \(x_{n+1}=\lambda_{n+1}y+(1-\lambda_{n+1})T_{n+1}x_n\) for infinitely/finitely many nonexpansive mappings \(T_i\) in a Hilbert space converges to \(Py\), where \(P\) is the projection onto the intersection of the fixed point sets of the \(T_i\)’s. This generalizes the result of T. Shimizu and W. Takahashi [J. Math. Anal. Appl. 211, 71–83 (1997; Zbl 0883.47075)], a complementary result to a result of H. H. Bauschke [J. Math. Anal. Appl. 202, 150–159 (1996; Zbl 0956.47024)], by introducing a condition on the sequence of parameters (\(\lambda_n\)). This condition improves P.-L. Lions’ condition [C. R. Acad. Sci., Paris, Sér. A 284, 1357–1359 (1977; Zbl 0349.47046)].

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
65J15 Numerical solutions to equations with nonlinear operators
PDFBibTeX XMLCite
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] Bruck, R. E., A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32, 107-116 (1979) · Zbl 0423.47024
[3] Bruck, R. E., On the convex approximation property and the asymptotic behaviour of nonlinear contractions in Banach spaces, Israel J. Math., 38, 304-314 (1981) · Zbl 0475.47037
[4] Deutsch, F.; Yamada, I., Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numer. Funct. Anal. Optim., 19, 1,2, 33-56 (1998) · Zbl 0913.47048
[5] K. Goebel, W.A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, Cambridge, 1990.; K. Goebel, W.A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, Vol. 28, Cambridge University Press, Cambridge, 1990. · Zbl 0708.47031
[6] Halpern, B., Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73, 957-961 (1967) · Zbl 0177.19101
[7] Lions, P., Approximation de points fixes de contractions, C.R. Acad. Sci. Paris Sèr. A-B, 284, 1357-1359 (1977) · Zbl 0349.47046
[8] Reich, S., Some problems and results in fixed point theory, Contemp. Math., 21, 179-187 (1983) · Zbl 0531.47048
[9] Reich, S., Approximating fixed points of nonexpansive mappings, Panamerican Math. J., 4, 2, 23-28 (1994) · Zbl 0856.47032
[10] Shimizu, T.; Takahashi, W., Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl., 211, 71-83 (1997) · Zbl 0883.47075
[11] 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
[12] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036
[13] Xu, H. K., An iterative approach to quadratic optimization, J. Optim. Theory Appl., 116, 3, 659-678 (2003) · Zbl 1043.90063
[14] Yamada, I.; Ogura, N.; Yamashita, Y.; Sakaniwa, K., Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces, Numer. Funct. Anal. Optim., 19, 1, 165-190 (1998) · Zbl 0911.47051
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.