×

Strong convergence of averaging iterations of nonexpansive nonself-mappings. (English) Zbl 1070.47056

Let \(H\) be a Hilbert space, \(C\) a nonempty closed convex subset of \(H\), and \(T\) a nonexpansive nonself-mapping \(C\) into \(H\). Two iteration processes \[ \begin{gathered} x_{n+1}= {1\over n+1} \sum^n_{j=0} (\alpha_n x+(1- \alpha_n)(PT)^j x_n),\quad n= 0,1,2,\dots,\tag{1}\\ y_{n+1}= {1\over n+1} \sum^n_{j=0} P(\alpha_n y+(1- \alpha_n)(TP)^j y_n),\quad n= 0,1,2,\dots,\tag{2}\end{gathered} \] where \(x_0\), \(x\), \(y_0\), \(y\) are elements of \(C\), \(0\leq \alpha_n\leq 1\), and \(P\) is the metric projection from \(H\) onto \(C\), are considered. Using the nowhere normal outward condition, it is proved that the sequences \(\{x_n\}\) and \(\{y_n\}\) generated by (1) and (2), respectively, converge strongly as \(n\to\infty\) to a fixed point of \(T\) provided that the set of all fixed points of \(T\) is nonempty.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
65J15 Numerical solutions to equations with nonlinear operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Halpern, B. R.; Bergman, G. M., A fixed-point theorem for inward and outward maps, Trans. Amer. Math. Soc., 130, 353-358 (1968) · Zbl 0153.45602
[2] Matsushita, S.; Kuroiwa, D., Approximation of fixed points of nonexpansive nonself-mappings, Sci. Math. Jpn., 57, 171-176 (2003) · Zbl 1035.47034
[3] Shimizu, T.; Takahashi, W., Strong convergence theorem for asymptotically nonexpansive mappings, Nonlinear Anal., 26, 265-272 (1996) · Zbl 0861.47030
[4] 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
[5] Shioji, N.; Takahashi, W., A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Arch. Math., 72, 354-359 (1999) · Zbl 0940.47045
[6] Takahashi, W., Convex Analysis and Approximation of Fixed Points (2000), Yokohama Publishers: Yokohama Publishers Yokohama, (in Japanese)
[7] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. Math., 58, 486-491 (1992) · Zbl 0797.47036
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.