×

zbMATH — the first resource for mathematics

Shrinking projection methods for firmly nonexpansive mappings. (English) Zbl 1238.47043
Summary: We study the shrinking projection method for firmly nonexpansive mappings. The method gives us a strong convergence iteration for a family of firmly nonexpansive mappings and also permit us to obtain a sufficient condition for the existence of a fixed point of a firmly nonexpansive mapping.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[2] Matsushita, S.; Takahashi, W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. approx. theory, 134, 257-266, (2005) · Zbl 1071.47063
[3] Martinez-Yanes, C.; Xu, H.K., Strong convergence of the CQ method for fixed point iteration processes, Nonlinear anal., 64, 2400-2411, (2006) · Zbl 1105.47060
[4] Kohsaka, F.; Takahashi, W., Approximating common fixed points of countable families of strongly nonexpansive mappings, Nonlinear stud., 14, 219-234, (2007) · Zbl 1147.47050
[5] Aoyama, K.; Takahashi, W., Strong convergence theorems for a family of relatively nonexpansive mappings in Banach spaces, Fixed point theory, 8, 143-160, (2007) · Zbl 1143.47044
[6] Solodov, M.V.; Svaiter, B.F., Forcing strong convergence of proximal point iterations in a Hilbert space, Math. program., 87, 189-202, (2000) · Zbl 0971.90062
[7] Bauschke, H.H.; Combettes, P.L., A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces, Math. oper. res., 26, 248-264, (2001) · Zbl 1082.65058
[8] Ohsawa, S.; Takahashi, W., Strong convergence theorems for resolvents of maximal monotone operators in Banach spaces, Arch. math. (basel), 81, 439-445, (2003) · Zbl 1067.47080
[9] Kamimura, S.; Takahashi, W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM J. optim., 13, 938-945, (2002), (electronic) (2003) · Zbl 1101.90083
[10] Kohsaka, F.; Takahashi, W., Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. appl. anal., 239-249, (2004) · Zbl 1064.47068
[11] Takahashi, W.; Takeuchi, Y.; Kubota, R., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. math. anal. appl., 341, 276-286, (2008) · Zbl 1134.47052
[12] Aoyama, K.; Kohsaka, F.; Takahashi, W., Strong convergence theorems by shrinking and hybrid projection methods for relatively nonexpansive mappings in Banach spaces, (), 7-26 · Zbl 1269.47050
[13] Matsushita, S.; Takahashi, W., A proximal-type algorithm by the hybrid method for maximal monotone operators in a Banach space, (), 355-365 · Zbl 1152.47056
[14] Goebel, K.; Kirk, W.A., Topics in metric fixed point theory, () · Zbl 0708.47031
[15] Bruck, R.E.; Reich, S., Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. math., 3, 459-470, (1977) · Zbl 0383.47035
[16] Kimura, Y.; Takahashi, W., A generalized proximal point algorithm and implicit iterative schemes for a sequence of operators on Banach spaces, Set-valued anal., 16, 597-619, (2008) · Zbl 1169.47050
[17] Takahashi, W., Nonlinear functional analysis, (2000), Yokohama Publ. Yokohama
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.