×

zbMATH — the first resource for mathematics

Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces. (English) Zbl 1140.47058
The author provides a complicated iteration process to obtain the strong convergence to a fixed point for a Lipschitzian pseudocontractive map with nonempty fixed point set defined on a closed convex subset of a Hilbert space.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Atsushiba, S.; Takahashi, W., Strong convergence of Mann’s-type iterations for nonexpansive semigroups in general Banach spaces, Nonlinear anal., 61, 881-899, (2005) · Zbl 1089.47052
[2] Browder, F.E., Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. natl. acad. sci. USA, 53, 1272-1276, (1965) · Zbl 0125.35801
[3] Browder, F.E., Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. ration. mech. anal., 24, 82-90, (1967) · Zbl 0148.13601
[4] 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
[5] Chidume, C.E.; Mutangadura, S.A., An example on the Mann iteration method for Lipschitz pseudo-contractions, Proc. amer. math. soc., 129, 2359-2363, (2001) · Zbl 0972.47062
[6] Halpern, B., Fixed points of nonexpansive maps, Bull. amer. math. soc., 73, 957-961, (1967) · Zbl 0177.19101
[7] Ishikawa, S., Fixed points by a new iteration method, Proc. amer. math. soc., 44, 147-150, (1974) · Zbl 0286.47036
[8] Kim, T.H.; Xu, H.K., Strong convergence of modified Mann iterations, Nonlinear anal., 61, 51-60, (2005) · Zbl 1091.47055
[9] Lan, K.Q.; Wu, J.H., Convergence of approximants for demicontinuous pseudo-contractive maps in Hilbert spaces, Nonlinear anal., 49, 737-746, (2002) · Zbl 1019.47040
[10] Lions, P.L., Approximation de points fixes de contractions, C. R. acad. sci. Sér. A-B, Paris, 284, 1357-1359, (1977) · Zbl 0349.47046
[11] Mann, W.R., Mean value methods in iterations, Proc. amer. math. soc., 4, 506-510, (1953) · Zbl 0050.11603
[12] Marino, G.; Xu, H.-K., Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. math. anal. appl., 329, 1, 336-346, (2007) · Zbl 1116.47053
[13] Nakajo, K.; Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. math. anal. appl., 279, 372-379, (2003) · Zbl 1035.47048
[14] Reich, S., Weak convergence theorems for nonexpansive mappings in Banach spaces, J. math. anal. appl., 75, 274-276, (1979) · Zbl 0423.47026
[15] Reich, S., Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. math. anal. appl., 75, 287-292, (1980) · Zbl 0437.47047
[16] Reich, S., Asymptotic behavior of contractions in Banach spaces, J. math. anal. appl., 44, 57-70, (1973) · Zbl 0275.47034
[17] Rhoades, B.E., Fixed point iterations using infinite matrices, Trans. amer. math. soc., 196, 162-176, (1974) · Zbl 0285.47038
[18] Wittmann, R., Approximation of fixed points of nonexpansive mappings, Arch. math., 58, 486-491, (1992) · Zbl 0797.47036
[19] Xu, H.-K., Remarks on an iterative method for nonexpansive mappings, Comm. appl. nonlinear anal., 10, 67-75, (2003) · Zbl 1035.47035
[20] Zhou, H.Y., Nonexpansive mappings and iterative methods in uniformly convex Banach spaces, Acta math. sinica, 20, 829-836, (2004) · Zbl 1083.47060
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.