Iterative construction of fixed points of asymptotically nonexpansive mappings. (English) Zbl 0734.47036

Let T be a completely continuous and asymptotically non-expansive self- mapping (in the sense of Goebel and Kirk) of a nonempty closed bounded and convex subset of a Hilbert space. The author gives conditions under which a fixed point of T may be obtained as limit of the Mann-type iterates \(x_{n+1}=\alpha_ nT^ n(x_ n)+(1-\alpha_ n)x_ n.\) A parallel result is obtained for a new class of operators (called “asymptotically pseudocontractive”) whose iterates admit a universal Lipschitz constant.


47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
Full Text: DOI


[1] Browder, F.E, Nonexpansive nonlinear operators in Banach space, (), 1041-1044 · Zbl 0128.35801
[2] Goebel, K; Kirk, W.A, A fixed point theorem for asymptotically nonexpansive mappings, (), 171-174, No. 1 · Zbl 0256.47045
[3] Goebel, K; Kirk, W.A, A fixed point theorem for transformations whose iterates have uniform Lipschitz constant, Studia math., 47, 137-140, (1973) · Zbl 0265.47044
[4] Göhde, D, Zum prinzip der kontraktiven abbildung, Math. nachr., 30, 251-258, (1965) · Zbl 0127.08005
[5] Groetsch, C.W, A note on segmenting Mann iterates, J. math. anal. appl., 40, 369-372, (1972) · Zbl 0244.47042
[6] Ishikawa, S, Fixed points by a new iteration method, (), 147-150, No. 1 · Zbl 0286.47036
[7] Ishikawa, S, Fixed points and iteration of a nonexpansive mapping in a Banach space, (), 65-71, No. 1 · Zbl 0352.47024
[8] Kirk, W.A, A fixed point theorem for mappings which do not increase distance, Amer. math. monthly, 72, 1004-1006, (1965) · Zbl 0141.32402
[9] Kirk, W.A, Krasnoselskii’s iteration process in hyperbolic space, Numer. fund. anal. optim., 4, No. 4, 371-381, (1982) · Zbl 0505.47046
[10] Qihou, L, On naimpally and Singh’s open questions, J. math. anal. appl., 124, 157-164, (1987) · Zbl 0625.47044
[11] Reinermann, J, Über fixpunkte kontrahierender abbildungen und schwach konvergente Toeplitz-verfahren, Arch. math., 20, 59-64, (1969) · Zbl 0174.19401
[12] Rhoades, B.E, Comments on two fixed point iteration methods, J. math. anal. appl., 56, 741-750, (1976) · Zbl 0353.47029
[13] Schöneberg, R, Fixpunktsätze für einige klassen kontraktionsartiger operatoren in banachräumen über einen fixpunktindex, eine zentrumsmethode und die fixpunkttheorie nichtexpansiver abbildungen, Dissertation RWTH, (1977), Aachen
[14] Vijayaraju, P, Fixed point theorems for asymptotically nonexpansive mappings, Bull. cal. math. soc., 80, 133-136, (1988) · Zbl 0667.47032
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.