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Fixed points of mappings satisfying semicontractivity conditions

Part of: Nonlinear operators and their properties

Published online by Cambridge University Press:  09 April 2009

Jürgen Schu
Jürgen Schu
Affiliation:
Nosenberger Straße 67 D-4000 Düsseldorf 30 Federal Republic of, Germany
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Abstract

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Let A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

Keywords

Fixed pointsemicontractive mappingasymptotically nonexpansive mappingasymptotically semicontractive mappinguniformly convex Banach spaceuniformly smooth Banach spaceweakly sequentially continuous duality mappingOpial's condition.

MSC classification

Secondary: 47H10: Fixed-point theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 1 , August 1992 , pp. 25 - 38
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Bose, S. C., ‘Weak convergence to the fixed point of an asymptotically nonexpansive map’, Proc. Amer. Math. Soc. 68 (1978), 305–308.Google Scholar
[2]Bose, S. C. and Laskar, S. K., ‘Fixed point theorems for certain class of mappings’, J. Math. Phys. Sci. 19 (1985), 503–509.Google Scholar
[3]Browder, F.E., ‘Fixed point theorems for nonlinear semicontractive mappings in Banach spaces’, Arch. Rational Mech. Anal. 21 (1966), 259–269.Google Scholar
[4]Diestel, J., Geometry of Banach Spaces - Selected Topics, Lecture Notes in Math. 485 (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[5]Goebel, K. and Kirk, W.A., ‘A fixed point theorem for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 35 (1) (1972), 171–174.CrossRefGoogle Scholar
[6]Goebel, K. and Reich, S., Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Pure Appl. Math. Vol. 83, (Marcel Dekker, Inc., New York and Basel, 1984).Google Scholar
[7]Górnicki, J., ‘Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces’, Comment. Math. Univ. Carolin. 30 (2) (1989), 249–252.Google Scholar
[8]Gossez, J.-P. and Dozo, E. Lami, ‘Some geometric properties related to the fixed point theory for nonexpansive mappings’, Pacific J. Math. 40 (1972), 565–573.CrossRefGoogle Scholar
[9]Hirano, N. and Takahashi, W., ‘Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces’, Kodai Math. J. 2 (1979), 11–25.CrossRefGoogle Scholar
[10]Opial, Z., ‘Weak convergence of the sequence of successive approximations for nonexpansive mappings’, Bull. Amer. Math. Soc. 73 (1967), 591–597.CrossRefGoogle Scholar
[11]Passty, G.B., ‘Construction of fixed points for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 84 (1982), 212–216.CrossRefGoogle Scholar
[12]Reich, S., ‘Strong convergence theorems for resolvents of accretive operators in Banach spaces’, J. Math. Anal. Appl. 75 (1980), 287–292.Google Scholar
[13]Roux, D. and Singh, S.P., ‘On some fixed point theorems’, Internat. J. Math. Math. Sci. 12 (1989), 61–64.Google Scholar
[14]Schu, J., ‘Approximation of fixed points of asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 112 (1991), 143–151.Google Scholar
[15]Vijayaraju, P., ‘Fixed point theorems for asymptotically nonexpansive mappings’, Bull. Calcutta Math. Soc. 80 (1988), 133–136.Google Scholar
[16]Zeidler, E., Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems, (Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986).Google Scholar