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The fixed point problem for generalised nonexpansive maps

Published online by Cambridge University Press:  17 April 2009

Michael A. Smyth
Michael A. Smyth
Affiliation:
Department of Mathematics, The University of Newcastle, New South Wales 2308, Australia Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
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Abstract

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This paper is concerned with extending the theory of the existence of fixed points for generalised nonexpansive maps as far as possible. This can be seen as a continuation of the work of Maurey on the extension of the fixed point theory for nonexpansive maps beyond the requirement of normal structure type conditions.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 55 , Issue 1 , February 1997 , pp. 45 - 61
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Alspach, D.E., ‘A fixed point free nonexpansive map’, Proc. Amer. Math. Soc. 82 (1981), 423–424.CrossRefGoogle Scholar
[2]Bae, Jong Sook, ‘Fixed point theorems of generalized nonexpansive maps’, J. Korean Math. Soc. 21 (1984), 233–248.Google Scholar
[3]Baker, J.W., ‘Dispersed images of topological spaces and uncomplemented subspaces of C(X), Proc. Amer. Math. Soc. 41 (1973), 309–314.Google Scholar
[4]Bogin, J., ‘A generalization of a fixed point theorem of Goebel, Kirk and Shimi’, Canad. Math. Bull. 19 (1976), 7–12.CrossRefGoogle Scholar
[5]Borwein, J. and Sims, B., ‘Non-expansive mappings on Banach lattices and related topics’, Houston J. Math. 10 (1983), 339–355.Google Scholar
[6]Diestel, J., Geometry of Banach spaces–Selected topics (Springer-Verlag, Berlin, Heidelberg, New York, 1975).CrossRefGoogle Scholar
[7]Elton, J., Lin, P.K., Odell, E. and Szarek, S., ‘Remarks on the fixed point problem for nonexpansive maps’, in Fixed points and nonexpansive mappings, (Sine, R., Editor), Contemp. Math. Amer. Math. Soc. 18 (Amer. Math. Soc, Providence RI, 1983).Google Scholar
[8]Engelking, R., Outline of general topology (North-Holland, Amsterdam, 1968).Google Scholar
[9]Goebel, K., ‘On the structure of minimal invariant sets for nonexpansive mappings’, Ann. Univ. Mariae. Curie-Sklodowska 29 (1975), 73–77.Google Scholar
[10]Goebel, K. and Kirk, W.A., Topics in metric fixed point theory (Cambridge University Press, Cambridge, 1990).CrossRefGoogle Scholar
[11]Goebel, K., Kirk, W.A. and Shimi, T., ‘A fixed point theorem in uniformly convex spaces’, Boll. Un. Mat. Ital. 7 (1973), 67–75.Google Scholar
[12]Jimenez-Melado, A. and Llorens-Fuster, E., ‘A geometric property of Banach spaces which implies the fixed point property for nonexpansive mappings’, (preprint).Google Scholar
[13]Jimenez-Melado, A. and Llorens-Fuster, E., ‘Stability of orthogonal convexity’, (preprint).Google Scholar
[14]Karlovitz, L., ‘Existence of fixed points for nonexpansive mappings in spaces without normal structure’, Pacific J. Math. 66 (1976), 153–156.CrossRefGoogle Scholar
[15]Lacey, H., The isometric theory of classical Banach spaces (Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[16]Lin, Pei-Kee, ‘Unconditional bases and fixed points of nonexpansive mappings’, Pacific J. Math. 116 (1985), 69–76.CrossRefGoogle Scholar
[17]Maurey, B., Points fixes des contractions sur un convexe forme de L1, Seminarie d' Analyse Fonctionnelle 1980–1981 (Expose no. 8) (Ecole Polytechnique, Palaiseau).Google Scholar
[18]Mazurkiewicz, S. and Sierpinski, W., ‘Contribution a la topologie des ensembles denombrables’, Fund. Math. 1 (1920), 17–27.CrossRefGoogle Scholar
[19]Pelczyński, A. and Semadeni, Z., ‘Spaces of continuous functions III’, Studia Math. 18 (1959), 211–222.Google Scholar
[20]Schaefer, H.H., Banach lattices and positive operators, Grundlehren der Math., Band 215 (Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[21]Schreier, J., ‘Ein Gegen beispiel zur Theorie der schwachen Konvergenz’, Studia Math. 2 (1930), 58–62.CrossRefGoogle Scholar
[22]Sims, B., ‘Othogonality and fixed points of nonexpansive maps’, in Proc. Centre Math. Anal. 20 (Australian National University, Canberra, ACT, 1988).Google Scholar
[23]Wong, C., ‘Close-to-normal structure and its applications’, J. Funct. Anal. 16 (1974), 353–358.CrossRefGoogle Scholar