On firmly nonexpansive mappings

Smarzewski, Ryszard

doi:10.1090/S0002-9939-1991-1050023-1

On firmly nonexpansive mappings
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by Ryszard Smarzewski

Proc. Amer. Math. Soc. 113 (1991), 723-725

DOI: https://doi.org/10.1090/S0002-9939-1991-1050023-1

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Abstract:

It is shown that any $\lambda$-firmly, $0 < \lambda < 1$ , nonexpansive mapping $T:C \to C$ has a fixed point in $C$ whenever $C$ is a finite union of nonempty, bounded, closed convex subsets of a uniformly convex Banach space.

References

Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
Kazimierz Goebel and W. A. Kirk, Topics in metric fixed point theory, Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. MR 1074005, DOI 10.1017/CBO9780511526152
Kazimierz Goebel and Simeon Reich, Uniform convexity, hyperbolic geometry, and nonexpansive mappings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 83, Marcel Dekker, Inc., New York, 1984. MR 744194
Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312
W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
J. Reinermann and R. Schöneberg, Some results and problems in the fixed point theory for nonexpansive and pseudocontractive mappings in Hilbert-space, Fixed point theory and its applications (Proc. Sem., Dalhousie Univ., Halifax, N.S., 1975) Academic Press, New York, 1976, pp. 187–196. MR 0451052