Based on the notion of asymptotically contractive mapping due to Penot [16], we propose in this paper a new method for the study of existence of fixed points for nonexpansive mappings defined on unbounded sets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Browder, F.E.: Nonexpansive Nonlinear Operators in a Banach Space. Proc. Nat. Acad. Sci., 54, 1041–1044 (1965)
Browder, F.E.: Nonlinear Operators and Nonlinear Equations of Evolution in Banach Spaces. Proc. Symp. Pure Math., 18, Amer. Math. Soc., Providence (1976)
Ciorănescu, I.: Geometry of Banach Spaces. Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers (1990)
Gatica, J.A., Kirk, W.A.: Fixed-Point Theorems for Contraction Mappings with Applications to Nonexpansive and Pseudo-Contractive Mappings. Rocky Mountain J. Math., 4, 69–79 (1974)
Goebel, K., Kirk, W.A.: Topics in Metric Fixed-point Theory. Cambridge University Press, Cambridge, U.K. (1990)
Ghde, D.: Zum Prinzip der Kontraktiven Abbildung. Math. Nach., 30, 251–258 (1965)
Isac, G: The Scalar Asymptotic Derivative and the Fixed-Point Theory on Cones. Nonliner Anal. Related Topics, 2, 92–97 (1999)
Isac, G., Németh, S.Z.: Scalar Derivatives and Scalar Asymptotic Derivatives: Properties and Some Applications. J. Math. Anal. Appl., 278, 149–170 (2003)
Isac, G., Németh, S.Z.: Scalar Derivatives and Scalar Asymptotic Derivatives. An Altman type fixed-point theorem on convex cones and some applications. J. Math. Anal. Appl., 290, 452–468 (2004)
Kirk, W.A.: A Fixed-Point Theorem for Mappings which do not Increase Distances. Amer. Math. Monthly, 72, 1004–1006 (1965)
Kirk, W.A.: Nonexpansive Mappings and Asymptotic Regularity. Nonlinear Anal., Theory Methods, Appl., 40, 323–332 (2002)
Kirk, W.A., Yanez, C.M., Shin, S.S: Asymptotically Nonexpansive Mappings. Nonlinear Anal., Theory Methods, Appl., 33, 1–12 (1998)
Luc, D.T.: Recessively Compact Sets: Uses and Properties. St-Valued Anal., 10, 15–35 (2002)
Németh, S.Z.: A Scalar Derivative for Vector Functions. Riv. Mat. Pura Appl., 10, 7–24 (1992)
Németh, S.Z.: Scalar Derivatives and Spectral Theory. Mathematica, 35, 49–58 (1993)
Penot, J.P.: A Fixed-Point Theorem for Asymptotically Contractive Mappings. Proc. Amer. Math. Soc., 131, 2371–2377 (2003)
Rhoades, B.E.: A Comparison of Various Definitions of Contractive Mappings. Trans. Amer. Math. Soc., 226, 257–290 (1977)
Rouhani, B.D., Kirk, W.A.: Asymptotic Properties of Non-Expansives Iterations in Reflexive Spaces. J. Math. Anal. Appl., 236, 281–289 (1999)
Rus, I.A., Petrusel, A., Petrusel, G.: Fixed-Point Theory (1950–2000). Romanian Contributions. House of the Book of Science, Cluj-Napoca (2002)
Singh, S., Watson, B., Srivastava, P.: Fixed-Point Theory and Best Approximation. The KKM-map Principle. Kluwer Academic Publishers (1997)
Takahashi, W.: Nonlinear Functional Analysis (Fixed-point Theory and its Applications). Yokohama Publishers (2000)
Zeidler, E.: Nonlinear Functional Analysis and its Applications, Part 1: Fixed-Point Theorems. Springer-Verlag, New York (1986)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Isac, G. (2008). Scalar Asymptotic Contractivity and Fixed Points for Nonexpansive Mappings on Unbounded Sets. In: Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds) Pareto Optimality, Game Theory And Equilibria. Springer Optimization and Its Applications, vol 17. Springer, New York, NY. https://doi.org/10.1007/978-0-387-77247-9_5
Download citation
DOI: https://doi.org/10.1007/978-0-387-77247-9_5
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-77246-2
Online ISBN: 978-0-387-77247-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)