Abstract
Chapter 9 is devoted to set-valued mappings. We study approximate fixed points of such mappings, existence of fixed points, and the convergence and stability of iterates of set-valued mappings. In particular, we consider a complete metric space of nonexpansive set-valued mappings acting on a closed and convex subset of a Banach space with a nonempty interior, and show that a generic mapping in this space has a fixed point. We then prove analogous results for two complete metric spaces of set-valued mappings with convex graphs. We also introduce the notion of a contractive set-valued mapping and study the asymptotic behavior of its iterates.
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
References
Caristi, J. (1976). Transactions of the American Mathematical Society, 215, 241–251.
Cohen, J. E. (1979). Bulletin of the American Mathematical Society, 1, 275–295.
Covitz, H., & Nadler, S. B. Jr. (1970). Israel Journal of Mathematics, 8, 5–11.
de Blasi, F. S., Myjak, J., Reich, S., & Zaslavski, A. J. (2009). Set-Valued and Variational Analysis, 17, 97–112.
Diamond, P., Kloeden, P. E., Rubinov, A. M., & Vladimirov, A. (1997). Set-Valued Analysis, 5, 267–289.
Goebel, K., & Kirk, W. A. (1990). Topics in metric fixed point theory. Cambridge: Cambridge University Press.
Kirk, W. A. (2001). In Handbook of metric fixed point theory (pp. 1–34). Dordrecht: Kluwer Academic.
Leizarowitz, A. (1985). SIAM Journal on Control and Optimization, 22, 514–522.
Leizarowitz, A. (1994). Set-Valued Analysis, 2, 505–527.
Lim, T.-C. (1974). Bulletin of the American Mathematical Society, 80, 1123–1126.
Nadler, S. B. Jr. (1969). Pacific Journal of Mathematics, 30, 475–488.
Radström, H. (1952). Proceedings of the American Mathematical Society, 3, 165–169.
Reich, S. (1972). Bollettino dell’Unione Matematica Italiana, 5, 26–42.
Reich, S. (1978). Journal of Mathematical Analysis and Applications, 62, 104–113.
Reich, S., & Zaslavski, A. J. (2002). In Set valued mappings with applications in nonlinear analysis (pp. 411–420). London: Taylor & Francis.
Reich, S., & Zaslavski, A. J. (2002). Set-Valued Analysis, 10, 287–296.
Ricceri, B. (1987). Atti Della Accademia Nazionale Dei Lincei. Rendiconti Della Classe Di Scienze Fisiche, Matematiche E Naturali, 81, 283–286.
Rockafellar, R. T. (1979). In Lecture notes in economics and mathematical systems. Convex analysis and mathematical economics (pp. 122–136). Berlin: Springer.
Rubinov, A. M. (1984). Journal of Soviet Mathematics, 26, 1975–2012.
Sach, P. H., & Yen, N. D. (1997). Set-Valued Analysis, 5, 37–45.
Zaslavski, A. J. (1996). Numerical Functional Analysis and Optimization, 17, 215–240.
Zaslavski, A. J. (2000). SIAM Journal on Control and Optimization, 39, 250–280.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this chapter
Cite this chapter
Reich, S., Zaslavski, A.J. (2014). Set-Valued Mappings. In: Genericity in Nonlinear Analysis. Developments in Mathematics, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9533-8_9
Download citation
DOI: https://doi.org/10.1007/978-1-4614-9533-8_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9532-1
Online ISBN: 978-1-4614-9533-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)