The characteristic of noncompact convexity and random fixed point theorem for set-valued operators
- 66 Downloads
Let (Ω, Σ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C. We prove that a set-valued nonexpansive mapping T: C → KC(C) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T: Ω × C → KC(C) has a random fixed point.
Keywordsrandom fixed point set-valued random operator measure of noncompactness
Unable to display preview. Download preview PDF.
- J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo: Measures of Noncompactness in Metric Fixed Point Theory; Advances and Applications Topics in Metric Fixed Point Theory. Birkhauser-Verlag, Basel 99, 1997.Google Scholar
- K. Deimling: Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1974.Google Scholar
- K. Goebel and W. A. Kirk: Topics in metric fixed point theorem. Cambridge University Press, Cambridge, 1990.Google Scholar
- W. A. Kirk: Nonexpansive mappings in product spaces, set-valued mappings, and k-uniform rotundity. Proceedings of the Symposium Pure Mathematics, Vol. 45, part 2, American Mathematical Society, Providence, 1986, pp. 51–64.Google Scholar
- K.-K. Tan and X. Z. Yuan: Some random fixed point theorems. Fixed Point Theory and Applications (K.-K. Tan, ed.). World Scientific, Singapore, 1992, pp. 334–345.Google Scholar