Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
random fixed point; set-valued random operator; measure of noncompactness
random fixed point; set-valued random operator; measure of noncompactness
Summary:
Let $(\Omega ,\Sigma )$ 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 $T\: C\rightarrow KC(C)$ has a fixed point. Furthermore, if $X$ is separable then we also prove that a set-valued nonexpansive operator $T\: \Omega \times C\rightarrow KC(C)$ has a random fixed point.
References:
[1] J. P. Aubin and H. Frankowska: Set-valued Analysis. Birkhäuser, Boston, 1990. MR 1048347
[2] 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. MR 1483889
[3] K. Deimling: Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1974. MR 0787404
[4] T. Domínguez Benavides and P. Lorenzo Ramírez: Fixed point theorem for multivalued nonexpansive mapping without uniform convexity. Abstr. Appl. Anal. 6 (2003), 375–386. DOI 10.1155/S1085337503203080 | MR 1982809
[5] T. Domínguez Benavides and P. Lorenzo Ramírez: Fixed point theorem for multivalued nonexpansive mapping satisfying inwardness conditions. J. Math. Anal. Appl. 291 (2004), 100–108. DOI 10.1016/j.jmaa.2003.10.019 | MR 2034060
[6] T. Domínguez Benavides, G. Lopez Acedo and H. K. Xu: Random fixed point of set-valued operator. Proc. Amer. Math. Soc. 124 (1996), 838–838. DOI 10.1090/S0002-9939-96-03062-6 | MR 1301487
[7] K. Goebel and W. A. Kirk: Topics in metric fixed point theorem. Cambridge University Press, Cambridge, 1990. MR 1074005
[8] S. Itoh: Random fixed point theorem for a multivalued contraction mapping. Pacific J. Math. 68 (1977), 85–90. DOI 10.2140/pjm.1977.68.85 | MR 0451228 | Zbl 0335.54036
[9] 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. MR 0843594 | Zbl 0594.47048
[10] P. Lorenzo Ramírez: Some random fixed point theorems for nonlinear mappings. Nonlinear Anal. 50 (2002), 265–274. DOI 10.1016/S0362-546X(01)00751-9 | MR 1904945
[11] P. Lorenzo Ramírez: Random fixed point of uniformly Lipschitzian mappings. Nonlinear Anal. 57 (2004), 23–34. DOI 10.1016/j.na.2004.01.003 | MR 2055985
[12] N. Shahzad and S. Latif: Random fixed points for several classes of 1-ball-contractive and 1-set-contractive random maps. J. Math. Anal. Appl. 237 (1999), 83–92. DOI 10.1006/jmaa.1999.6454 | MR 1708163
[13] 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. MR 1190049
[14] D.-H. Wagner: Survey of measurable selection theorems. SIAM J. Control Optim. 15 (1977), 859–903. DOI 10.1137/0315056 | MR 0486391 | Zbl 0407.28006
[15] H. K. Xu: Some random fixed point for condensing and nonexpansive operators. Proc. Amer. Math. Soc. 110 (1990), 395–400. DOI 10.1090/S0002-9939-1990-1021908-6 | MR 1021908
[16] H. K. Xu: Metric fixed point for multivalued mappings. Dissertationes Math. (Rozprawy Mat.) 389 (2000), 39. MR 1799531
[17] H. K. Xu: A random theorem for multivalued nonexpansive operators in uniformly convex Banach spaces. Proc. Amer. Math. Soc. 117 (1993), 1089–1092. DOI 10.1090/S0002-9939-1993-1123670-8 | MR 1123670
[18] H. K. Xu: Random fixed point theorems for nonlinear uniform Lipschitzian mappings. Nonlinear Anal. 26 (1996), 1301–1311. DOI 10.1016/0362-546X(95)00013-L | MR 1376105
[19] H. K. Xu: Multivalued nonexpansive mappings in Banach spaces. Nonlinear Anal. 43 (2001), 693–706. DOI 10.1016/S0362-546X(99)00227-8 | MR 1808203 | Zbl 0988.47034
[20] S. Reich: Fixed points in locally convex spaces. Math. Z. 125 (1972), 17–31. DOI 10.1007/BF01111112 | MR 0306989 | Zbl 0216.17302
[21] X. Yuan and J. Yu: Random fixed point theorems for nonself mappings. Nonlinear Anal. 26 (1996), 1097–1102. DOI 10.1016/0362-546X(94)00268-M | MR 1375652
Search
Partner of
