On deterministic and random fixed points

Tan, Kok-Keong; Yuan, Xian-Zhi

doi:10.1090/S0002-9939-1993-1169051-2

On deterministic and random fixed points
HTML articles powered by AMS MathViewer

by Kok-Keong Tan and Xian-Zhi Yuan

Proc. Amer. Math. Soc. 119 (1993), 849-856

DOI: https://doi.org/10.1090/S0002-9939-1993-1169051-2

PDF | Request permission

Abstract:

Based on an extension of Aumann’s measurable selection theorem due to Leese, it is shown that each fixed point theorem for $F(\omega , \cdot )$ produces a random fixed point theorem for $F$ provided the $\sigma$-algebra $\Sigma$ for $\Omega$ is a Suslin family and $F$ has a measurable graph (in particular, when $F$ is random continuous with closed values and $X$ is a separable metric space). As applications and illustrations, some random fixed points in the literature are obtained or extended.

References

Robert J. Aumann, Measurable utility and the measurable choice theorem, La Décision, 2: Agrégation et Dynamique des Ordres de Préférence (Actes Colloq. Internat., Aix-en-Provence, 1967) Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1969, pp. 15–26 (English, with French summary). MR 0261938
A. T. Bharucha-Reid, Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc. 82 (1976), no. 5, 641–657. MR 413273, DOI 10.1090/S0002-9904-1976-14091-8
Gh. Bocşan, A general random fixed point theorem and applications to random equations, Rev. Roumaine Math. Pures Appl. 26 (1981), no. 3, 375–379. MR 627283
C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
Shih-Sen Chang, Some random fixed point theorems for continuous random operators, Pacific J. Math. 105 (1983), no. 1, 21–31. MR 688405
Yuan Shih Chow and Henry Teicher, Probability theory, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1988. Independence, interchangeability, martingales. MR 953964, DOI 10.1007/978-1-4684-0504-0
David Downing and W. A. Kirk, Fixed point theorems for set-valued mappings in metric and Banach spaces, Math. Japon. 22 (1977), no. 1, 99–112. MR 473934
Heinz W. Engl, Random fixed point theorems for multivalued mappings, Pacific J. Math. 76 (1978), no. 2, 351–360. MR 500323
Heinz W. Engl, A general stochastic fixed-point theorem for continuous random operators on stochastic domains, J. Math. Anal. Appl. 66 (1978), no. 1, 220–231. MR 513495, DOI 10.1016/0022-247X(78)90279-2
Massimo Furi and Alfonso Vignoli, A fixed point theorem in complete metric spaces, Boll. Un. Mat. Ital. (4) 2 (1969), 505–509. MR 0256378
C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, DOI 10.4064/fm-87-1-53-72
A. D. Ioffe, Single-valued representation of set-valued mappings, Trans. Amer. Math. Soc. 252 (1979), 133–145. MR 534114, DOI 10.1090/S0002-9947-1979-0534114-6
Shigeru Itoh, A random fixed point theorem for a multivalued contraction mapping, Pacific J. Math. 68 (1977), no. 1, 85–90. MR 451228

Random fixed point theorem with an application to random differential equations in Banach spaces

67

Anna Kucia and Andrzej Nowak, Some results and counterexamples on random fixed points, Transactions of the Tenth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Vol. B (Prague, 1986) Reidel, Dordrecht, 1988, pp. 75–82. MR 1136311
S. J. Leese, Multifunctions of Souslin type, Bull. Austral. Math. Soc. 11 (1974), 395–411. MR 417365, DOI 10.1017/S0004972700044038
Tzu-Chu Lin, Random approximations and random fixed point theorems for non-self-maps, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1129–1135. MR 954994, DOI 10.1090/S0002-9939-1988-0954994-0
Tzu-Chu Lin and Ch’i Lin Yen, Applications of the proximity map to fixed point theorems in Hilbert space, J. Approx. Theory 52 (1988), no. 2, 141–148. MR 929300, DOI 10.1016/0021-9045(88)90053-6
Zuo Shu Liu and Shao Zhong Chen, On fixed-point theorems of random set-valued maps, Kexue Tongbao (English Ed.) 28 (1983), no. 4, 433–435. MR 740971
Andrzej Nowak, Random fixed points of multifunctions, Uniw. Śląski w Katowicach Prace Nauk.-Prace Mat. 11 (1981), 36–41 (English, with Polish summary). MR 638585
Nikolaos S. Papageorgiou, Random fixed point theorems for measurable multifunctions in Banach spaces, Proc. Amer. Math. Soc. 97 (1986), no. 3, 507–514. MR 840638, DOI 10.1090/S0002-9939-1986-0840638-3

Real analysis

Longin E. Rybiński, Random fixed points and viable random solutions of functional-differential inclusions, J. Math. Anal. Appl. 142 (1989), no. 1, 53–61. MR 1011408, DOI 10.1016/0022-247X(89)90163-7

On the existence of von Neumann-Aumann’s theorem

17

Stanisław Saks, Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. English translation by L. C. Young; With two additional notes by Stefan Banach. MR 0167578
V. M. Sehgal and S. P. Singh, On random approximations and a random fixed point theorem for set valued mappings, Proc. Amer. Math. Soc. 95 (1985), no. 1, 91–94. MR 796453, DOI 10.1090/S0002-9939-1985-0796453-1
V. M. Sehgal and Charles Waters, Some random fixed point theorems for condensing operators, Proc. Amer. Math. Soc. 90 (1984), no. 3, 425–429. MR 728362, DOI 10.1090/S0002-9939-1984-0728362-7

Some random fixed points

Kok-Keong Tan and Xian-Zhi Yuan, Some random fixed point theorems, Fixed point theory and applications (Halifax, NS, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 334–345. MR 1190049

Random fixed point theorems and approximation

Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
Hong Kun Xu, Some random fixed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110 (1990), no. 2, 395–400. MR 1021908, DOI 10.1090/S0002-9939-1990-1021908-6