Random approximations and random fixed point theorems for non-self-maps
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Abstract:
Recently, Sehgal and Singh [18] and Papageorgiou [16] considered different random versions of a very interesting theorem of Fan [4]. Instead of compact convex domain, here we consider a continuous condensing or non-expansive random map defined on a closed ball (or closed convex set with bounded range). We prove it is true for certain spaces. As applications of our theorems, some random fixed point theorems of non-self-maps are derived.References
- 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
- C. Castaing and M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics, Vol. 580, Springer-Verlag, Berlin-New York, 1977. MR 0467310
- Ward Cheney and Allen A. Goldstein, Proximity maps for convex sets, Proc. Amer. Math. Soc. 10 (1959), 448–450. MR 105008, DOI 10.1090/S0002-9939-1959-0105008-8
- Ky Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240. MR 251603, DOI 10.1007/BF01110225
- Ky Fan, Some properties of convex sets related to fixed point theorems, Math. Ann. 266 (1984), no. 4, 519–537. MR 735533, DOI 10.1007/BF01458545
- Chung Wei Ha, Extensions of two fixed point theorems of Ky Fan, Math. Z. 190 (1985), no. 1, 13–16. MR 793344, DOI 10.1007/BF01159159
- C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53–72. MR 367142, DOI 10.4064/fm-87-1-53-72
- Shigeru Itoh, Random fixed-point theorems with an application to random differential equations in Banach spaces, J. Math. Anal. Appl. 67 (1979), no. 2, 261–273. MR 528687, DOI 10.1016/0022-247X(79)90023-4
- W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
- K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397–403 (English, with Russian summary). MR 188994
- Tzu Chu Lin, A note on a theorem of Ky Fan, Canad. Math. Bull. 22 (1979), no. 4, 513–515. MR 563767, DOI 10.4153/CMB-1979-067-x
- Tzu-Chu Lin, Convex sets, fixed points, variational and minimax inequalities, Bull. Austral. Math. Soc. 34 (1986), no. 1, 107–117. MR 847978, DOI 10.1017/S000497270000455X
- Tzu-Chu Lin, Approximation theorems and fixed point theorems in cones, Proc. Amer. Math. Soc. 102 (1988), no. 3, 502–506. MR 928968, DOI 10.1090/S0002-9939-1988-0928968-X
- 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
- Roger D. Nussbaum, The fixed point index for local condensing maps, Ann. Mat. Pura Appl. (4) 89 (1971), 217–258. MR 312341, DOI 10.1007/BF02414948
- 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
- Simeon Reich, Approximate selections, best approximations, fixed points, and invariant sets, J. Math. Anal. Appl. 62 (1978), no. 1, 104–113. MR 514991, DOI 10.1016/0022-247X(78)90222-6
- 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
- Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 1129-1135
- MSC: Primary 47H10; Secondary 47H09, 60H25
- DOI: https://doi.org/10.1090/S0002-9939-1988-0954994-0
- MathSciNet review: 954994