Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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RANDOM FIXED POINT THEOREMS FOR *-NONEXPANSIVE OPERATORS IN FRECHET SPACES

  • Abdul, Rahim-Khan (Department of Mathematical Sciences King Fahd University of Petroleum and Minerals) ;
  • Nawab, Hussain (Centre for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University)
  • Published : 2002.01.01
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Abstract

Some random fixed point theorems for nonexpansive and *-nonexpansive random operators defined on convex and star-shaped sets in a Frechet space are proved. Our work extends recent results of Beg and Shahzad and Tan and Yaun to noncontinuous multivalued random operators, sets analogue to an earlier result of Itoh and provides a random version of a deterministic fixed point theorem due to Singh and Chen.

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References

  1. G. C. Ahuja, T. D. Narang, and S. Trehan, Best approximation on convex sets in metric linear spaces, Math. Nachr. 78 (1977), 125-130 https://doi.org/10.1002/mana.19770780110
  2. I. Beg, Random fixed points of nonself maps and random approximations, J. Appl. Math. Stoch. Anal. 10 (1997), no. 2, 127-130 https://doi.org/10.1155/S1048953397000154
  3. I. Beg and N. Shahzad, A general fixed point theorem for a class of continuous random operators, New Zealand J. Math. 26 (1997), 21-24
  4. I. Beg and N. Shahzad, Random fixed point of weakly inward operators in conical shells, J. Appl. Math. Stoch. Anal. 8 (1995), no. 3, 261-264 https://doi.org/10.1155/S1048953395000232
  5. F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc. 74 (1968), 660-665 https://doi.org/10.1090/S0002-9904-1968-11983-4
  6. C. J. Himmelberg, Measurable relations, Fund. Math. 87 (1975), 53-72 https://doi.org/10.4064/fm-87-1-53-72
  7. T. Husain and A. Latif, Fixed points of multivalued nonexpansive maps, Math. Japon. 33 (1988), no. 3, 385-391
  8. S. 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 https://doi.org/10.1016/0022-247X(79)90023-4
  9. K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 397-403
  10. T. C. Lin, Random approximations and random fixed point theorems for continuous 1-set contractive random maps, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1167-1176 https://doi.org/10.1090/S0002-9939-1995-1227521-4
  11. S. Massa, Some remarks on Opial's spaces, Bolletino, U.M.I. (6) 2-A (1983), 65-69
  12. W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973
  13. V. M. Sehgel and S. P. Singh, A generalization to multifunctions of Fan's best approximation theorem, Proc. Amer. Math. Soc. 102 (1988), no. 3, 534-537 https://doi.org/10.1090/S0002-9939-1988-0928974-5
  14. V. M. Sehgel and S. P. Singh, On random approximation and a random fixed point theorem for setvalued mappings, Proc. Amer. Math. Soc. 95 (1985), 91-94 https://doi.org/10.1090/S0002-9939-1985-0796453-1
  15. N. Shahzad, Random fixed point theorems for various classes of l-set contractive maps in Banach spaces, J. Math. Anal. Appl, 203 (1996), 712-718 https://doi.org/10.1006/jmaa.1996.0407
  16. K. L. Singh and Y. Chen, Fixed points for nonexpansive multivalued mapping inlocally convex spaces, Math. Japon. 36 (1991), no. 3, 423-425
  17. K. K. Tan and X. Z. Yaun, Random fixed point theorems and approximation, Stoch. Anal. Appl. 15 (1997), 103-123 https://doi.org/10.1080/07362999708809466
  18. K. K. Tan and X. Z. Yaun, Random fixed point theorems and approximation in cones, J. Math. Anal. Appl. 185 (1994), 378-390 https://doi.org/10.1006/jmaa.1994.1256
  19. A. B. Thaheem, Existence of best approximations, Port. Math. 42 (1983-84), 435-440
  20. G. E. F. Thomas, Integration on functions with values in locally convex Suslin spaces, Trans. Amer. Math. Soc. 212 (1975), 61-81 https://doi.org/10.1090/S0002-9947-1975-0385067-1
  21. H. K. Xu, On weakly nonexpansive and *-nonexpansive multivalued mappings, Math. Japon. 36 (1991), no. 3, 441-445
  22. H. K. Xu, Some random fixed point theorems for condensing and nonexpansive operators, Proc. Amer. Math. Soc. 110 (1990), no. 2, 395-400 https://doi.org/10.1090/S0002-9939-1990-1021908-6
  23. H. W. Yi and Y. C. Zhao, Fixed point theorems for weakly inward multivalued mappings and their randomizations, J. Math. Anal. Appl. 183 (1994), no. 3, 613-619 https://doi.org/10.1006/jmaa.1994.1167
  24. S. Zhang, Star-shaped sets and fixed points of multivalued mappings, Math. Japon. 36 (1991), no. 2, 327-334

Cited by

  1. Random Coincidence Point Theorem in Fréchet Spaces with Applications vol.22, pp.1, 2004, https://doi.org/10.1081/SAP-120028028