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L0-Convex Compactness and Random Normal Structure in L0 (F, B)

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Abstract

Let (B, ∥ · ∥) be a Banach space, (Ω, F, P) a probability space, and L0 (F, B) the set of equivalence classes of strong random elements (or strongly measurable functions) from (Ω, F, P) to (B, ∥ · ∥). It is well known that L0 (F, B) becomes a complete random normed module, which has played an important role in the process of applications of random normed modules to the theory of Lebesgue-Bochner function spaces and random operator theory. Let V be a closed convex subset of B and L0 (F, V) the set of equivalence classes of strong random elements from (Ω, F, P) to V. The central purpose of this article is to prove the following two results: (1) L0 (F, V) is L0-convexly compact if and only if V is weakly compact; (2) L0 (F, V) has random normal structure if V is weakly compact and has normal structure. As an application, a general random fixed point theorem for a strong random nonexpansive operator is given, which generalizes and improves several well known results. We hope that our new method, namely skillfully combining measurable selection theorems, the theory of random normed modules, and Banach space techniques, can be applied in the other related aspects.

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References

  1. Bharucha-Reid A T. Random Integral Equations. New York: Academic Press, 1972

    MATH  Google Scholar 

  2. Bharucha-Reid A T. Fixed point theorems in probabilistic analysis. Bull Amer Math Soc, 1976, 82(5): 641–657

    Article  MathSciNet  Google Scholar 

  3. Brodskil M S, Mil’man D P. On the center of a convex set. Dokl Akad Nauk SSSR, 1968, 59(5): 837–840

    MathSciNet  Google Scholar 

  4. Browder F E. Nonexpansive nonlinear operators in a Banach space. Proc Nat Acad Sci USA, 1965, 54(4): 1041–1044

    Article  MathSciNet  Google Scholar 

  5. Dunford N, Schwartz J T. Linear operators. New York: Interscience, 1958

    MATH  Google Scholar 

  6. Goebel K, Reich S. Uniform convexity, hyperbolic geometry, and nonexpansive mappings. New York, Basel: Marcel Dekker, 1984

    MATH  Google Scholar 

  7. Göhde D. Zum Prinzip der kontraktiven Abbildung. Math Nach, 1965, 30(3/4): 251–258

    Article  MathSciNet  Google Scholar 

  8. Guo T X. The theory of probabilistic metric spaces with applications to random functional analysis[Master’s Thesis]. Xi’an: Xi’an Jiaotong University, 1989

    Google Scholar 

  9. Guo T X. Random metric theory and its applications[Ph D Thesis]. Xi’an: Xi’an Jiaotong University, 1992

    Google Scholar 

  10. Guo T X. A new approach to probabilistic functional nalysis//Feng E B. Proceedings of the first China postdoctoral academic conference. Beijing: The China National Defense and Industry Press, 1993: 1150–1154

    Google Scholar 

  11. Guo T X. Extension theorems of continuous random linear operators on random domains. J Math Anal Appl, 1995, 193: 15–27

    Article  MathSciNet  Google Scholar 

  12. Guo T X. The Radon-Nikodym property of conjugate spaces and the w*-equivalence theorem for immeasurable functions. Sci China Math Ser A, 1996, 39: 1034–1041

    MATH  Google Scholar 

  13. Guo T X. The relation of Banach-Alaoglu theorem and Banach-Bourbaki-Kakutani-Smulian theorem in complete random normed modules to stratification structure. Sci China Math Ser A, 2008, 51(9): 1651–1663

    Article  Google Scholar 

  14. Guo T X. Relations between some basic results derived from two kinds of topologies for a random locally convex module. J Funct Anal, 2010, 258(9): 3024–3047

    Article  MathSciNet  Google Scholar 

  15. Guo T X. On some basic theorems of continuous module homomorphisms between random normed modules. J Funct spaces Appl, 2013, Article ID 989102, 13 pages

    Google Scholar 

  16. Guo T X, Lin S B. The James theorem in complete random normed modules. J Math Anal Appl, 2005, 308(1): 257–265

    Article  MathSciNet  Google Scholar 

  17. Guo T X, You Z Y. The Riesz representation theorem on complete random inner product modules and its applications. Chinese Ann Math Ser A (in Chinese), 1996, 17(3): 361–364

    MATH  Google Scholar 

  18. Guo T X, Zeng X L. Random strict convexity and random uniform convexity in random normed modules. Nonlinear Anal TMA, 2010, 73(5): 1239–1363

    Article  MathSciNet  Google Scholar 

  19. Guo T X, Zeng X L. An L0(F, R)-valued function’s intermediate value theorem and its applications to random uniform convexity. Acta Math Sin, 2012, 28(5): 909–924

    Article  MathSciNet  Google Scholar 

  20. Guo T X, Zhang E X, Wang Y C, et al. Two fixed point theorems in complete random normed modules and their applications to backward stochastic equations. J Math Anal Appl. https://doi.org/10.1016/j.jmaa.2019.123644

  21. Guo T X, Zhang E X, Wang Y C, et al. L0-convex compactness and its applications to random convex optimization and random variational inequalities. arXiv: 1709.07137V6

  22. Guo T X, Zhang E X, Wu M Z, et al. On random convex analysis. J Nonlinear Conv Anal, 2017, 18(11): 1967–1996

    MathSciNet  MATH  Google Scholar 

  23. Guo T X, Zhao S E, Zeng X L. The relations among the three kinds of conditional risk measures. Sci China Math, 2014, 57(8): 1753–1764

    Article  MathSciNet  Google Scholar 

  24. Guo T X, Zhao S E, Zeng X L. Random convex analysis (I): separation and Fenchel-Moreau duality in random locally convex modules. Sci Sin Math (in Chinese), 2015, 45(12): 1960–1980

    Google Scholar 

  25. Guo T X, Zhao S E, Zeng X L. Random convex analysis(II): continuity and subdifferentiability in L0-pre-barreled random locally convex modules. Sci Sin Math (in Chinese), 2015, 45(5): 647–662

    Google Scholar 

  26. Hans O. Reduzierende zufallige transformationen. Czechoslovak Math J, 1957, 7(1): 154–158

    MathSciNet  MATH  Google Scholar 

  27. Hans O. Random fixed point theorems//Spacek A. Transactions of the First Prague Conference on Information Theory, Statistical Decision Functions, Random Processes. Prague: Czechoslovak Acad Sci, 1957: 105–125

    Google Scholar 

  28. Hans O. Random operator equations//Neyman J. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol.11, Probability Theory. Berkeley: Univ California Press, 1961: 185–202

    Google Scholar 

  29. He S W, Wang J G, Yan J A. Semimartingale Theory and Stochastic Calculus. Boca Raton: CRC Press, 1992

    MATH  Google Scholar 

  30. James R C. Weakly compact sets. Trans Amer Math Soc, 1964, 113(1): 129–140

    Article  MathSciNet  Google Scholar 

  31. Kirk W A. A fixed point theorem for mappings which do not increase distances. Amer Math Monthly, 1965, 72(9): 1004–1006

    Article  MathSciNet  Google Scholar 

  32. Kirk W A. Fixed point theory for nonexpansive mappings//Fadell, Edward, Fournier, et al. Fixed point theory. Berlin: Springer, 1981: 484–505

    Google Scholar 

  33. Landes T. Permanence properties of normal structure. Paci J Math, 1984, 110(1): 125–143

    Article  MathSciNet  Google Scholar 

  34. Lin T C. Random approximations and random fixed point theorems for non-self-maps. Proc Amer Math Soc, 1988, 103(4): 1129–1135

    Article  MathSciNet  Google Scholar 

  35. Schweizer B, Sklar A. Probabilistic Metric Spaces. New York: Dover Publications, 2005

    MATH  Google Scholar 

  36. Smith M A, Turett B. Normal Structure in Bochner Lp spaces. Paci J Math, 1990, 142(2): 347–356

    Article  Google Scholar 

  37. Spacëk A. Zufallige Gleichungen. Czechoslovak Math J, 1955, 5(4): 462–466

    MathSciNet  MATH  Google Scholar 

  38. Wagner D H. Survey of measurable selection theorems. SIAM J Control Opti, 1977, 15(5): 859–903

    Article  MathSciNet  Google Scholar 

  39. Wang Z K. Introduction to probabilistic functional analysis. Adv Math (in Chinese), 1962, 5(1): 45–71

    Google Scholar 

  40. Xu H K. Some random fixed point theorems for condensing and nonexpansive operators. Proc Amer Math Soc, 1990, 110(2): 395–400

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Tiexin Guo, Erxin Zhang & Yachao Wang

  2. Centre for Financial Engineering, Soochow University, Suzhou, 215006, China

    George Yuan

Authors
  1. Tiexin Guo
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  2. Erxin Zhang
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  3. Yachao Wang
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  4. George Yuan
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Correspondence to Tiexin Guo.

Additional information

This work was supported by National Natural Science Foundation of China (11571369).

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Guo, T., Zhang, E., Wang, Y. et al. L0-Convex Compactness and Random Normal Structure in L0 (F, B). Acta Math Sci 40, 457–469 (2020). https://doi.org/10.1007/s10473-020-0211-9

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