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Nonlinear random equations involving completely closed operators

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Abstract

We introduce the concept of completely closed operators in Banach spaces and then obtain the existence of random solutions of operator equations involving such operators. As simple corollaries we obtain the existence theorems for random operator equations involving monotone operators as well as operators of type (M).

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References

  1. Amann H, Existence theorems for equations of Hammerstein type,Appl. Anal. 2 (1972) 385–397

    Article  MathSciNet  Google Scholar 

  2. Bharucha-Reid A T, On random solutions of integral equations in Banach space, in:Trans. II Prague conference on information theory, statistical decision function, random processes (Prague: Academe of Science) 27–48 (1960)

    Google Scholar 

  3. Bharucha-Reid A T, On theory of random equations, in:Proc. symposia in applied mathematics (Rhode Island: Am. Math. Soc.) 40–69 (1964)

    Google Scholar 

  4. Bharucha-Reid A T,Random integral equations (New York: Academic Press) (1972)

    MATH  Google Scholar 

  5. Bharucha-Reid A T, Fixed point theorems in probabilistic functional analysis,Bull. Am. Math. Soc. 182 (1976) 641–657

    Article  MathSciNet  Google Scholar 

  6. Billingsley P,Probability and measures (New York: John Wiley) (1986)

    Google Scholar 

  7. Browder F E, Remarks on nonlinear functional equations,Proc. Natl. Acad. Sci. (USA) 51 (1964) 985–989

    Article  MATH  MathSciNet  Google Scholar 

  8. Browder F E,Problemes nonlineares (Montreal: Les Presses) (1966)

    Google Scholar 

  9. Browder F E, Nonlinear functional analysis and nonlinear integral equations of Hammerstein and Urysohn type, in:Contributions to nonlinear functional analysis (ed) E Zarantonello (New York: Academic Press) (1971)

    Google Scholar 

  10. Hans O, Random fixed point theorems, in:Trans. I Prague conference on information theory, statistical decision theory, random processes (Prague: Academe of Science) (1957)

    Google Scholar 

  11. Hans O, Inverse and adjoint transforms of linear bounded random transformation, in:Trans. I Prague conference on information theory, statistical decision functions, random processes (Prague: Academe of Science) 127–133 (1957)

    Google Scholar 

  12. Hans O, Random operator equations, in:Fourth Berkeley Symposium on mathematical statistics and probability (Berkeley: University Press) Vol. 2. 185–202 (1961)

    Google Scholar 

  13. Itoh S, Nonlinear random equations with monotone operators in Banach spaces,Math. Ann. 236 (1978) 133–146

    Article  MATH  MathSciNet  Google Scholar 

  14. Joshi M, Measurability of inverses of random operators and existence theorems,Proc. Indian Acad. Sci. (Math. Sci.) 89 (1980) 95–100

    Article  MATH  Google Scholar 

  15. Joshi M, Nonlinear random equations involving operators of type (M),J. Math. Anal. Appl. 94 (1983) 460–469

    Article  MATH  MathSciNet  Google Scholar 

  16. Kalianpur G,Stochastic filtering theory (New York: Springer Verlag) (1980)

    Google Scholar 

  17. Kannan R, Random operator equations, in:Dynamical systems, Proceedings of the University of Florida international symposium (New York: Academic Press) (1977)

    Google Scholar 

  18. Kannan R and Salehi H, Random nonlinear equation and monotone nonlinearities,J. Math. Anal. Appl. 57 (1977) 234–256

    Article  MATH  MathSciNet  Google Scholar 

  19. Kuratowski K and Ryll-Nardzewski C, A general theorem on selectors,Bull. Acad. Pol. Sci. Ser. Math. Sci. Astronom. Phys. 13 (1965) 397–403

    MATH  MathSciNet  Google Scholar 

  20. Nashed M S and Salehi H, Measurability of generalized inverses of random linear operators,SIAM J. Appl. Math. 25 (1973) 681–692

    Article  MathSciNet  MATH  Google Scholar 

  21. Petryshyn W V, A characterization of strict convexity of Banach spaces and other uses of duality mapping,J. Func. Anal. 6 (1970) 282–291

    Article  MATH  MathSciNet  Google Scholar 

  22. Petryshyn W V and Fitzpatrick P M, New existence theorems for nonlinear equations of Hammerstein type,Trans. Am. Math. Soc. 160 (1971) 39–63.

    Article  MATH  MathSciNet  Google Scholar 

  23. Specek A, Zufallige gleichungen,Czechoslovak. Math. J. 5 (1955) 462–466

    MathSciNet  Google Scholar 

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Joshi, M. Nonlinear random equations involving completely closed operators. Proc. Indian Acad. Sci. (Math. Sci.) 96, 75–85 (1987). https://doi.org/10.1007/BF02887133

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  • DOI: https://doi.org/10.1007/BF02887133

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