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Positivity

, Volume 22, Issue 5, pp 1445–1459 | Cite as

On surjective second order non-linear Markov operators and associated nonlinear integral equations

  • Farrukh Mukhamedov
  • Otabek Khakimov
  • Ahmad Fadillah Embong
Article
  • 12 Downloads

Abstract

It was known that orthogonality preserving property and surjectivity of nonlinear Markov operators, acting on finite dimensional simpleces, are equivalent. It turns out that these notions are no longer equivalent when such kind of operators are considered over on infinite dimensional spaces. In the present paper, we find necessary and sufficient condition to be equivalent of these notions, for the second order nonlinear Markov operators. To do this, we fully describe all surjective second order nonlinear Markov operators acting on infinite dimensional simplex. As an application of this result, we provided some sufficient conditions for the existence of positive solutions of nonlinear integral equations whose domain are not compact.

Keywords

Non-linear Markov operator Quadratic stochastic operator Orthogonality preserving Surjective Integral equation 

Mathematics Subject Classification

46L35 46L55 46A37 

Notes

Acknowledgements

The present work is supported by the UAEU “Start-Up” Grant No. 31S259.

References

  1. 1.
    Akin, H., Mukhamedov, F.: Orthogonality preserving infinite dimensional quadratic stochastic operators. AIP Conf. Proc. 1676, 020008 (2015)CrossRefGoogle Scholar
  2. 2.
    Arutyunov, A.V.: Two problems of the theory of quadratic maps. Funct. Anal. Appl. 46, 225–227 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bernstein, S.N.: The solution of a mathematical problem concerning the theory of heredity. Ann. Math. Stat. 13, 53–61 (1924)CrossRefGoogle Scholar
  4. 4.
    Ganikhodzhaev, R., Mukhamedov, F., Rozikov, U.: Quadratic stochastic operators and processes: results and open problems. Infin. Dimens. Anal. Quant. Prob. Relat. Top. 14, 279–335 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Jamilov, U.U.: Quadratic stochastic operators corresponding to graphs. Lobachevskii J. Math. 34, 148–151 (2013)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations. Cambridge Univ. Press, New York (2010)CrossRefGoogle Scholar
  7. 7.
    Li, C.-K., Zhang, S.: Stationary probability vectors of higher-order Markov chains. Linear Algebra Appl. 473, 114–125 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Lyubich, YuI: Mathematical Structures in Population Genetics. Springer, Berlin (1992)CrossRefGoogle Scholar
  9. 9.
    Mukhamedov, F.M.: On infinite dimensional Volterra operators. Russ. Math. Surv. 55, 1161–1162 (2000)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Mukhamedov, F., Akin, H., Temir, S.: On infinite dimensional quadratic Volterra operators. J. Math. Anal. Appl. 310, 533–556 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Mukhamedov, F., Embong, A.F.: On non-linear Markov operators: surjectivity vs orthogonality preserving property. Linear Multilinear Algebra.  https://doi.org/10.1080/03081087.2017.1389849 MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mukhamedov, F., Embong, A.F., Rosli, A.: Orthogonality preserving and surjective cubic stochastic operators. Ann Funct. Anal. 8, 490–501 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mukhamedov, F., Ganikhodjaev, N.: Quantum Quadratic Operators and Processes. Springer, Berlin (2015)CrossRefGoogle Scholar
  14. 14.
    Mukhamedov, F., Taha, M.H.: On Volterra and orthoganality preserving quadratic stochastic operators. Miskloc Math. Notes 17, 457–470 (2016)CrossRefGoogle Scholar
  15. 15.
    Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Raftery, A.: A model of high-order Markov chains. J. R. Stat. Soc. 47, 528–539 (1985)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Saburov, M.: On the surjectivity of quadratic stochastic operators acting on the simplex. Math. Notes 99, 623–627 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Farrukh Mukhamedov
    • 1
  • Otabek Khakimov
    • 1
  • Ahmad Fadillah Embong
    • 2
  1. 1.Department of Mathematical Sciences, and College of ScienceThe United Arab Emirates UniversityAl Ain, Abu DhabiUAE
  2. 2.Department of Computational and Theoretical Sciences, Faculty of ScienceInternational Islamic University MalaysiaKuantanMalaysia

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