Skip to main content

On Regularity of Diagonally Positive Quadratic Doubly Stochastic Operators

Abstract

The classical Perron–Frobenius theorem says that a trajectory of a linear stochastic operator associated with a positive square stochastic matrix always converges to a unique fixed point. In general, an analogy of the Perron–Frobenius theorem does not hold for a quadratic stochastic operator associated with a positive cubic stochastic matrix. Namely, its trajectories may converge to different fixed points depending on initial points or may not converge at all. In this paper, we show regularity of quadratic doubly stochastic operators associated with diagonally positive cubic stochastic matrices. This is a nonlinear analogy of the Perron–Frobenius theorem for positive doubly stochastic matrices.

This is a preview of subscription content, access via your institution.

References

  1. Ganikhodjaev, N., Zanin, D.: On a necessary condition for the ergodicity of quadratic operators defined on the two-dimensional simplex. Russ. Math. Surv. 59(3), 571–572 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  2. Ganikhodzhaev, R.: The study on quadratic stochastic operators. Doctor of Science Thesis, Tashkent (1993)

  3. Ganikhodzhaev, R.: On the definition of quadratic doubly stochastic operators. Russ. Math. Surv. 48(4), 244–246 (1992)

    MathSciNet  Article  Google Scholar 

  4. Ganikhodzhaev, R., Mukhamedov, F.: Linear Algebra Appl. Saburov G-decompositions of matrices and related problems I 436, 1344–1366 (2012)

    Google Scholar 

  5. Ganikhodzhaev, R., Mukhamedov, F., Rozikov, U.: Quadratic stochastic operators and processes: results and open problems. Inf. Dim. Anal. Quan. Probl. Rel. Top. 14(2), 279–335 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  6. Gantmacher, F.: The Theory of Matrices, vol. 2. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  7. Kolokoltsov, V.: Nonlinear Markov Processes and Kinetic Equations. Cambridge University, Cambridge (2010)

    Book  MATH  Google Scholar 

  8. Li, W., Ng, M.K.: On the limiting probability distribution of a transition probability tensor. Linear Multilinear Algebra 62, 362–385 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  9. Li, C.-K., Zhang, S.: Stationary probability vectors of higher-order Markov chains. Linear Algebra Appl. 473, 114–125 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  10. Lyubich, Yu.: Mathematical Structures in Population Genetics. Springer, Berlin (1992)

    Book  MATH  Google Scholar 

  11. Marshall, A., Olkin, I., Arnold, B.: Inequalities: Theory of Majorization and its Applications. Springer, Berlin (2011)

    Book  MATH  Google Scholar 

  12. Saburov, M.: A class of nonergodic Lotka–Volterra operators. Math. Notes 97(5–6), 759–763 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  13. Saburov, M.: On divergence of any order Cesaro mean of Lotka–Volterra operators. Ann. Funct. Anal. 6(4), 247–254 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  14. Saburov, M.: Ergodicity of nonlinear Markov operators on the finite dimensional space. Nonlinear Anal. Theory Methods 143, 105–119 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  15. Saburov, M., Yusof, N.A.: Counterexamples to the conjecture on stationary probability vectors of the second-order Markov chains. Linear Algebra Appl. 507, 153–157 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  16. Saburov, M., Yusof, N.A.: On quadratic stochastic operators having three fixed points. J. Phys. Conf. Ser. 697, 012012 (2016)

    Article  Google Scholar 

  17. Seneta, E.: Nonnegative Matrices and Markov Chains. Springer, New York (1981)

    Book  MATH  Google Scholar 

  18. Ulam, S.: A Collection of Mathematical Problems. New-York, London (1960)

  19. Zakharevich, M.: On the behavior of trajectories and the ergodic hypothesis for quadratic mappings of a simplex. Russ. Math. Surv. 33(6), 207–208 (1978)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mansoor Saburov.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Saburov, M. On Regularity of Diagonally Positive Quadratic Doubly Stochastic Operators. Results Math 72, 1907–1918 (2017). https://doi.org/10.1007/s00025-017-0723-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-017-0723-3

Mathematics Subject Classification

  • 47H25
  • 47H60
  • 37A30

Keywords

  • Quadratic doubly stochastic operator
  • cubic stochastic matrix
  • regularity