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

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

  • Farrukh MukhamedovEmail author
  • Otabek Khakimov
  • Ahmad Fadillah Embong


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.


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

Mathematics Subject Classification

46L35 46L55 46A37 



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


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Farrukh Mukhamedov
    • 1
    Email author
  • 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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