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

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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.

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematical Sciences, and College of Science, The United Arab Emirates University, P.O. Box 15551, Al Ain, Abu Dhabi, UAE

    Farrukh Mukhamedov & Otabek Khakimov

  2. Department of Computational and Theoretical Sciences, Faculty of Science, International Islamic University Malaysia, Kuantan, Pahang, Malaysia

    Ahmad Fadillah Embong

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  1. Farrukh Mukhamedov
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  2. Otabek Khakimov
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  3. Ahmad Fadillah Embong
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Correspondence to Farrukh Mukhamedov.

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Mukhamedov, F., Khakimov, O. & Embong, A.F. On surjective second order non-linear Markov operators and associated nonlinear integral equations. Positivity 22, 1445–1459 (2018). https://doi.org/10.1007/s11117-018-0587-0

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  • DOI: https://doi.org/10.1007/s11117-018-0587-0

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