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

Mukhamedov, Farrukh; Khakimov, Otabek; Embong, Ahmad Fadillah

doi:10.1007/s11117-018-0587-0

Cited by 11 publications

(9 citation statements)

References 17 publications

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“…We emphasize that there is a big difference between finite and infinite dimensional settings. It is known [22] that in the infinite dimensional setting, some implication of Theorem 2.4 fails.…”

Section: Discrete Quadratic Stochastic Operatorsmentioning

confidence: 99%

“…We refer the reader to [11,18,27] as the exposition of the recent achievements and open problems in the theory of the QSO can be further researched. In [22,23] the surjectivity of DQSO and its relation with orthogonal preserving property of V have been investigated.…”

Section: Introductionmentioning

confidence: 99%

“…The main aim of the present paper is to investigate projective surjectivity of such kind of continuum analogous of QSO. First, in Section 3, we revise the results of [22], and in section 4, we apply them to the projective surjectivity of QSO acting on the set of probability measures. We notice that very particular cases of QSO have been studied (see for example, [3,5,6,10]).…”

Section: Introductionmentioning

confidence: 99%

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Projective surjectivity of quadratic stochastic operators $L_1$ and its application

Mukhamedov,

Khakimov,

Embong

2021

Preprint

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A nonlinear Markov chain is a discrete time stochastic process whose transitions depend on both the current state and the current distribution of the process. The nonlinear Markov chain over a infinite state space can be identified by a continuous mapping (the so-called nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex). In the present paper, we consider a continuous analogue of the mentioned mapping acting on L 1 -spaces. Main aim of the current paper is to investigate projective surjectivity of quadratic stochastic operators (QSO) acting on the set of all probability measures. To prove the main result, we study the surjectivity of infinite dimensional nonlinear Markov operators and apply them to the projective surjectivity of a QSO. Furthermore, the obtained result has been applied for the existence of positive solution of some Hammerstein integral equations.

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Section: Discrete Quadratic Stochastic Operatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Projective surjectivity of quadratic stochastic operators $L_1$ and its application

Mukhamedov,

Khakimov,

Embong

2021

Preprint

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“…In Mukhamedov et al, 31 OP and surjective QSOs defined on the infinite-dimensional simplex have been studied.…”

Section: Satisfies the Following Conditionsmentioning

confidence: 99%

“…We point out that the results on surjectivity of nonlinear Markov operators that we considered here open new insight to the theory of nonlinear operators 8 . In infinite-dimensional setting, it turns out that the surjectivity and orthogonal preserveness are not necessarily the same, while they coincide in the finite-dimensional case (see Mukhamedov and Embong [9][10][11][12][13]21,28,29,31 ).…”

Section: Introductionmentioning

confidence: 99%

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

Mukhamedov

Khakimov

Embong

2020

Math Methods in App Sciences

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In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.

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A class of bijective Lotka–Volterra operators and its application

Mukhamedov

Hee

Rosli

2023

Math Methods in App Sciences

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It is well known that any classical Lotka-Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved.

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

Cited by 11 publications

References 17 publications

Projective surjectivity of quadratic stochastic operators $L_1$ and its application

Projective surjectivity of quadratic stochastic operators $L_1$ and its application

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

A class of bijective Lotka–Volterra operators and its application

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