On fixed points of strictly positive stochastic operators on a one-dimensional simplex
https://doi.org/10.26907/2949-3919.2026.2.118-133
Abstract
A class of strictly positive stochastic operators of degree n acting on a one-dimensional simplex is investigated. The main focus is on the problem of the number of fixed points of such operators. It is shown that this problem reduces to studying the number of roots of the corresponding degree n polynomial on the interval (0,1). The existence of at least one fixed point is proved for arbitrary n.
For the quadratic case (n = 2), the uniqueness of the fixed point is established. For a cubic operator (n = 3), conditions for the existence of one, two, or three fixed points are obtained. For a quartic operator (n = 4), criteria determining the number of fixed points are proved in terms of the operator’s parameters.
Keywords
Supporting agencies: This research was supported by an applied grant from the Agency for Innovative Development under the Ministry of Higher Education, Science and Innovations of the Republic of Uzbekistan (Grant No. AL-9224093956).
About the Authors
Sh. D. Nodirov
Karshi State University; Kazan Federal University, Volga Region Mathematical Center
Uzbekistan
Uzbekistan
Shohruh Dilmurodovich Nodirov,
17, Kuchabag str., Karshi, 180119;
18, Kremlyovskaya str., Kazan 420008 (Russia).
Yu. Kh. Eshkabilov
Karshi State University
Uzbekistan
Uzbekistan
Yusup Khalbayevich Eshkabilov,
17, Kuchabag str., Karshi, 180119.
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Review
For citations:
Nodirov Sh.D., Eshkabilov Yu.Kh. On fixed points of strictly positive stochastic operators on a one-dimensional simplex. Mathematics and Theoretical Computer Science. 2026;4(2):118-133. (In Russ.) https://doi.org/10.26907/2949-3919.2026.2.118-133
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