Existence of positive bounded solutions of system of three dynamic equations with neutral term on time scales

Year, pages: 2019, 123 - 137

In this paper, the system of three dynamic equations with neutral term in the following form \begin{equation*} \begin{cases} & (x(t)+p(t) \, x(u_1(t)))^{\Delta} = a(t) \, f(y(u_2(t))), \\ & y^{\Delta}(t)= b(t) \, g(z(u_3(t))), \\ & z^{\Delta}(t)= c(t)\, h(x(u_4(t))) \end{cases} \end{equation*} on time scales is considered. The aim of this paper is to present sufficient conditions for the existence of positive bounded solutions of the considered system for $0 < p(t) \leq const < 1$. The main tool of the proof of presented here result is Krasnoselskii's fixed point theorem. Also, the useful generalization of the Arzel{\'a}-Ascoli theorem on time scales to the three-dimensional case is proved.

Ostaszewska, U., Schmeidel, E., Zdanowicz, M. 2019. Existence of positive bounded solutions of system of three dynamic equations with neutral term on time scales. In Tatra Mountains Mathematical Publications, vol. 71, no.1, pp. 123-137. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2018-0011

Ostaszewska, U., Schmeidel, E., Zdanowicz, M. (2019). Existence of positive bounded solutions of system of three dynamic equations with neutral term on time scales. Tatra Mountains Mathematical Publications, 71(1), 123-137. 1210-3195. DOI: https://doi.org/10.2478/tmmp-2018-0011

Publisher: MÚ SAV