2019
Том 71
№ 9

# Полищук Є. М.

Публікацій: 2
Стаття (українською)

### Интегральные представления аналитических функционалов

Укр. мат. журн. - 1974. - 26, № 4. - С. 487–495

Стаття (російською)

### О дифференциальных уравнениях с функциональными параметрами

Укр. мат. журн. - 1963. - 15, № 1. - С. 13-24

The author investigates a differential equation of the form $$y^m = f(t, y, y',..., y^{m-1})\varphi(x_1(t), ...x_p(t))\quad (1)$$ where $t$ is an independent variable, $y$ is an unknown function, $x_1,...x_p$ are parameters depending on $t$. It is shown that under definite conditions in respect to $f$ and $\varphi$, if the function $\varphi$ satisfies the differential equation $$\sum_{k=1}^pa_k\frac{\partial^2\varphi}{\partial x^2_k}$$ then on varying the functions $x_1,...x_p$ the integral $Y[x_1,...x_p; t]$ of equation (1), considered as a functional of $x_1,...x_p$ satisfies the equation $\sum a_k\delta_k Y = 0$, where $\delta_k$ is a partial functional Laplace operator in respect to the function $x_k(t),\; k = 1, 2,.., p$. If equation (1) has the form $$y^m = f_1(t, y, y',..., y^{m-1}) + f_2(t, y, y',..., y^{m-1})$$ then on varying $x$, its integral $Y [x/t]$ satisfies the equation $\delta Y = 0$ (i. e. is a harmonic functional of $x$). As examples the Ricatti and Schwarz integrals are presented as discussed by Fantappie in his theory of analytical functionals.