# Evtukhov V. M.

### On the asymptotic of solutions of second-order differential equations with rapidly varying nonlinearities

Chernikova A. G., Evtukhov V. M.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 73-91

We establish the conditions of existence for one class of monotone solutions of two-term nonautonomous differential equations of the second order with rapidly varying nonlinearities and the asymptotic representations of these solutions and their first-order derivatives as $t \uparrow \omega (\omega \leq +\infty )$.

### Asymptotic behavior of the solutions of second-order differential equations with rapidly varying nonlinearities

Chernikova A. G., Evtukhov V. M.

Ukr. Mat. Zh. - 2017. - 69, № 10. - pp. 1345-1363

We establish conditions for the existence of one class of solutions of two-term nonautonomous differential equations of the second-order with rapidly varying nonlinearities and the asymptotic representations for these solutions and their first-order derivatives as и $t \uparrow \omega (\omega \leq +\infty )$.

### Asymptotic representation of solutions of differential equations with rightly varying nonlinearities

Evtukhov V. M., Korepanova Е. S.

Ukr. Mat. Zh. - 2017. - 69, № 9. - pp. 1198-1216

The conditions of existence of some types of power-mode solutions of a binomial nonautonomous ordinary differential equation with regularly varying nonlinearities are established.

### Asymptotic Representations for Some Classes of Solutions of Ordinary Differential Equations of Order $n$ with Regularly Varying Nonlinearities

Ukr. Mat. Zh. - 2013. - 65, № 3. - pp. 354-380

Existence conditions and asymptotic (as $t \uparrow \omega (\omega \leq +\infty)$) representations are obtained for one class of monotone solutions of an $n$th-order differential equation whose right-hand side contains a sum of terms with regularly varying nonlinearities.

### Asymptotics of Solutions of Nonautonomous Second-Order Ordinary Differential Equations Asymptotically Close to Linear Equations

Ukr. Mat. Zh. - 2012. - 64, № 10. - pp. 1346-1364

Asymptotic representations are obtained for a broad class of monotone solutions of nonautonomous binary differential equations of the second order that are close in a certain sense to linear equations.

### Asymptotic representations of solutions of essentially nonlinear systems of ordinary differential equations with regularly and rapidly varying nonlinearities

Evtukhov V. M., Shlepakov O. R.

Ukr. Mat. Zh. - 2012. - 64, № 9. - pp. 1165-1185

We obtain asymptotic representations for one class of solutions of systems of ordinary differential equations more general than systems of the Emden – Fowler type.

### Existence criteria and asymptotics for some classes of solutions of essentially nonlinear second-order differential equations

Ukr. Mat. Zh. - 2011. - 63, № 7. - pp. 924-938

We establish existence theorems and asymptotic representations for some classes of solutions of second-order differential equations whose right-hand sides contain nonlinearities of a more general form than nonlinearities of the Emden - Fowler type.

### Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point

Evtukhov V. M., Samoilenko A. M.

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 52 - 80

We establish conditions for the existence of solutions vanishing at a singular point for various classes of systems of quasilinear differential equations appearing in the investigation of the asymptotic behavior of solutions of essentially nonlinear nonautonomous differential equations of higher orders.

### Asymptotic representations of solutions of essentially nonlinear two-dimensional systems of ordinary differential equations

Ukr. Mat. Zh. - 2009. - 61, № 12. - pp. 1597-1611

We establish asymptotic representations for one class of solutions of two-dimensional systems of ordinary differential equations that are more general than systems of the Emden–Fowler type.

### Asymptotic representations of the solutions of essentially nonlinear nonautonomous second-order differential equations

Belozerova M. A., Evtukhov V. M.

Ukr. Mat. Zh. - 2008. - 60, № 3. - pp. 310–331

We establish asymptotic representations for the solutions of a class of nonlinear nonautonomous second-order differential equations.

### Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1363–1375

We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type.

### Conditions of oscillatory or nonoscillatory nature of solutions for a class of second-order semilinear differential equations

Evtukhov V. M., Vasileva N. S.

Ukr. Mat. Zh. - 2007. - 59, № 4. - pp. 458–466

For a class of second-order semilinear differential equations, we prove the theorems on oscillatory or nonoscillatory nature of all proper solutions. These theorems are analogs of the well-known Kneser theorems for linear differential equations.

### Asymptotic behavior of unbounded solutions of essentially nonlinear second-order differential equations. II

Evtukhov V. M., Kas'yanova V. A.

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 901–921

We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type.

### Asymptotic Behavior of Unbounded Solutions of Essentially Nonlinear Second-Order Differential Equations. I

Evtukhov V. M., Kas'yanova V. A.

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 338–355

We establish asymptotic representations for one class of unbounded solutions of second-order differential equations whose right-hand sides contain a sum of terms with nonlinearities of a more general form than nonlinearities of the Emden-Fowler type.

### Asymptotic Representations for Solutions of One Class of Systems of Quasilinear Differential Equations

Ukr. Mat. Zh. - 2003. - 55, № 12. - pp. 1658-1668

We establish asymptotic representations for solutions of one class of systems of differential equations appearing in the investigation of the asymptotic behavior of *n*th-order quasilinear differential equations.

### On Some Problems of the Asymptotic Theory of Linear Differential Equations of the nth Order

Ukr. Mat. Zh. - 2002. - 54, № 1. - pp. 20-42

We investigate smoothness properties of the roots of algebraic equations with almost constant coefficients and construct a transformation, which may be efficiently used for the investigation of the asymptotic behavior of a fundamental family of solutions of a broad class of nonautonomous linear differential equations of the *n*th order.

### On conditions for oscillation and nonoscillation of the solutions of a semilinear second-order differential equation

Ukr. Mat. Zh. - 1994. - 46, № 7. - pp. 833–841

We establish sufficient conditions for oscillation and nonoscillation of regular solutions of the secondorder differential equation $y'' + p(t)|y|^{1-\lambda} |y'|^{\lambda} \,{\text {\rm sign }}\, y = 0$, where $λ < 1$ and $p: [a,ω[→ ℝ,\, −∞ < a < ω ≤ +∞$ is a locally summable function.