2019
Том 71
№ 11

# Evtukhov V. M.

Articles: 17
Article (Russian)

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

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 $t \uparrow \omega (\omega \leq +\infty )$.

Article (Russian)

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

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 $t \uparrow \omega (\omega \leq +\infty )$.

Article (Russian)

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

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.

Article (Russian)

### 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.

Article (Russian)

### 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.

Article (Russian)

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

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.

Article (Russian)

### 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.

Article (Russian)

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

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.

Article (Russian)

### 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.

Article (Russian)

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

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.

Article (Russian)

### 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.

Article (Russian)

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

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.

Article (Russian)

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

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.

Article (Russian)

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

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.

Article (Russian)

### 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 nth-order quasilinear differential equations.

Article (Russian)

### 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 nth order.

Article (Ukrainian)

### 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.