2019
Том 71
№ 11

All Issues

Vitrychenko I. E.

Articles: 11
Brief Communications (Ukrainian)

On a special critical case of stability of a nonautonomous essentially nonlinear system

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1711–1718

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous essentially nonlinear differential system in a special critical case.

Brief Communications (Ukrainian)

Critical Cases of the π-Stability of a Nonautonomous Quasilinear Equation of the nth Order

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 264-270

We establish sufficient conditions for the π-stability of the trivial solution of a quasilinear equation of the nth order.

Article (Ukrainian)

Global λ-Stability of One Nonautonomous Quasilinear Second-Order Equation

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2002. - 54, № 9. - pp. 1172-1189

We establish sufficient conditions for the λ-stability of the trivial solution of one quasilinear differential equation of the second order.

Article (Russian)

On the functional polystability of certain essentially nonlinear nonautonomous differential systems

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2000. - 52, № 2. - pp. 197-207

For essentially nonlinear differential systems with the limit matrix of coefficients of the first-approximation system, we establish sufficient conditions for functional polystability, which generalizes the notion of exponential polystability.

Article (Russian)

A critical case of stability of one quasilinear difference equation of the second order

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 12. - pp. 1593–1603

We obtain sufficient conditions for the Perron stability of the trivial solution of a real difference equation of the form $$y_{n + 1} - 2\lambda _n y_n + y_{n - 1} = F(n,y_n ,\Delta y_{n - 1} ), n \in N$$ where \(y_n \in \left] { - 1,1} \right[,\left| {F(n,y_n ,\Delta y_{n - 1} )} \right| \le L_n \left( {\left| {y_n \left| + \right|\Delta y_{n - 1} } \right|} \right)^{1 + \alpha } ,L_n \ge 0\) and \(\alpha \in \left] {0, + \infty } \right[\) . The resuits obtained are valid for the case where \(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \) .

Brief Communications (Russian)

Functional polystability of some nonautonomous quasilinear differential systems

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 7. - pp. 989–995

For quasilinear differential systems with a boundary matrix of coefficients of the system of the first approximation, we obtain sufficient conditions of functional polystability, which generalizes the notion of exponential polystability.

Brief Communications (Russian)

On the instability of one nonautonomous essentially nonlinear equation of thenth order

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 835–841

We establish sufficient conditions for the Lyapunov instability of the trivial solution of a nonautonomous equation of thenth order in the case where its limit characteristic equation has a multiple zero root. The instability is determined by nonlinear terms.

Brief Communications (Ukrainian)

Critical cases of stability of one nonautonomous essentially nonlinear equation of the nth order

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 720–724

We establish sufficient conditions of the Lyapunov stability of the trivial solution of a nonautonomous ordinary differential equation of the nth order in the case where its characteristic equation has a multiple zero root. The stability is determined by nonlinear terms.

Brief Communications (Russian)

On stability of annth-order equation in a critical case

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1995. - 47, № 8. - pp. 1138–1143

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomousnth-order equation in the case where the root of the boundary characteristic equation is equal to zero and has multiplicity greater than one.

Brief Communications (Russian)

On stability of the trivial solution of a nonautonomous quasilinear system whose characteristic equation has multiple roots

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1994. - 46, № 8. - pp. 1072–1079

For $t \uparrow \omega, \quad \omega \leq +\infty$, we obtain sufficient conditions for Lyapunov stability of the zero solution of a specific nonautonomous quasilinear differential system in the case where the matrix of the first-degree approximation has the Jordan form with triangular blocks. Methods to reduce certain classes of general differential systems to differential systems of special type are given.

Article (Ukrainian)

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Vitrychenko I. E.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1994. - 46, № 4. - pp. 347–356

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for $t = ω$) solutions of the nonautonomous quasilinear equation $y'' \pm \lambda (t)y = F(t,y,y'),$, where $t ∈ Δ = [a, ω[,-∞ < a < ω ≤ + ∞, λ(t) > 0, λ(t) ∈ C_Δ^{(1)},$ $ |F((t,x,y))| ≤ L(t)(|x|+|y|)^{1+α}, L(t) ≥ -0, α ∈ [0,+∞[,$ $ F: Δ × R^2 →R, F ∈ C_{Δ × R^2}, R$ is the set of real numbers, and $R^2$ is the two-dimensional real Euclidean space.