# Vitrychenko I. E.

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

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.

### Critical Cases of the π-Stability of a Nonautonomous Quasilinear Equation of the *n*th Order

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 *n*th order.

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

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.

### On the functional polystability of certain essentially nonlinear nonautonomous differential systems

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.

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

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 \(\left| {\lambda _n } \right| = 1 + o(1), n \to + \infty \) .

### Functional polystability of some nonautonomous quasilinear differential systems

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.

### On the instability of one nonautonomous essentially nonlinear equation of the*n*th order

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 the*n*th order in the case where its limit characteristic equation has a multiple zero root. The instability is determined by nonlinear terms.

### Critical cases of stability of one nonautonomous essentially nonlinear equation of the *n*th order

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 *n*th order in the case where its characteristic equation has a multiple zero root. The stability is determined by nonlinear terms.

### On stability of an*n*th-order equation in a critical case

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

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

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

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.

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

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.