# 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

We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous differential system of a special form as*t ?* ?, ? ? + ?. For this system, the coefficient matrix of a differential system of the first approximation has almost Jordan form with triangular blocks. We indicate methods that enable one to reduce certain classes of differential systems of the general form to special differential systems.

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

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