# Volume 60, № 2, 2008

### Forced oscillations of an infinite-dimensional oscillator under impulsive perturbations

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 155–166

Existence and uniqueness theorems for the impulsive differential operator equation $$ \frac{d^2}{dt^2}[Au(t)] + Bu(t) = f(t, u(t))$$ are obtained. The operator A is allowed to be noninvertible. The results are applied to differential algebraic equations and partial differential equations, which are not equations of Kovalevskaya type.

### General conditions for the unique solvability of initial-value problem for nonlinear functional differential equations

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 167–172

We establish general conditions for the unique solvability of the Cauchy problem for systems of nonlinear functional differential equations.

### Localization of the limit set of trajectories of the Euler-Bernoulli equation with control

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 173–182

We investigate a differential equation in a Hilbert space that describes vibrations of the Euler-Bernoulli elastic beam with feedback control. The relative compactness of positive semitrajectories of the considered equation is proved. Constructing a Lyapunov functional in explicit form and using the invariance principle, we obtain representations of limit sets.

### Controllability in oscillation dynamical systems

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 183–191

We consider the problem of controllability in oscillation dynamical systems. A solution of the local control problem is obtained for one class of systems of differential equations. An example of application of the main results is given.

### Well-posedness of boundary-value problems for multidimensional hyperbolic systems

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 192–203

By the method of characteristics, we investigate the well-posedness of local (the Cauchy problem, mixed problems) and nonlocal (with nonseparable and integral boundary conditions) problems for some multidimensional almost linear first-order hyperbolic systems. Reducing these problems to the systems of integral operator equations, we prove the existence and uniqueness of classical solutions.

### On stability of linear hybrid mechanical systems with distributed components

Martynyuk A. A., Slyn'ko V. I.

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 204–216

We present a new approach to the solution of problems of stability of hybrid systems based on the constructive determination of elements of a matrix-valued functional.

### On properties of solutions of a limit problem for systems of nonlinear functional differential equations of neutral type

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 217–224

For a class of systems of nonlinear differential-functional equations, we study asymptotic characteristics of their solutions continuously differentiable and bounded for
*t* >* T *> 0 (along with the first derivative).

### Solvability of semilinear differential equations with singularity

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 225–239

Local theorems on the existence of solutions of the Cauchy problem for the singular equations of the form $$ \frac{d}{dt}(Au(t)) + Bu(t) = f(t, u)$$ in Banach spaces are proved. The conditions for the solvability depend on a type of the singularity of the sheaf $\lambda A + B$ of closed linear operators $A, B$. Examples and applications to finite-dimensional differential algebraic equations, infinite systems of differential equations, and partial differential equations of non-Kovalevskaya type are presented.

### Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 240–259

The paper presents existence principles for the nonlocal boundary-value problem $$ (\phi (u^(p-1)))' = g(t, u,...,u^{(p-1)}), \alpha_k(u)=0, 1 \leq k \leq p-1$$ where $p\geq2,\quad \phi: {\mathbb R}\rightarrow{\mathbb R}$ is an increasing and odd homeomorphism, $g$ is a Caratheodory function which is either regular or has singularities in its space variables and $\alpha_k: C^{p-1}[0,T]\rightarrow{\mathbb R}$ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems $(-1)^n(\phi(u^{(2n-1)}))' = f (t,u,...,u^{(2n-1)}),\quad u^{(2k)}(0) = 0,\quad$ $a_ku^{(2k)}(T) + b_k u^{(2k+1)}(T)=0,\quad 0\leq k\leq n-1$ is given.

### Global attractor for the autonomous wave equation in *R*_{n } with continuous nonlinearity

_{n }

Horban’ N. V., Stanzhitskii A. N.

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 260–267

We investigate the dynamics of solutions of an autonomous wave equation in ℝn with continuous nonlinearity. A priori estimates are obtained. We substantiate the existence of an invariant global attractor for an m-semiflow.

### On the smoothness of conjugation of circle diffeomorphisms with rigid rotations

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 268–282

We prove that any *C ^{3+β }*-smooth orientation-preserving circle diffeomorphism with rotation number
from the Diophantine class

*D*0 < β < δ < 1, is

_{δ},*C*-smoothly conjugate to the rigid rotation of the circle by appropriate angle.

^{ 2+β-δ }### Some remarks on linear functional differential inequalities of hyperbolic type

Ukr. Mat. Zh. - 2008. - 60, № 2. - pp. 283–292

It is proved that for the validity of a theorem on differential inequalities for the hyperbolic equation $$ \frac{\partial^2u(t, x)}{\partial t \partial x} = l(u)(t,x)+q(t,x)$$ with a nonincreasing linear operator $l: {C}([a, b]\times [c,d];\mathbb{R})\rightarrow{L}([a, b]\times [c,d];\mathbb{R})$, it is necessary that the operator indicated be an $(a, c)$-Volterra operator.