2017
Том 69
№ 9

Volume 60, № 2, 2008

Article (Russian)

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.

Article (Ukrainian)

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.

Article (Ukrainian)

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.

Article (Russian)

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.

Article (Ukrainian)

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.

Article (Russian)

On stability of linear hybrid mechanical systems with distributed components

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.

Article (Russian)

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).

Article (Russian)

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.

Article (English)

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.

Article (Ukrainian)

Global attractor for the autonomous wave equation in Rn with continuous nonlinearity

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.

Article (Ukrainian)

On the smoothness of conjugation of circle diffeomorphisms with rigid rotations

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

We prove that any C3+β -smooth orientation-preserving circle diffeomorphism with rotation number from the Diophantine class Dδ , 0 < β < δ < 1, is C 2+β-δ -smoothly conjugate to the rigid rotation of the circle by appropriate angle.

Article (Russian)

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.