# Volume 51, № 6, 1999

### Some problems in the theory of nonoverlapping domains

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 723–731

We generalize some results concerning extremal problems of nonoverlapping domains with free poles on the unit circle.

### Estimates for the modulus of a Cauchy-type integral and its derivatives

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 732–743

In a domain bounded by a closed rectifiable Jordan curve, we obtain estimates for the modulus of a Cauchy-type integral and its derivatives in terms of the contour moduli of smoothness of the integrand and a metric characteristic of the curve.

### On the renewal of a Wiener field on a plane with the use of its values on closed curves

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 744–752

For*w(u, v), (u, v)*∉ γ (here,*w(x, y), x≥0, y≥0*, is a Wiener field and γ is a certain closed curve on a plane), we construct the best mean-square estimate on the basis of the values of*w(x, y)* for (*x, y*)∈ γ. We also calculate the error of this estimate.

### Nonsymmetric problem of divisors in an arithmetic progression

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 753–761

We investigate the distribution of values of a nonsymmetric divisor function*d(a,b; n)* in an arithmetic progression with increasing difference.

### Variational method for the solution of problems of transmission with the principal conjugation condition

Komarenko A. N., Trotsenko V. A.

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 762–775

We prove the existence of a solution of a variational minimax problem that is equivalent to the problem of transmission. We propose an algorithm for the construction of approximate solutions and prove its convergence.

### Equations for second moments of solutions of a system of linear differential equations with random semi-Markov coefficients and random input

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 776–783

We derive equations that determine second moments of a random solution of a system of Itô linear differential equations with coefficients depending on a finite-valued random semi-Markov process. We obtain necessary and sufficient conditions for the asymptotic stability of solutions in the mean square with the use of moment equations and Lyapunov stochastic functions.

### Stability analysis of linear impulsive differential systems under structural perturbation

Martynyuk A. A., Stavroulakis I. P.

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 784–795

The stability and asymptotic stability of solutions of large-scale linear impulsive systems under structural perturbations are investigated. Sufficient conditions for stability and instability are formulated in terms of the fixed signs of special matrices.

### On deviations and defects of meromorphic functions of finite lower order

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 796–803

We obtain estimates for the sum of deviations and sum of defects to power 1/2 in terms of the Valiron defect of the derivative at zero. In particular, the Fuchs hypothesis (1958) is verified.

### Optimal stopping times for solutions of nonlinear stochastic differential equations and their application to one problem of financial mathematics

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 804–809

We solve the problem of finding the optimal switching time for two alternative strategies at the financial market in the case where a random process*X* _{ t },*t ∈ [0, T]*, describing an investor's assets satisfies a nonlinear stochastic differential equation. We determine this switching time τ∈[0,*T*] as the optimal stopping time for a certain process*Y* _{ t } generated by the process*X* _{ t } so that the average investor's assets are maximized at the final time, i.e.,*EX* _{ T }.

### Dynamics of solutions of the simplest nonlinear boundary-value problems

Romanenko O. Yu., Sharkovsky O. M.

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 810–826

We investigate two classes of essentially nonlinear boundary-value problems by using methods of the theory of dynamical systems and two special metrics. We prove that, for boundary-value problems of both these classes, all solutions tend (in the first metric) to upper semicontinuous functions and, under sufficiently general conditions, the asymptotic behavior of almost every solution can be described (by using the second metric) by a certain stochastic process.

### On periodic solutions of the equation of a nonlinear oscillator with pulse influence

Samoilenko A. M., Samoilenko V. G., Sobchuk V. S.

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 827–834

We study periodic solutions and the behavior of phase trajectories of the differential equation of a nonlinear oscillator with pulse influence at unfixed moments of time.

### 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.

### On certain exact relations for sojourn probabilities of a wiener process

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 842–846

New exact relations are proved for the sojourn probability of a Wiener process between two time-de-pendent boundaries. The proof is based on the investigation of the heat-conduction equation in the domain determined by these functions-boundaries. The relations are given in the form of series.

### Limit behavior of the distribution of the ruin moment of a modified risk process

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 847–853

For modified risk process with instantaneous reflection at the point $B > 0$ under which the considered process $$\zeta(t) = \zeta_{B, \mu}(t),\; \zeta(0) = u,\; 0 \leq u \leq B,$$ returns in the initial state $u$, we investigate the limit behavior of generating function of the first ruin moment as $u \rightarrow B$ and $B \rightarrow \infty$.

### On the Chernikova theorem on well-factorizable groups

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 854–855

We describe finite groups *G* every subgroup of which that does not belong to the Frattini subgroup ϕ (*G*) has a complement.

### Integrability of Riccati equations and stationary Korteweg-de vries equations

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 856–860

By using the Lie infinitesimal method, we establish the correspondence between the integrability of a one-parameter family of Riccati equations and the hierarchy of the higher Korteweg-de Vries equations.

### Asymptotics of solutions of discontinuous singularly perturbed boundary-value problems

Ukr. Mat. Zh. - 1999. - 51, № 6. - pp. 861–864

We construct an asymptotic expansion of a boundary-value problem for a singularly perturbed system of differential equations with the right-hand side discontinuous at certain surface.