# Volume 62, № 9, 2010

### On impulsive Sturm–Liouville operators with Coulomb potential and spectral parameter linearly contained in boundary conditions

Amirov R. Kh., Güldü Y., Topsakal N.

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1155–1172

The Sturm–Liouville problem with linear discontinuities is investigated in the case where an eigenparameter appears not only in a differential equation but also in boundary conditions. Properties and the asymptotic behavior of spectral characteristics are studied for the Sturm–Liouville operators with Coulomb potential that have discontinuity conditions inside a finite interval. Moreover, the Weyl function for this problem is defined and uniqueness theorems are proved for a solution of the inverse problem with respect to this function.

### Quasilinear hyperbolic stefan problem with nonlocal boundary conditions

Andrusyak R. V., Burdeina N. O., Kirilich V. M.

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1173–1199

Using the method of contracting mappings, we prove, for small values of time, the existence and uniqueness of a generalized Lipschitz solution of a mixed problem with unknown boundaries for a hyperbolic quasilinear system of first-order equations represented in terms of Riemann invariants with nonlocal (nonseparated and integral) boundary conditions.

### On the problem of determining the parameter of a parabolic equation

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1200–1210

We study the boundary-value problem of determining the parameter p of a parabolic equation $$v′(t)+Av(t)=f(t)+p,\;0⩽t⩽1,v(0)=φ,\;v(1)=ψ,$$ with strongly positive operator $A$ in an arbitrary Banach space $E$. The exact estimates are established for the solution of this problem in Hölder norms. In applications, the exact estimates are obtained for the solutions of the boundary-value problems for parabolic equations.

### Solvability of boundary-value problems for nonlinear fractional differential equations

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1211–1219

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations $$D^{α}u(t)+λ[f(t,u(t))+q(t)]=0,\; 0 < t < 1, \; u(0) = 0,\; u(1) = βu(η),$$ where $λ > 0$ is a parameter, $1 < α ≤ 2,\; η ∈ (0, 1),\; β ∈ \mathbb{R} = (−∞,+∞),\; βη^{α−1} ≠ 1,\; D^{α}$ is a Riemann–Liouville differential operator of order $α$, $f: (0,1)×\mathbb{R}→\mathbb{R}$ is continuous, $f$ may be singular for $t = 0$ and/or $t = 1$, and $q(t) : [0, 1] → [0, +∞)$. We give some sufficient conditions for the existence of nontrivial solutions to the formulated boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. In particular, we do not use the assumption of nonnegativity and monotonicity of $f$ essential for the technique used in almost all available literature.

### Elements of a non-gaussian analysis on the spaces of functions of infinitely many variables

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1220–1246

We present a review of some results of the non-Gaussian analysis in the biorthogonal approach and consider elements of the analysis associated with the generalized Meixner measure. The main objects of our interest are stochastic integrals, operators of stochastic differentiation, elements of theWick calculus, and related topics.

### Banach algebra generated by a finite number of bergman polykernel operators, continuous coefficients, and a finite group of shifts

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1247–1255

We study the Banach algebra generated by a finite number of Bergman polykernel operators with continuous coefficients that is extended by operators of weighted shift that form a finite group. By using an isometric transformation, we represent the operators of the algebra in the form of a matrix operator formed by a finite number of mutually complementary projectors whose coefficients are Toeplitz matrix functions of finite order. Using properties of Bergman polykernel operators, we obtain an efficient criterion for the operators of the algebra considered to be Fredholm operators.

### Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations

Ral’chenko K. V., Shevchenko H. M.

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1256–1268

We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion.

### On one class of extreme extensions of a measure

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1269–1279

We consider a relationship between two sets of extensions of a finite finitely additive measure $μ$ defined on an algebra $\mathfrak{B}$ of sets to a broader algebra $\mathfrak{A}$. These sets are the set $\text{ex} S_{μ}$ of all extreme extensions of the measure $μ$ and the set $H_{μ}$ of all extensions defined as $λ(A) = \widehat{\mu}(h(A)), A ∈ \mathfrak{A}$, where $\widehat{\mu}$ is a quotient measure on the algebra $\mathfrak{B}/μ$ of the classes of $μ$-equivalence and $h: \mathfrak{A} →\mathfrak{B}/μ$ is a homomorphism extending the canonical homomorphism $\mathfrak{B}$ to $\mathfrak{B}/μ$. We study the properties of extensions from $H_{μ}$ and present necessary and sufficient conditions for the existence of these extensions, as well as the conditions under which the sets $\text{ex} S_{μ}$ and $H_{μ}$ coincide.

### On Fourier multipliers and absolute convergence of Fourier integrals of radial functions

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1280–1293

We obtain sufficient conditions for the representability of a function in the form of an absolutely convergent Fourier integral. These conditions are given in terms of the joint behavior of the function and its derivatives at infinity, and their efficiency and exactness are verified with the use of a known example. We also consider radial functions of an arbitrary number of variables.

### Quantization of Lyapunov functions

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1294–1296

For a state of equilibrium of an autonomous system of differential equations, we propose discrete conditions for stability and asymptotic stability in the sense of Lyapunov.