# Gomilko A. M.

### Inverse Sturm-Liouville problem on a figure-eight graph

Gomilko A. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2008. - 60, № 9. - pp. 1168–1188

We study the inverse problem for the Strum-Liouville equation on a graph that consists of two quasione-dimensional loops of the same length having a common vertex. As spectral data, we consider the set of eigenvalues of the entire system together with the sets of eigenvalues of two Dirichlet problems for the Sturm-Liouville equations with the condition of total reflection at the vertex of the graph. We obtain conditions for three sequences of real numbers that enable one to reconstruct a pair of real potentials from L 2 corresponding to each loop. We give an algorithm for the construction of the entire set of potentials corresponding to this triple of spectra.

### On a criterion for the uniform boundedness of a *C*_{0}-semigroup of operators in a Hilbert space

Gomilko A. M., Wróbel I., Zemanek J.

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 853-858

Let $T(t),\quad t ≥ 0$, be a $C_0$-semigroup of linear operators acting in a Hilbert space $H$ with norm $‖·‖$. We prove that $T(t)$ is uniformly bounded, i.e., $‖T(t)‖ ≤ M, \quad t ≥ 0$, if and only if the following condition is satisfied: $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds < ∞$$ forall $x ∈ H$, where $T^{*}$ is the adjoint operator.

### Cayley transform of the generator of a uniformly bounded $C_0$-semigroup of operators

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1018-1029

We consider the problem of estimates for the powers of the Cayley transform $V = (А + I)(А - I)^{-1}$ of the generator of a uniformly bounded $C_0$-semigroup of operators $e^{tA} , t \geq 0$, that acts in a Hilbert space $H$. In particular, we establish the estimate $\sup_{n \in N}\left(||V^n||/\ln(n + 1)\right) < \infty$. We show that the estimate $\sup_{n ∈ N} ∥V^n∥ < ∞$ is true in the following cases: (a) the semigroups $e^{tA}$ and $e^{tA^{−1}}$ are uniformly bounded; (b) the semigroup etA uniformly bounded for $t ≥ ∞$ is analytic (in particular, if the generator of the semigroup is a bounded operator).

### On the Boundedness of a Recurrence Sequence in a Banach Space

Gomilko A. M., Gorodnii M. F., Lagoda O. A.

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1410-1418

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space *B*: \(x_n = \sum\limits_{k = 1}^\infty {A_k x_{n - k} + y_n } ,{ }n \geqslant 1,{ }x_n = {\alpha}_n ,{ }n \leqslant 0,\) where |*y* _{n}} and |α_{ n }} are sequences bounded in *B*, and *A* _{k}, *k* ≥ 1, are linear bounded operators. We prove that if, for any ε > 0, the condition \(\sum\limits_{k = 1}^\infty {k^{1 + {\varepsilon}} \left\| {A_k } \right\| < \infty } \) is satisfied, then the sequence |*x* _{n}} is bounded for arbitrary bounded sequences |*y* _{n}} and |α_{ n }} if and only if the operator \(I - \sum {_{k = 1}^\infty {\text{ }}z^k A_k } \) has the continuous inverse for every *z* ∈ *C*, | *z* | ≤ 1.

### Asymptotics of Solutions of an Infinite System of Linear Algebraic Equations in Potential Theory

Gomilko A. M., Koval'chuk V. F.

Ukr. Mat. Zh. - 2001. - 53, № 9. - pp. 1184-1193

For bounded solutions *x* _{k} of an infinite system of linear algebraic equations arising in potential theory in the course of investigation of an axially symmetric problem in the exterior of two spheres with equal radii, we obtain asymptotic formulas with respect to a parameter that characterizes the approach of the spheres to one another and for *k* → ∞.

### Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

Gomilko A. M., Pivovarchik V. N.

Ukr. Mat. Zh. - 2001. - 53, № 6. - pp. 742-757

On a finite segment [0, *l*], we consider the differential equation $$\left( {a\left( x \right)y\prime \left( x \right)} \right)\prime + \left[ {{\mu \rho }_{\text{1}} \left( x \right) + {\rho }_{2} \left( x \right)} \right]y\left( x \right) = 0$$ with a parameter μ ∈ *C*. In the case where *a*(*x*), ρ(*x*) ∈ *L* _{∞}[0, *l*], ρ_{ j }(*x*) ∈ *L* _{1}[0, *l*], *j* = 1, 2, *a*(*x*) ≥ *m* _{0} > 0 and ρ(*x*) ≥ *m* _{1} > 0 almost everywhere, and *a*(*x*)ρ(*x*) is a function absolutely continuous on the segment [0, *l*], we obtain exponential-type asymptotic formulas as \(\left| {\mu } \right| \to \infty\) for a fundamental system of solutions of this equation.

### Integral equations in the linear theory of elasticity in semiinfinite domains

Ukr. Mat. Zh. - 1998. - 50, № 5. - pp. 613–622

We investigate linear integral equations on a semiaxis that appear in the course of construction of solutions of boundary-value problems in the theory of elasticity in such domains as a semiinfinite strip or a cylinder. By using the Mellin transformation and the theory of perturbations of linear operators, we establish general results concerning the solvability and asymptotic properties of solutions of the equations considered. We give examples of application of the general statements obtained to specific integral equations in the theory of elasticity.

### Recurrent relations for the solutions of an infinite system of linear algebraic equations

Ukr. Mat. Zh. - 1995. - 47, № 10. - pp. 1328–1332

We obtain recurrent relations for bounded solutions of the system of equations $$X_k - \sum\limits_{n = 0}^\infty {\frac{{(k + n)!}}{{k!n!}}} \alpha ^{k + n + 1} x_n = f_{k,} k = 0,1,..., \alpha \in (0,1/2),$$ with right-hand sides {*f* _{ k }} _{ k=0} ^{ ∞ } ={δ_{ kj }} _{ k=0} ^{ ∞ } ,*j*=0,1,..., where δ_{ kj } is the Kronecker symbol.

### Expansion of a bundle of fourth-order differential operators in a part of its eigenfunctions

Ukr. Mat. Zh. - 1993. - 45, № 12. - pp. 1601–1612

A bundle of differential operators $$\mathcal{L}(\lambda ),\lambda \in \mathbb{C}:\mathcal{L}(\lambda )y(x) = y^{(4)} (x) - 2\lambda ^2 y^{(2)} (x) + \lambda ^4 y(x),|x| \leqslant 1,y( \pm 1) = y\prime ( \pm 1) = 0,$$ is considered. In various function spaces, we establish the facts about the expansions of a pair of functions $f(x)$ and $g(x)$ in the system $\{y_k (x),\; iλ_k y_k (x)|}_{k=1}^{ ∞}$, where $y_k(x), k = 1,2,...,$ are the eigenfunctions of the bundle $L (λ)$ corresponding to the eigenvalues $λ_k$, with $\Im λ_k > 0$.

### The Kontorovich-Lebedev integral transform

Ukr. Mat. Zh. - 1991. - 43, № 10. - pp. 1356–1361

### Continuous maps in scales of banach spaces

Ukr. Mat. Zh. - 1989. - 41, № 8. - pp. 1130–1134

### Law of asymptotic expressions in the theory of functional equations in *K*_{σ}-spaces

Ukr. Mat. Zh. - 1987. - 39, № 5. - pp. 551–554