# Mokhon'ko O. A.

### On the Order of Growth of the Solutions of Linear Differential Equations in the Vicinity of a Branching Point

Mokhon'ko A. Z., Mokhon'ko O. A.

↓ Abstract

Ukr. Mat. Zh. - 2015. - 67, № 1. - pp. 139-144

Assume that the coefficients and solutions of the equation $f^{(n)}+p_{n−1}(z)f^{(n−1)} +...+ p_{s+1}(z)f^{(s+1)} +...+ p_0(z)f = 0$ have a branching point at infinity (e.g., a logarithmic singularity) and that the coefficients $p_j , j = s+1, . . . ,n−1$, increase slower (in terms of the Nevanlinna characteristics) than $p_s(z)$. It is proved that this equation has at most $s$ linearly independent solutions of finite order.

### Malmquist Theorem for the Solutions of Differential Equations in the Vicinity of a Branching Point

Mokhon'ko A. Z., Mokhon'ko O. A.

Ukr. Mat. Zh. - 2014. - 66, № 9. - pp. 1286–1290

An analog of the Malmquist theorem on the growth of solutions of the differential equation $f' = P(z, f)/Q(z, f)$, where $P(z, f)$ and $Q(z, f)$ are polynomials in all variables, is proved for the case where the coefficients and solutions of this equation have a branching point in infinity (e.g., a logarithmic singularity).

### Deficiency Values for the Solutions of Differential Equations with Branching Point

Mokhon'ko A. Z., Mokhon'ko O. A.

Ukr. Mat. Zh. - 2014. - 66, № 7. - pp. 939–957

We study the distribution of values of the solutions of an algebraic differential equation *P*(*z, f, f′, . . . , f* ^{(s)}) = 0 with the property that its coefficients and solutions have a branching point at infinity (e.g., a logarithmic singularity). It is proved that if *a* ∈ ℂ is a deficiency value of *f* and *f* grows faster than the coefficients, then the following identity takes place: *P*(*z, a,* 0*, . . . ,* 0) ≡ 0*, z* ∈ {*z* : *r* _{0} ≤ *|z| <* ∞}*.* If *P*(*z, a,* 0*, . . . ,* 0) is not identically equal to zero in the collection of variables *z* and *a,* then only finitely many values of *a* can be deficiency values for the solutions *f* ∈ *M* _{ b } with finite order of growth.

### Nonisospectral flows on semiinfinite unitary block Jacobi matrices

Ukr. Mat. Zh. - 2008. - 60, № 4. - pp. 521–544

It is proved that if the spectrum and spectral measure of a unitary operator generated by a semiinfinite block Jacobi matrix $J(t)$ vary appropriately,
then the corresponding operator $\textbf{J}(t)$ satisfies the generalized
Lax equation $\dot{\textbf{J}}(t) = \Phi(\textbf{J}(t), t) + [\textbf{J}(t), A(\textbf{J}(t), t)]$,
where $\Phi(\lambda, t)$ is a polynomial in $\lambda$ and $\overline{\lambda}$ with $t$-dependent coefficients and $A(J(t), t) = \Omega + I + \frac12 \Psi$ is a skew-symmetric matrix.

The operator $J(t$) is analyzed in the space ${\mathbb C}\oplus{\mathbb C}^2\oplus{\mathbb C}^2\oplus...$.
It is mapped into the unitary operator of multiplication $L(t)$ in the isomorphic space $L^2({\mathbb T}, d\rho)$, where ${\mathbb T} = {z: |z| = 1}$.
This fact enables one to construct an efficient algorithm for solving the block lattice of differential equations generated by the Lax equation.
A procedure that allows one to solve the corresponding Cauchy problem by the Inverse-Spectral-Problem method is presented.

The article contains examples of block difference-differential lattices and the corresponding flows that are analogues of the Toda and van Moerbeke lattices
(from self-adjoint case on ${\mathbb R}$)
and some notes about applying this technique for Schur flow (unitary case on ${\mathbb T}$ and OPUC theory).

### On the Malmquist Theorem for Solutions of Differential Equations in the Neighborhood of an Isolated Singular Point

Ukr. Mat. Zh. - 2005. - 57, № 4. - pp. 505–513

The statement of the Malmquist theorem (1913) about the growth of meromorphic solutions of the differential equation \(f' = \frac{{P(z,f)}}{{Q(z,f)}}\), where *P*(*z, f*) and *Q*(*z, f*) are polynomials in all variables, is proved in the case of solutions with isolated singular point at infinity.

### On Some Solvable Classes of Nonlinear Nonisospectral Difference Equations

Ukr. Mat. Zh. - 2005. - 57, № 3. - pp. 356–365

We investigate different measure transformations of the mapping-multiplication type in the cases where the corresponding chains of differential equations can be efficiently found and integrated.

### Malmquist theorem for solutions of differential equations in a neighborhood of a logarithmic singular point

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 476–483

The Malmquist theorem (1913) on the growth of meromorphic solutions of the differential equation *f ′ = P(z,f) / Q(z,f)*, where *P(z,f)* and *Q(z,f)* are polynomials in all variables, is proved for the case of meromorphic solutions with logarithmic singularity at infinity.