2018
Том 70
№ 12

# Gorodnii M. F.

Articles: 13
Anniversaries (Ukrainian)

### Mykola Oleksiiovych Perestyuk (on his 70th birthday)

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 142-144

Anniversaries (Ukrainian)

### Dmytro Ivanovych Martynyuk (on the 70th anniversary of his birthday)

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 571-573

Brief Communications (Ukrainian)

### On the boundedness of one recurrent sequence in a banach space

Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1293-1296

We establish necessary and sufficient conditions under which a sequence $x_0 = y_0,\; x_{n+1} = Ax_n + y_{n+1},\; n ≥ 0$, is bounded for each bounded sequence $\{y_n : n ⩾ 0\} ⊂ \left\{x ∈ ⋃^{∞}_{n=1} D(A_n)|\sup_{n ⩾ 0} ∥A^nx∥ < ∞\right\}$, where $A$ is a closed operator in a complex Banach space with domain of definition $D(A)$.

Article (Ukrainian)

### On the invertibility of the operator d/dt + A in certain functional spaces

Ukr. Mat. Zh. - 2007. - 59, № 8. - pp. 1020–1025

We prove that the operator $\cfrac{d}{dt} + A$ constructed on the basis of a sectorial operator $A$ with spectrum in the right half-plane of $ℂ$ is continuously invertible in the Sobolev spaces $W_p^1 (ℝ, D_{α}),\; α ≥ 0$. Here, $D_{α}$ is the domain of definition of the operator $A^{α}$ and the norm in $D_{α}$ is the norm of the graph of $A^{α}$.

Obituaries (Ukrainian)

### Anatolii Yakovych Dorogovtsev

Ukr. Mat. Zh. - 2004. - 56, № 8. - pp. 1151-1152

Brief Communications (Russian)

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

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 zC, | z | ≤ 1.

Article (Ukrainian)

### Stability of Bounded Solutions of Differential Equations with Small Parameter in a Banach Space

Ukr. Mat. Zh. - 2003. - 55, № 7. - pp. 889-900

For a sectorial operator A with spectrum σ(A) that acts in a complex Banach space B, we prove that the condition σ(A) ∩ i R = Ø is sufficient for the differential equation $\varepsilon x_\varepsilon^\prime\prime(t)+x_\varepsilon^\prime(t)=Ax_\varepsilon(t)+f(t), t \in R,$ where ε is a small positive parameter, to have a unique bounded solution x ε for an arbitrary bounded function f: RB that satisfies a certain Hölder condition. We also establish that bounded solutions of these equations converge uniformly on R as ε → 0+ to the unique bounded solution of the differential equation x′(t) = Ax(t) + f(t).

Brief Communications (Ukrainian)

### $l_p$-Solutions of One Difference Equation in a Banach Space

Ukr. Mat. Zh. - 2003. - 55, № 3. - pp. 425-430

We establish a criterion for the existence and uniqueness of solutions of a linear difference equation with an unbounded operator coefficient belonging to the space $l_p(B)$ of sequences of elements of a Banach space $B$.

Article (Ukrainian)

### Bounded Solutions for Some Classes of Difference Equations with Operator Coefficients

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1495-1500

We obtain necessary and sufficient conditions for the existence and uniqueness of bounded solutions for some classes of linear one- and two-parameter difference equations with operator coefficients in a Banach space.

Article (Ukrainian)

### On Bounded Solutions of Some Classes of Two-Parameter Difference Equations in a Banach Space

Ukr. Mat. Zh. - 2000. - 52, № 12. - pp. 1610-1614

We obtain criteria for the existence of bounded solutions of some classes of linear two-parameter difference equations with operator coefficients in a Banach space.

Article (Ukrainian)

### Approximation of a bounded solution of one difference equation with unbounded operator coefficient by solutions of the corresponding boundary-value problems

Ukr. Mat. Zh. - 2000. - 52, № 4. - pp. 548-552

We investigate the problem of approximation of a bounded solution of a difference analog of the differential equation $$x^{(m)}(t) + A_1x^{(m-1)}(t) + ... + A_{m-1}x'(t)) = Ax(t) +f(0), t \in R$$ by solutions of the corresponding boundary-value problems. Here, A is an unbounded operator in a Banach space B, {A 1,...,A m-1} ⊂L(B) and f:ℝ→B is a fixed function.

Brief Communications (Ukrainian)

### On the approximation of a bounded solution of a linear differential equation in a Banach space

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1268–1271

We investigate the problem of approximation of a bounded solution of a linear differential equation by solutions of the corresponding difference equations in a Banach space.

Article (Ukrainian)

### Bounded and periodic solutions of a difference equation and its stochastic analogue in Banach space

Ukr. Mat. Zh. - 1991. - 43, № 1. - pp. 41-46