# Gorodnii M. F.

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

Boichuk О. A., Gorbachuk M. L., Gorodnii M. F., Khruslov E. Ya., Lukovsky I. O., Makarov V. L., Parasyuk I. O., Samoilenko A. M., Samoilenko V. G., Sharkovsky O. M., Shevchuk I. A., Slyusarchuk V. Yu., Stanzhitskii A. N.

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

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

Danilov V. Ya., Gorodnii M. F., Kirichenko V. V., Perestyuk N. A., Samoilenko A. M.

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

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

Gorodnii M. F., Vyatchaninov O. V.

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 $D(A)$.

### 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 $A^{α}$.

### Anatolii Yakovych Dorogovtsev

Buldygin V. V., Gorodnii M. F., Gusak D. V., Korolyuk V. S., Samoilenko A. M.

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

### 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.

### 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*: **R** → *B* 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*).

### $l_p$-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 $B$.

### 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.

### 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.

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

Gorodnii M. F., Romanenko V. N.

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.

### 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.

### Bounded solutions of one class of nonlinear operator difference equations

Dorogovtsev A. Ya., Gorodnii M. F.

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 890–896

Sufficient conditions for existence of bounded solutions of operator difference equations with quadratic nonlinearity are given.

### Approximation of a bounded solution of a linear difference equation on Z^{2} by solutions of corresponding boundary-value problems in a Banach space

Ukr. Mat. Zh. - 1995. - 47, № 4. - pp. 535–541

We study the problem of existence and uniqueness of solutions of boundary-value difference problems in a Banach space that correspond to a certain difference equation on Z^{2}. We prove a theorem on approximation of the unique bounded solution of the considered equation by solutions of the corresponding boundary-value problems.

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

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