2019
Том 71
№ 11

# Lagoda O. A.

Articles: 3
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)

### 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.