# Investigation of the Structure of a Set of Continuous Solutions of Difference Equations Systems

## DOI:

https://doi.org/10.20535/1810-0546.2015.4.50445## Keywords:

Difference equations, Continuous limited solutions## Abstract

**Background. **We consider the structure of a set of continuous solutions of equations systems

[x(t+1)=A(t)x(t)+B(t)x(qt)+F(t)\] (1)

in a number of cases depending on the hypotheses for the matrices *A*, *B*, number *q* and their properties.

**Objective.** To study existence of continuous limited solutions for** **\[t\in \mathbb{R}\], study the structure of their set and also developing the method of their construction.

**Methods.** We use methods of the theory of differential and difference equations.

**Results. **The existence of the family of continuous limited solutions for [t\geqslant 0\] which depends on \[\bar{n}=\sum_{i=1}^{k}n_{i}\] arbitrary continuous one-periodic functions at some conditions is proved in theorem 1. Similar theorem is proved for case \[t\leq 0\] (the theorem 2), and is proved the theorem 3 about the existence of the continuous limited solution of homogeneous system of the equation (1) is also proved.

**Conclusions. **New conditions for the existence of continuous solutions of difference equations systems (1) are established, we proposed the method of constructing these solutions and investigated the structure of their set.

## References

G.D. Birkhoff, “General theory of linear difference equations”,* Trans. Amer. Math. Soc.*, vol. 12, pp. 243–284, 1911.

W.J. Trjitzinsky, “Analytic theory of linear q-difference equations”, *Trans. Amer. Math. Soc.*, vol. 61, pp. 1–38, 1933.

G.P. Pelyukh and O.A. Sivak, “A study of the structure of the set of continuous solutions to systems of linear functional-difference equations”, *Neliniyni Kolyvannya*, vol. 12, no. 3, pp. 307–335, 2009 (in Ukrainian).

G.P. Pelyukh and O.A. Sivak, “On the structure of the set of continuous solutions of functional-difference equations with linearly transformed argument”, *Neliniyni Kolyvannya*, vol. 13, no. 1, pp. 75–95, 2010 (in Ukrainian).

O.A. Sivak,* *“The structure of a set of continuous solutions of systems of linear functional difference equations”, *Naukovi Visti NTUU KPI*, no. 4, pp. 81–87, 2011 (in Ukrainian).

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2017 NTUU KPI Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work