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Open AccessFeature PaperArticle

# Solvability of the Class of Two-Dimensional Product-Type Systems of Difference Equations of Delay-Type (1, 3, 1, 1)

by 1,2
1
Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
2
Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Symmetry 2017, 9(10), 200; https://doi.org/10.3390/sym9100200
Received: 1 September 2017 / Revised: 21 September 2017 / Accepted: 21 September 2017 / Published: 25 September 2017
This paper essentially presents the last and important steps in the study of (practical) solvability of two-dimensional product-type systems of difference equations of the following form $z n = α z n - k a w n - l b ,$ $w n = β w n - m c z n - s d ,$ $n ∈ N 0 ,$ where $k , l , m , s ∈ N$ , $a , b , c , d ∈ Z$ , and where $α , β$ and the initial values are complex numbers. It is devoted to the most complex case which has not been considered so far (the case $k = l = s = 1$ and $m = 3$ ). Closed form formulas for solutions to the system are found in all possible cases. The structure of the solutions to the system is considered in detail. The following five cases: (1) $b = 0$ ; (2) $c = 0$ ; (3) $d = 0$ ; (4) $a c ≠ 0$ ; (5) $a = 0$ , $b c d ≠ 0$ , are considered separately. Some of the situations appear for the first time in the literature. View Full-Text