{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:15:00Z","timestamp":1760242500412,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2017,9,25]],"date-time":"2017-09-25T00:00:00Z","timestamp":1506297600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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  = \u03b1  z  n - k  a   w  n - l  b  ,         w n  = \u03b2  w  n - m  c   z  n - s  d  ,        n \u2208  N 0  ,     where     k , l , m , s \u2208 N    ,     a , b , c , d \u2208 Z    , and where     \u03b1 , \u03b2     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 \u2260 0    ; (5)     a = 0    ,     b c d \u2260 0    , are considered separately. Some of the situations appear for the first time in the literature.<\/jats:p>","DOI":"10.3390\/sym9100200","type":"journal-article","created":{"date-parts":[[2017,9,26]],"date-time":"2017-09-26T04:28:01Z","timestamp":1506400081000},"page":"200","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solvability of the Class of Two-Dimensional Product-Type Systems of Difference Equations of Delay-Type (1, 3, 1, 1)"],"prefix":"10.3390","volume":"9","author":[{"given":"Stevo","family":"Stevi\u0107","sequence":"first","affiliation":[{"name":"Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36\/III, 11000 Beograd, Serbia"},{"name":"Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,9,25]]},"reference":[{"key":"ref_1","first-page":"387","article-title":"Further properties of the rational recursive sequence xn+1 = axn\u22121\/(b+cxnxn\u22121)","volume":"26","author":"Migda","year":"2006","journal-title":"Opusc. Math."},{"key":"ref_2","first-page":"1","article-title":"On the rational recursive sequence xn+1 = axn\u22121\/(b+cxnxn\u22121)","volume":"43","author":"Migda","year":"2009","journal-title":"Tatra Mt. Math. Publ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1080\/10236190410001726421","article-title":"On impulsive Beverton-Holt difference equations and their applications","volume":"10","author":"Berezansky","year":"2004","journal-title":"J. Differ. Equ. 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