Unbounded rational systems with nonconstant coefficients

Abstract

We show the existence of unbounded solutions to difference equations of the form { x n + 1 = c ′ n x n B n y n , y n + 1 = b n x n + c n y n A n + C n y n f o r n = 0 , 1 , … , \left\{ {\matrix{{{x_{n + 1}} = {{{{c'}_n}{x_n}} \over {{B_n}{y_n}}},} \hfill \cr {{y_{n + 1}} = {{{b_n}{x_n} + {c_n}{y_n}} \over {{A_n} + {C_n}{y_n}}}} \hfill \cr } \,\,\,\,\,for} \right.\,\,\,n = 0,1, \ldots , where { c ′ n } n = 0 ∞ \left\{ {{{c'}_n}} \right\}_{n = 0}^\infty , { B ′ n } n = 0 ∞ \left\{ {{{B'}_n}} \right\}_{n = 0}^\infty , { b n } n = 0 ∞ \left\{ {{b_n}} \right\}_{n = 0}^\infty , { c n } n = 0 ∞ \left\{ {{c_n}} \right\}_{n = 0}^\infty , and { A n } n = 0 ∞ \left\{ {{A_n}} \right\}_{n = 0}^\infty are all bounded above and below by positive constants, and { C n } n = 0 ∞ \left\{ {{C_n}} \right\}_{n = 0}^\infty is either bounded above and below by positive constants or is identically zero. In the latter case, we give an example which can be reduced to a system of the form { x n + 1 = x n y n , y n + 1 = x n + γ n y n f o r n = 0 , 1 , … , \left\{ {\matrix{ {{x_{n + 1}} = {{{x_n}} \over {{y_n}}},} \hfill \cr {{y_{n + 1}} = {x_n} + {\gamma _n}{y_n}} \hfill \cr } \,\,\,\,\,for} \right.\,\,\,n = 0,1, \ldots , where 0 < γ′ < γ n < γ < 1 for some constants γ and γ′ for all n. This provides a counterexample to the main result of the 2021 paper by Camouzis and Kotsios.