Unbounded rational systems with nonconstant coefficients

Abstract

We show the existence of unbounded solutions to difference equations of the form {xn+1=c′nxnBnyn,yn+1=bnxn+cnynAn+Cnyn for 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, {bn}n=0∞\left\{ {{b_n}} \right\}_{n = 0}^\infty, {cn}n=0∞\left\{ {{c_n}} \right\}_{n = 0}^\infty, and {An}n=0∞\left\{ {{A_n}} \right\}_{n = 0}^\infty are all bounded above and below by positive constants, and {Cn}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

Keywords

• rational difference equations
• systems of difference equations
• boundedness
• 39a22