On the Dynamic of a Nonautonomous Difference Equation

Nonlinear difference equations of higher order are important in applications; such equations appear naturally as discrete analogues of differential and delay differential equations which model various diverse phenomena in biology, ecology, economics, physics and engineering. The study of dynamical properties of such equations is of great importance in many areas. The autonomous difference equations have been studied extensively. The situation is more complicated when the considered model is nonautonomous. In this work we study global attractivity and boundedness of solutions of the following nonautonomous difference equation of order , , in which is a positive bounded sequence. This equation is nonautonomous form of the logistic type difference equation with several delays. We prove that if , then every positive solution is bounded and persistence. Furthermore we prove that when we have a positive solution such that , then for all positive solutions , .

Nonlinear difference equations of higher order are important in applications; such equations appear naturally as discrete analogues of differential and delay differential equations which model various diverse phenomena in biology, ecology, economics, physics and engineering. The study of dynamical properties of such equations is of great importance in many areas. The autonomous difference equations have been studied extensively. The situation is more complicated when the considered model is nonautonomous. In this work we study global attractivity and boundedness of solutions of the following nonautonomous difference equation of order , , in which is a positive bounded sequence. This equation is nonautonomous form of the logistic type difference equation with several delays. We prove that if , then every positive solution is bounded and persistence. Furthermore we prove that when we have a positive solution such that , then for all positive solutions , .