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 delaydifferential equations which model various diverse phenomena in biology,ecology, economics, physics and engineering. the study of dynamical propertiesof such equations is of great importance in many areas. the autonomousdifference equations have been studied extensively. the situation is morecomplicated when the considered model is nonautonomous. in this work we studyglobal attractivity and boundedness of solutions of the following nonautonomousdifference equation of order k + 1, x 11+1 = anx n , in which {all} is a1+ x n + ... + x n-kpositive bounded sequence. this equation is nonautonornous form of the logistictype difference equation with several delays. we prove that if lim inf an > 1, thenn----)xevery positive solution is bounded and persistence. furthermore we prove thatwhen we have a positive solution {xn } sucll that k (k +1)lim sup f, ::; 1, then forall positive solutions {x n}' lim xn =1.il----)if)