Abstract

We study the global behaviours of solutions of the fourth-order difference equation xn+1=xnxn−1xn−2xn−3+f(xn,xn−1,xn−2,xn−3)xk+1n−3+xmn−3+axnxn−1xn−2+f(xn,xn−1,xn−2,xn−3)xkn−3+xmn−3+a, n=0,1,2,…, where f:(0,∞)4→(0,∞) is arbitrary and differentiable, m,k∈[0,∞) and the initial values x−3,x−2,x−1,x0∈(0,∞). We also study the positive nonoscillatory solutions of the following difference equation xn+1=A1xn+xn−1+A2xn−1+xn−2+⋯+Ak−1xn−k+2+xn−k+1−1xn−k, n=0,1,2,…, where A1,A2,…,Ak−1∈[0,∞) and A=∑k−1i=1Ai−2>0.