Abstract

The object of the first part of the paper is the following sixth order difference equation xn+1=f(xn,…,xn−5)g(xn,…,xn−5)xn−2xn−5+f(xn,…,xn−5)xn−2f(xn,…,xn−5)g(xn,…,xn−5)(xn−2+xn−5)+f(xn,…,xn−5)xn−2+g(xn,…,xn−5)xn−5+g(xn,…,xn−5)xn−5+f(xn,…,xn−5)g(xn,…,xn−5)f(xn,…,xn−5)g(xn,…,xn−5)(xn−2+xn−5)+f(xn,…,xn−5)xn−2+g(xn,…,xn−5)xn−5 for which the global behavior of the solutions is considered. If f,g:(0,∞)6→(0,∞) are continuously differentiable and the initial conditions x−5,…,x−1,x0 are strictly positive, the positive equilibrium of the equation is globally asymptotically stable. In the second part of the paper the equation xn+1=A−xpnxqn−1xrn−2,n=0,1,2,… with strictly positive A,p,q,r, p+q−r>0 and strictly positive initial conditions x−2,x−1,x0 is considered. In this case there exists a positive nonoscillatory solution of the equation, converging to the positive equilibrium x¯, the positive solution of x¯+x¯p+q−r−A=0.