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We consider the family of nonlinear difference equations: xn+1=(∑i=13fi(xn,…,xn−k)+f4(xn,…,xn−k)f5(xn,…,xn−k))/(f1(xn,…,xn−k)f2(xn,…,xn−k)+∑i=35fi(xn,…,xn−k)), n=0,1,…, where fi∈C((0,+∞)k+1,(0,+∞)), for i∈{1,2,4,5}, f3∈C([0,+∞)k+1,(0,+∞)), k∈{1,2,…} and the initial values x−k,x−k+1,…,x0∈(0,+∞). We give sufficient conditions under which the unique equilibrium x¯=1 of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references.