Global behavior of solutions of the generalized Lyness difference equations under quadratic perturbations

We study the global asymptotic stability of solutions of the following two difference equationsxn+2xn=a+bxn+1+(1-c)xn+12+cxn2,n=0,1,2,…andxn+2xn=a+bxn+1+d(1-c)xn+12d+xn+1+cxn2,n=0,1,2,…,where a∈(0,+∞),d∈[0,+∞),c∈(0,1]a∈(0,+∞),d∈[0,+∞),c∈(0,1] and the initial values x0,x1∈(0,+∞)x0,x1∈(0,+∞). Bastien and Rogalski (2004) showed if c=0c=0 then there exist all the possible periods for the solutions of the above equations. Using an extension of the quasi-Lyapunov method, we prove that the sequences generated by the first difference equation are globally asymptotically stable where 0b>-2a(1-c) and the initial values x0,x1∈(0,+∞). The global convergence property of the second difference equation has also been obtained for b>0b>0 and 00b>0.