Periodicity, convergence, and boundedness of some difference equations

We investigate the periodicity, the boundedness, and the convergence of the solutions of several difference equations. The first equation is xn+1= a+b xn-1g +xn, n= 0,1,&ldots; For this equation, we give a detailed description of the trichotomy behavior of solutions, which depends on the relative values of the parameters beta and gamma.; The second equation is xn+1 =a+ bxn+ gxn-1 A+xn , n=0,1,&ldots; For this equation, the trichotomy nature of the solutions is shown to hold, where the trichotomy depends upon the values of gamma, beta, and A.; We also investigate the nature of the solutions of the general equation xn+1= a+b xn+g xn-1+ dxn-2 A+Bx n+Dxn -2, n= 0,1,&ldots; For this equation, we present many open problems and conjectures related to the existence of solutions for which a trichotomy character holds.; For all equations, the parameters and initial conditions are taken to be nonnegative real numbers.