On The Dynamics Of Three Class Of Second Order Rational Difference Equations

In this paper, we investigated the global stability character, the oscillation, and the boudedness of solutions of three rational difference equations with nonnegative parameters and with nonnegative initial conditions.We solved some opening problems which were raised by G. Ladas in 2008, see the reference [2].The first chapter, we briefly introduced the history, current situation, and frontier of development of rational difference equation. Then we briefly introduced some theories about the global stability character of rational difference equation. At last we gave our main results in this paper.The second chapter, we discussed the dynamics properties of a rational difference equation with nonnegative parameters and with nonnegative initial conditions raised by G. Ladas. We gave a sufficient condition of global stability of the equation. Then we confirmed that every solution are oscillatory for all values of the parameters a> A, by using the subsequence. And all the solutions are bounded because of having an invariant and attracting interval.In the third chapter, we discussed the dynamics properties of a rational difference equation with nonnegative parameters and with nonnegative initial conditions. We confirmed that it has a unique positive equilibrium which is locally asymptotically stable. And it has an invariant and attracting interval. We also gave a sufficient condition of global stability of the equation. The solutions of the equation are oscillatory, by using the subsequence. And all the solutions are bounded because of having an invariant and attracting interval.In the forth chapter, we discussed the dynamics properties of a rational difference equation with nonnegative parameters and with nonnegative initialconditions raised by G. Ladas. We confirmed that it has a unique positive equilibrium which is locally stable. We also gave a sufficient condition of global stability of the equation. And all the solutions are bounded because of having an invariant and attracting interval. The solutions of the equation are oscillatory, by analyzing the subsequence.In the fifth chapter, we gave some examples of the rational difference equations to test and verify our conclusions.