| In my first manuscript, I investigate the global stability character of the equilibrium points and the period-two solutions of yn+1=pyn+yn-1 r+qyn+yn-1 n=0,1,&ldots; with positive parameters and nonnegative initial conditions. I show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist I give a precise description of the basins of attraction of all points. My results give an affirmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 from [1].;In my second manuscript, I prove fixed point theorems for monotone mappings in partially ordered complete metric spaces which satisfy a weaker contraction condition for all points that are related by a given ordering. I also give a global attractivity result for all solutions of the difference equation zn+1=Fzn, zn-1,n=2,3,&ldots; where F satisfies certain monotonicity conditions with respect to the given ordering.;In my third manuscript, I investigate the global dynamics of solutions of two distinct competitive rational systems of difference equations in the plane. I show that the basins of attraction of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of non-hyperbolic equilibrium points. My results give the complete answer to Open Problem 1 posed recently in [2].;[1] M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman and Hall/CRC Press, 2001. [2] E. Camouzis, M. R. S. Kulenovic, G. Ladas and O. Merino, Rational Systems in the Plane - Open Problems and Conjectures, J. Differ. Equations Appl., 15(2009), 303-323. |
