Research on comparisons of Global Properties of solutions of Non-linear Difference Equations and solutions of Nonlinear Phenomena.

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1.There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions of nonlinear second order difference equations.2.One of the main work of this research is obtaining representations of analytic general solutions which converge to an equilibrium point of nonlinear second order difference equations with new methods of complex analysis.3.For analytic solutions of a nonlinear first order difference equation, T.Kimura studied the cases in which eigenvalue equal to 1,furthermore N.Yanagihara studied the cases in which the absolute value of the eigenvalue equal to 1. For nonlinear second order difference equation in which the absolute values of the all eigenvalues equal to 1, we consider analytic solutions of it in this study.4.Furthermore we apply the our theory of nonlinear higher order difference equations to Dynamics of Economic systems, for example, Nonlinear Cournot Dynamics with Heterogenous Duopolists. Generally, economist consider only equilibrium points for Mathematical Models. They call the equilibrium points as "solution of the model", and ignore orbits of the true solutions of the Models. However the equilibrium points are not "solution" in mathematical means, furthermore the orbits are important in the systems. Therefore in this study, we derive the true solutions and represent of solutions of Cournot Dynamics.5.In other studies, we treat economic Models with time delay. One of them is a dynamic economic model in isolated island. This Model is nonlinear differential equation with delayed terms, and studied stability of it. Next we treat an singular perturbation problem for economic models of R.Goodwin.