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

1.非线性差分方程解没有一般的存在性定理，所以必须按照模型逐一证明解的存在性。在我们的工作中，我们发现了非线性二阶差分方程解析解的存在性定理。本研究的主要工作之一是用新的复分析方法得到非线性二阶差分方程收敛于平衡点的解析一般解的表示。对于非线性一阶差分方程的解析解，T.Kimura研究了特征值等于1的情况，N.Yanagihara研究了特征值绝对值等于1的情况。对于所有特征值的绝对值都等于1的非线性二阶差分方程，本文研究了它的解析解。此外，我们将非线性高阶差分方程理论应用于经济系统动力学，例如具有异质双极体的非线性古诺动力学。一般来说，经济学家只考虑数学模型的平衡点。他们把平衡点称为“模型的解”，而忽略了模型真正解的轨道。但是平衡点在数学上并不是“解”，而且轨道在系统中也很重要。因此，在本研究中，我们导出了古诺动力学的真解和解的表示。在其他研究中，我们用时滞来处理经济模型。其中之一是孤岛上的动态经济模型。该模型是一类具有时滞项的非线性微分方程，并研究了它的稳定性。接下来，我们处理R.Goodwin经济模型的奇异摄动问题。

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.