New developments of functional differential equations combined with difference equations, and studies of related topics

关于课题题目的结果：我们考虑了一个线性非齐次差分方程，其线性系数为一般矩阵，通过将其解分解为系数矩阵的广义特征空间的分量，得到了其关于离散时间和初值的解的一般公式。在系数矩阵是矩阵的指数函数的情况下，我们进一步详细讨论了这个公式与公式之间的关系。这个新公式有三个应用。首先，我们得到了由初值决定线性周期微分方程解的渐近性质的公式。其次，通过对微分方程的变换，我们发现解可以分解为周期函数和指数函数。我们建立了一个差分方程，直接说明了差分法和微分法的关系。作为第三个应用，我们发现用差分法得到的这个新公式适用于计算线性微分方程解的李雅普诺夫数。结果，我们成功地计算了非齐次线性周期微分方程解的李雅普诺夫数。这个数值不是用经典方法得到的。关于微分方程和差分方程的结果：利用基本解的概要性和特征算子，得到了Bamach空间上Volterra差分方程解的稳定性。利用Schawder不动点定理和收缩原理，得到了泛函微分方程解的有界性和周期性的渐近结果。得到了二阶非齐次差分方程在系数矩阵特征值为1时解析解的存在性。其他相关研究：得到了一类高阶退化偏微分方程初值的Gevray适定性质。利用有限元方法构建了语音生成问题的数值方法，并提出了一种构建语音表单的算法。将有限元法与基本解法相结合，研究了二维空间外区域的约简波问题。得到了黎曼曲面到复射影空间的光滑映射空间的能量函数奇异空间的同伦理论。少

Results about title of project : We considered a linear non-homogeneous difference equations with general matrix as a linear coefficient, and obtained a general formula representing its solutions with respect to discrete time and initial values by decomposing the solutions into the component of generalized eigen-spaces of the coefficient matrix. We followed up the detailed relation between this formula and the formula for the case that the coefficient matrix is an exponential function of a matrix. The new formula has three applications. Firstly, we obtained a formula determining by the initial values the asymptotic behavior of the solutions to the linear periodic differential equations. Secondary by transform of the results to the differential equations, we observed that the solutions are decomposed to the periodic functions and exponential functions. We made a difference equations which shows directly the relations of difference method and differential methods. As the third applicatio … More n, we observed that this new formula obtained by using difference methods is applicable to compute the Lyapunov numbers of solutions of linear differential equations. As a result, we succeeded to compute the Lyapunov numbers of solutions of non-homogeneous linear periodic differential equations. This number are not obtained by the classical methods.Results about differential equations and difference equations : The stability of solutions of Volterra difference equations on a Bamach space are obtained by using the summarbility of fundamental solutions and the characteristic operator. The asymptotic results of bounded and periodic property of solutions to functional differential equations are obtained by using Schawder fixed point theorem and contraction principle. Results about the existence of analytic solutions are obtained for the second order, non-homogeneous difference equations such that the eigen-values of the coefficient matrix are oneOther related researches : The Gevray well-posed property of initial values are obtained for a class of higher order, degenerated partial differential equations. Numerical methods for voice generation problems are constructed by using the finite element methods, and proposed an algorithm for the construction of the voice form. Finite element method and fundamental solution method are combined for the study of the reduced wave problem in the exterior region in two dimensional space. A homotopy theory are obtained about the space of singularity of energy functions of the space of smooth mappings from Riemann surface to complex projective spaces. Less