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

项目题目的结果：我们考虑了一般矩阵为线性系数的线性非齐次差分方程组，通过将解分解到系数矩阵的广义特征空间的分量，得到了关于离散时间和初值的解的一般公式。在系数矩阵为矩阵的指数函数的情况下，给出了该公式与该公式之间的具体关系。新配方有三种用途。首先，我们得到了一个由初值决定线性周期微分方程解的渐近行为的公式。其次，通过将结果转化为微分方程组，我们观察到解被分解为周期函数和指数函数。我们建立了一个差分方程组，直观地反映了差分法和微分法的关系。作为…的第三个应用此外，我们还观察到用差分法得到的新公式适用于计算线性微分方程解的Lyapunov数。作为结果，我们成功地计算了非齐次线性周期微分方程解的Lyapunov数。关于微分方程组和差分方程组的结果：利用基本解的概化和特征算子，得到了Bamach空间上Volterra差分方程解的稳定性。利用Schawder不动点定理和压缩原理，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解的存在性结果，使得系数矩阵的特征值是另一个相关研究：得到了一类高阶退化偏微分方程初值的Gevray适定性.利用有限元方法构造了语音生成问题的数值方法，并提出了一种语音形式的构造算法。将有限元方法和基本解方法相结合，研究了二维空间外部区域的约化波问题。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性空间的同伦理论。较少

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