Research on Global Properties and Correlation of Solutions to Functioanl Equations and Difference Equations

1.利用差分方程，得到了带周期强迫函数的线性微分方程有界解的存在性条件。周期解映射的迭代得到一个序列，该序列是一个线性差分方程的解。该序列的一般项由初始项、周期强迫函数的平均值和系数矩阵的特征值表示。解的有界性和周期性完全由强迫函数的初值和平均值决定。2.将Banach空间中的线性泛函微分方程转化为有界连续函数空间中的算子方程。计算了解的谱，计算了谱满足周期条件的解的存在性，得到了谱满足周期条件的解的存在性。3.完整地得到了常数的一般变分公式，进一步讨论了线性周期泛函微分方程的相空间。该公式适用于周期解的存在性。4.差分方程的解析解应用于人口科学和环境科学的研究。5.本文提出了基于有限元法的数值模拟算法，利用人工边界将无界区域的辐射散射问题简化为有界区域的辐射散射问题。6.研究了圆盘外域约简波问题的一种基本解法。从理论分析和数值实验两方面研究了源点和配点等距等相位排列的情况。7.研究了结构声系统的特征值的可调性。在Gevrey课上研究了高阶弱双曲型方程的柯西问题的适定性。不动点定理应用于积分微分方程解的稳定性理论。少

1.The existence conditions of bounded solutions of the linear differential equation with a periodic forcing function are obtained by using the difference equations. The itteration of the period solution map makes a sequence which is a solutions of a linear difference equation. The general term of this sequence is represented by the initial term, the mean of the periodic forcing function and the eigenvalues of the coefficient matrix. The boundedness and periodicity of solutions are determined completely by initial values ant the mean of the forcing functions. 2.Linear functional differential equations in a Banach space are transformed to opertor equations on the space of bounded continuous functions. The spectrum of the solutions are computed and the existence of solutions whose spectrum satisfies the period conditions are computed and the existence of solutions whose spectrum satisfies the period conditions are obtained. 3.The general variation of constants formula is completely obtain … More ed on the phase space for linear periodic functional differential equations. The formula is applied for the existence of periodic solutions. 4.Analytic solutions of difference equations are applied to the research in the population and environmental sciences. 5.The numerical simulation algorithm is developed based on the finite element method that reduces the radiation and scattering problem in an unbounded region into the one in a bounded region using aritificial boundary. 6.A fundamental solution method applied to reduced wave problems in the exterior domain of disc has been investigated. The case of equi-distant equally phased arrangement of source points and collocation points has been studied from the view points of both theoretical analysis and numerical experiment. 7.Pertabations of eigenvalues are studied for structural-acoustic system. Well-posedness of the Cauchy problem is studied in Gevrey class for some weakly hyperbolic equations of higher order. Fixed point theorems are applied to the stability theory of solutions of integral differential equations. Less