Formal adjoint equation for equations with time delay and its applications

首席调查员和五名调查员研究了泛函微分方程解、积分微分方程解和Volterra差分方程解的定性性质，得到了如下所述的许多结果。对于具有可积核的线性积分微分方程解，利用特征算子的谱分布和预解的可积性刻画了零解的一致渐近稳定性。作为结果的应用，对于具有概周期扰动的方程，我们得到了概周期解存在的一个充分条件，并分析了概周期解的谱。此外，将该方法应用于Volterra差分方程解的稳定性，得到了一些关于分段连续时滞偏微分方程解的稳定性的结果。利用时滞线性方程相空间中的常量变分公式，证明了某些非线性方程的不变流形(如局部稳定流形、局部中心流形等)的存在性，并将结果应用于稳定性问题。此外，通过一些更精细的考虑，我们研究了不变流形的光滑性。对于具有扰动的线性泛函差分方程解的渐近行为，利用解算子的谱分析，将相空间分解为稳定子空间和不稳定流形的直和，得到了常微分方程解的Perron定理的推广.

Head investigator and five investigators studied qualitative properties of solutions of functional differential equations, integrodifferential equations and Volterra difference equations which are typical ones of equations with delay, and obtained many results on the subject as cited below.1. For linear integrodifferential equations with integrable kernels, we characterized uniform asymptotic stability property of the zero solution in terms of the distribution of spectrum of the characteristic operator as wellas the integrability of the resolvent. As an application of the result, for equations with almost periodic perturbation we obtained a sufficient condition for the existence of almost periodic solutions, and analyzed the spectrum of the almost periodic solutions. Furthermore, applying the method to Volterra difference equations, we obtained some result on the stability property of the solution of partial differential equations with piecewise continuous delay.2. Using a variation-of-constants formula in the phase space for linear equations with delay, we established the existence of invariant manifolds (such as local stable manifold, local center manifold and so on) for some nonlinear equations, and applied the result to the stability problem. Also, through some finer considerations, we investigated the smoothness of the invariant manifolds.3. For linear functional difference equations with perturbations, we investigated the asymptotic behavior of solutions by decomposing the phase space into the direct sum of the stable subspace and the unstable manifold by means of the spectrum analysis of the solution operator, and obtained an extension of the Perron theorem for ordinary differential equations.