Spectral analysis of an operator associated with equations with time delay

主要研究员和8名研究员研究了时滞方程解的一些性质，得到了许多结果。首先，我们分析了泛函差分方程和泛函微分方程这两类典型的时滞方程所对应的算子谱的一些性质。作为结果的应用，我们研究了一些函数空间解的渐近等价性和可容许性，得到了一些结果。这些结果在有限维方程中几乎是最好的结果。其次，我们处理了一个抽象的无穷维泛函微分方程，建立了一个常数变分公式，它表示相空间中的解段。这个公式在定性性质的研究中是至关重要的，因为利用这个公式可以将无穷维方程的研究简化为有限维方程的研究。实际上，利用这个公式，我们建立了无穷维方程容许性理论中的一个结果。

Head investigator and 8 investigators studied some properties of solutions in equations with time delay, and obtained many results on the subject. The contents of a part of results obtained are summarized in the following :First we analyzed some prperties of spectrum of the operator associated with functional difference equations and functional differential equations which are typical ones as equations with time delay. As applications of the result, we investigated the asymptotic equivalence of solutions and admissibility of some function spaces, and obtained some results on the subjects. These results are almost best possible ones in equations of finite dimension.Next we treated an abstract functional differential equation which is the one of infinite dimension and established a variation-of-constants formula which represents the segment of solutions in the phase space. This formula is crucial in the study of qualitative properties, because one can reduce the study of inifinite dimensional equations to the study of finite dimensional equations by using the formula. Indeed, by using the formula we established a result in admissibility theory for infinite dimensional equations.