A representation formula for solutions of equations with delay in the phase space and its applications

课题负责人和10名研究人员对时滞方程的定性性质进行了研究，并取得了许多成果。某一部分的内容是关于这个主题的。对于一个抽象的无限维泛函微分方程，我们建立了其解在相空间中的表示公式，并对其进行了分解。该公式在定性性质的研究中起着重要的作用，因为利用该公式可以将无限维方程的研究简化为有限维方程的研究。实际上，通过应用该公式，我们建立了一些函数空间可容许性的结果和关于线性泛函微分方程概周期解存在性的Massera型结果。同时，我们建立了非线性泛函微分方程的局部不变流形，并将结果应用于一些线性化方程的稳定性问题。进一步，利用极限方程研究了渐近概周期泛函微分方程渐近概周期解的存在性。

Head investigator and 10 investigators studied qualitative properties for equations with time delay, and obtained many results on the subject. The contents of a part of results on the subject. The contents of a part of results obtained are summarized in the following:For an abstract functional differential equation which is the one of infinite dimension, we established a representation formula of solutions in the phase space, together with the decomposed formula. The formula plays an important role in the study of qualitive properties, because one can reduce the study of infinite dimensional equations to the study of finite dimensional equations by using the formula. Indeed, by applying the formula we established some results on the admissibility of some function spaces and Massera's type results on the existence of almost periodic solutions for linear functional differential equations. Also, we established some local invariant manifolds for nonlinear functional differential equations, and applied the results to some stability problems via linearized equations. Furthermore, for asymptotically almost periodic functional differential equations we studied the existence of asymptotically almost periodic solutions by means of limiting equations.