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

首席调查员和10名调查员研究了时滞方程的定性性质，并获得了许多结果。内容的一部分内容就是关于这个课题的结果。对一类无限维抽象泛函微分方程，建立了相空间中解的表示式，并给出了分解公式。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归结为有限维方程的研究。实际上，通过应用该公式，我们建立了关于某些函数空间的可容许性的一些结果，以及关于线性泛函微分方程概周期解的存在性的Massera型结果。此外，我们还建立了一些非线性泛函微分方程的局部不变流形，并通过线性化方程将这些结果应用于一些稳定性问题。此外，对于渐近概周期型泛函微分方程，我们利用极限方程的方法研究了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.