PSEUDO-DIFFERENTIAL EQUATIONS IN COMPLEX DOMAINS

本研究的目的如下：[1]全纯函数微微分方程组的非特征柯西问题.[2]无穷阶(伪)微分方程组的基本原理.对于问题[1]，利用M.Kashiwara和P.Schapira建立的伪微分算子的作用，证明了在微局部方向p上的Cauchy-Kosalevskaya定理作为派生范畴D_b(X；p)中的同构.通过这笔赠款，我们邀请巴黎第六大学的P.夏皮拉教授进行讨论并接受建议。事实上，这对研究、他的建议以及与他的多次讨论都是相当重要的。对于问题[2]，我们研究了复域上卷积方程全纯解的连续性。我们将算子的特征集定义为微分算子情形的自然推广，并利用这一概念证明了由特征集决定的解在任意方向上的解析延拓。我们将这一理论应用于微分差分方程解的情形，并给出了它的特征集。在这项研究中，我们还在抽象泛函微分方程的稳定性和概周期解的存在性的研究、2-微双曲型拟微分方程解的研究、具有大参数的三阶常微分方程组的Stokes几何的确定、Grothendieck留数的研究以及计算对偶性的算法等方面得到了许多结果。

The aims of this research were as follows :[1] The non-charasteristic Cauchy problem of the system of micro-differential equations for holomporphic functions.[2] The fundemental principle for the systems of (pseudo-)differential equations of infinite order.For the problem [1], using the action of pseudo-differential operators established by M.Kashiwara and P.Schapira, we proved the Cauchy-Kowalevskaya theorem at a micro-local direction p as the isomorphism in the derived category D_b (X ; p). By this grant, we invited Professor P.Schapira of Universite Paris VI to have discussions and to be sugested. In fact, it was quite important for the research, his sugestions and many discussions with him. For the problem [2], we studied the continuation of holomorphic solutions for convolution equations in the complex domains. We defined the characteristic set of the operator as a natural generalization of the case of differential operators and using this notion, we proved the analytic continuation of solutions to any direction determined by the characteristic set. We applied this theory to the differential-difference equation case and we gave explicitly its characteristic set. We constructed also an example of the convolution equations having all good properties.In this research, we also obtained many results concerning the study of stabilities and the existence of almost periodic solutions in abstract functional differential equations, the study of the 2-micro-hyperbolic pseudodifferential equations, the study of determination of Stokes geometry of third order ordinary differential equations having large parameters, the study of Grothendieck residue and the algorithm computing the duality.