PSEUDO-DIFFERENTIAL EQUATIONS IN COMPLEX DOMAINS

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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.