Synthetic research on differential equations

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We studied differential equations, in the range from ordinary differential equations to partial differential equations and from linear ones to nonlinear ones. We cooperated with the Division of Functional Equations, Mathematical Scociety of Japan.As the result of a study including all the researchers of this field in Japan, we clarified the tendency of the forcecoming subjects. As one of the main results of our synthetic research, Atsushi Yagi published a monograph on evolution equations in Banach spaces. His theory made possible a systematic treatment of nonlinear problems in a form of evolution equations in Banach spaces. One point we want to emphasize is that his theory includes degenerate problems. As to another main subject of research, scattering theory, the Head of Investigaor is writing a monograph which represents the results of synthetic studies of partial differential equations related to scattering theory. Other subject, for instance, hypoellipticity of equations, by the studies in the space of analytic functions or of Gevrey class, the meanings of the conditions posed up to now has been made clear. In ordinary differential equations, by the studies of asymptotic behaviors of hypergeometric functions of congruent type made clear the structure of solutions. As to inverse problems of scattering problems, which are important even in the engineering, a theorem which guarantees the indentification of scatterers from scattering informations of a fixed enegy.We prepared the synthetic discussions on differential equations, but could not reach to results which sythesize the many subject of differential equations.