Research of equations with time delay

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Head investigator and 8 investigators studied some qualitative properties of solutions in equations with time delay, and obtained many results on the subject. The contents of a part of results obtained are summarized in the following :First we treated functional differential equations which are typical ones as equations with time delay, and obtained some results on stability properties and the existence of almost periodic solutions under some conditions. Moreover, applying the degree theory we established the existence of periodic solutions for some functional differential equations with diffusion which appear as crucial models in the field of mathematical biology.Next we treated functional difference equations and characterized the summability of the fundamental solution in connection with the uniform asymptotic stability property for the solu tion. Also, we established the representation theorem of solutions in phase space, and discussed the existence of bounded solutions and the asymptotic equivalence of solutions by applying the representation theorem.Furthermore, we treated processes which belong to a more wide class than equations with time delay, and developing a part of qualitative theory for processes, we applied general results obtained for processes to get some stability properties for functional differential equations and to establish the existence of almost periodic solutions for wave equations.