New developments of functional differential equations combined with difference equations, and studies of related topics

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Results about title of project : We considered a linear non-homogeneous difference equations with general matrix as a linear coefficient, and obtained a general formula representing its solutions with respect to discrete time and initial values by decomposing the solutions into the component of generalized eigen-spaces of the coefficient matrix. We followed up the detailed relation between this formula and the formula for the case that the coefficient matrix is an exponential function of a matrix. The new formula has three applications. Firstly, we obtained a formula determining by the initial values the asymptotic behavior of the solutions to the linear periodic differential equations. Secondary by transform of the results to the differential equations, we observed that the solutions are decomposed to the periodic functions and exponential functions. We made a difference equations which shows directly the relations of difference method and differential methods. As the third applicatio … More n, we observed that this new formula obtained by using difference methods is applicable to compute the Lyapunov numbers of solutions of linear differential equations. As a result, we succeeded to compute the Lyapunov numbers of solutions of non-homogeneous linear periodic differential equations. This number are not obtained by the classical methods.Results about differential equations and difference equations : The stability of solutions of Volterra difference equations on a Bamach space are obtained by using the summarbility of fundamental solutions and the characteristic operator. The asymptotic results of bounded and periodic property of solutions to functional differential equations are obtained by using Schawder fixed point theorem and contraction principle. Results about the existence of analytic solutions are obtained for the second order, non-homogeneous difference equations such that the eigen-values of the coefficient matrix are oneOther related researches : The Gevray well-posed property of initial values are obtained for a class of higher order, degenerated partial differential equations. Numerical methods for voice generation problems are constructed by using the finite element methods, and proposed an algorithm for the construction of the voice form. Finite element method and fundamental solution method are combined for the study of the reduced wave problem in the exterior region in two dimensional space. A homotopy theory are obtained about the space of singularity of energy functions of the space of smooth mappings from Riemann surface to complex projective spaces. Less