Formal adjoint equation for equations with time delay and its applications

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Head investigator and five investigators studied qualitative properties of solutions of functional differential equations, integrodifferential equations and Volterra difference equations which are typical ones of equations with delay, and obtained many results on the subject as cited below.1. For linear integrodifferential equations with integrable kernels, we characterized uniform asymptotic stability property of the zero solution in terms of the distribution of spectrum of the characteristic operator as wellas the integrability of the resolvent. As an application of the result, for equations with almost periodic perturbation we obtained a sufficient condition for the existence of almost periodic solutions, and analyzed the spectrum of the almost periodic solutions. Furthermore, applying the method to Volterra difference equations, we obtained some result on the stability property of the solution of partial differential equations with piecewise continuous delay.2. Using a variation-of-constants formula in the phase space for linear equations with delay, we established the existence of invariant manifolds (such as local stable manifold, local center manifold and so on) for some nonlinear equations, and applied the result to the stability problem. Also, through some finer considerations, we investigated the smoothness of the invariant manifolds.3. For linear functional difference equations with perturbations, we investigated the asymptotic behavior of solutions by decomposing the phase space into the direct sum of the stable subspace and the unstable manifold by means of the spectrum analysis of the solution operator, and obtained an extension of the Perron theorem for ordinary differential equations.