Toshiki Naito

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For A(t) and f (t, x, y) T -periodic in t , we consider the following evolution equation with infinite delay in a general Banach space X: u′(t)+A(t)u(t)= f (t, u(t), ut ), t > 0, u(s)= φ(s), s 0, (0.1) where the resolvent of the unbounded operator A(t) is compact, and ut (s) = u(t + s), s 0. By utilizing a recent asymptotic fixed point theorem of Hale and(More)
We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form _ x = A(t)x+f(t) (), with f having precompact range, which will be then applied to nd new spectral criteria for the existence of almost periodic solutions with speciic spectral properties in the resonnant case where e(More)
We first give a criterion for positivity of the solution semigroup of linear Volterra integro-differential systems. Then, we offer some explicit conditions under which the solution of a positive linear Volterra system is exponentially stable or (robustly) lies in L2[0,+∞). Mathematics Subject Classification (2000). Primary 34A30; Secondary 34K20.
where L is a bounded linear operator mapping a uniform fading memory space B = B((−∞, 0];C) into C, and study the admissibility of Eq. (1) for a translation invariant function space M which consists of functions whose spectrum is contained in a closed set Λ in R. In case of Λ = R or Λ = {2kπ/ω : k ∈ Z}, the problem for the admissibility is reduced to the(More)
This paper is concerned with equations of the form: u′ = A(t)u + f(t) , where A(t) is (unbounded) periodic linear operator and f is almost periodic. We extend a central result on the spectral criteria for almost periodicity of solutions of evolution equations to some classes of periodic equations which says that if u is a bounded uniformly continuous mild(More)
This paper is concerned with a general existence and continuous dependence of mild solutions to semilinear functional differential equations with infinite delay in Banach spaces. In particular, our results are applicable to the equations whose Co-semigroups and nonlinear operators, defined on an open set, are noncompact. Introduction. Let £ be a Banach(More)
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