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)
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), u t , t >0, u(s) = φ(s), s 0, (0.1) where the resolvent of the unbounded operator A(t) is compact, and u t (s) = u(t + s), s 0. By utilizing a recent asymptotic fixed point theorem of Hale and… (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 L 2 [0, +∞).
For linear functional differential equations with infinite delay in a Banach space, a variation-of-constants formula is established in the phase space. As an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line. A variation of constants formula for an abstract… (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)
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)