Almost Periodicity of Mild Solutions of Inhomogeneous Periodic Cauchy Problems

Almost Periodic Solutions of First- and Second-Order Cauchy Problems

Almost periodicity of solutions of firstand second-order Cauchy problems on the real line is proved under the assumption that the imaginary (resp. real) spectrum of the underlying operator is countable. Related results have been obtained by Ruess Vu~ and Basit. Our proof uses a new idea. It is based on a factorisation method which also gives a short proof (of the vector-valued version) of Loomi...

Almost Periodicity of Inhomogeneous Parabolic Evolution Equations

Bounded Convolutions and Solutions of Inhomogeneous Cauchy Problems

Let X be a complex Banach space, be bounded functions, and suppose that the singular points of the Laplace transforms of T and f do not coincide. Under various supplementary assumptions, we show that the convolution T f is bounded. When T(t) = I, this is a classical result of Ingham. Our results are applied to mild solutions of inhomogeneous Cauchy problems on R + : u 0 (t) = Au(t) + f(t) (t 0)...

Non-autonomous Inhomogeneous Boundary Cauchy Problems

In this paper we prove existence and uniqueness of classical solutions for the non-autonomous inhomogeneous Cauchy problem d dt u(t) = A(t)u(t) + f(t), 0 ≤ s ≤ t ≤ T, L(t)u(t) = Φ(t)u(t) + g(t), 0 ≤ s ≤ t ≤ T, u(s) = x. The solution to this problem is obtained by a variation of constants formula.

Periodicity and Almost-periodicity