Investigation of one linear differential equation by using generalized functions with values in a Banach space (Q2754807)

Under certain conditions on the bounded operator \(A\), the author gives a description of solutions with sub-exponential growth to the equation \(x'(t)=Ax(t-1)\), \(t\in \mathbb{R}\). The main tool is the Fourier transform on a certain space of operator-valued distributions. The corresponding space of test functions consists of smooth functions with values in the closure of the set of holomorphic functions of \(A\), admitting analytic continuation into a strip around the real axis and satisfying some conditions at infinity.