The null space and the range of a convolution operator in a fading memory space

Abstract:

We study the convolution equation $(\ast )$ \[ \mu \; \ast \;x’(t) + v\; \ast \;x(t) = f(t) \quad ( - \infty < t < \infty )\], as well as a perturbed version of $(\ast )$, namely $(\ast \ast )$ \[ \mu \;\ast \;x’(t) + v \ast \;x(t) = F(x) (t)\quad ( - \infty < t < \infty ).\] Here $x$ is a ${{\mathbf {R}}^n}$-valued function on $( - \infty ,\infty ),x’(t) = dx(t)/dt$, and $\mu$ and $\nu$ are matrix-valued measures. If $\mu$ and $\nu$ are supported on $[0,\infty )$, with $\mu$ atomic at zero, then $(\ast )$ can be regarded as a linear, autonomous, neutral functional differential equation with infinite delay. However, most of the time we do not consider the ordinary Cauchy problem for the neutral equation, i.e. we do not suppose that $\mu$ and $\nu$ are supported on $[0,\infty )$, prescribe an initial condition of the type $x(t) = \xi (t) (t \leqslant 0)$, and require $(\ast )$ and $(\ast \ast )$ to hold only for $t \geqslant 0$. Instead we permit $(\ast )$ and $(\ast \ast )$ to be of "Fredholm" type, i.e. $\mu$ and $\nu$ need not vanish on $( - \infty ,0)$, we restrict the growth rate of $x$ and $f$ at plus and minus infinity, and we look at the problem of the existence and uniqueness of solutions of $(\ast )$ and $(\ast \ast )$ on the whole real line, satisfying conditions like $|x(t)| \leqslant C\eta (t)\;( - \infty < t < \infty )$, where $C$ is a constant, depending on $x$, and $\eta$ is a predefined function. Some authors use the word "admissible" when discussing problems of this type. In the case when the homogeneous version of $(\ast )$ has nonzero solutions, we decompose the solutions into components with different exponential growth rates, and give a priori bounds on the growth rates of the solutions. As an application of the basic theory, we look at the Cauchy problem for a neutral functional differential equation, and prove the existence of stable and unstable manifolds.