Abstract convolution systems on the line. (English) Zbl 0892.45008

The author studies a fairly general system of abstract integral and integrodifferential equations \[ Au(t)+(K*u)(t)=f(t),\quad t\in \mathbb R\qquad (\gamma u)'(t)-Bu(t)- (H*u)(t)=g(t),\quad t\in \mathbb R, \tag{1} \] where \(A, B\) and \(\gamma \) are certain operators acting (and bounded) on appropriate Banach spaces, and \(H,K\) are integrable vector valued convolution kernels. Necessary and sufficient conditions for (1) to be well posed are sought for in different function space contexts. Various examples and applications are pointed out, including the Dirichlet problem for the Laplace operator in an angle or in a strip, and an elliptic integrodifferential boundary value problem with dynamic boundary conditions. The latter example had inspired the author to study the problem in such a general setting.


45N05 Abstract integral equations, integral equations in abstract spaces
47G20 Integro-differential operators