Abstract Volterra equation

Summary

An abstract Volterra equation is a Volterra integral equation whose unknown takes values in a Banach space. Its memory term involves a linear operator, often unbounded. Its standard form is $u(t) = f(t) + \int_0^t a(t - s) A u(s) \, ds$ with a scalar kernel $a$ and a closed operator $A$. It contains abstract first-order and second-order evolution equations as the cases $a = 1$ and $a(t) = t$. Other kernels model heat conduction and viscoelasticity with memory, where $A$ is a differential operator in space.