Uniqueness of solutions for weakly degenerate cordial Volterra integral equations

Abstract

We investigate the uniqueness of solutions to cordial Volterra integral equations in the sense of Vainikko in the case where the kernel function $\mathcal K(\theta ) \equiv \mathcal K(y/x)$ vanishes on the diagonal $x=y$. When, in addition, $\mathcal K$ is sufficiently regular, is strictly positive on $(0,1)$, and $\theta ^{-k} \, \mathcal K'(\theta )$ is nonincreasing for some $k\in \mathbb{R} $, we prove that the solution to the corresponding Volterra integral equation of the first kind is unique in the class of functions which are continuous on the positive real axis and locally integrable at the origin. Alternatively, we obtain uniqueness in the class of locally integrable functions with locally integrable mean. We further discuss a uniqueness-of-continuation problem where the conditions on the kernel need only be satisfied in some neighborhood of the diagonal. We illustrate with examples the necessity of the conditions on the kernel and on the uniqueness class, and sketch the application of the theory in the context of a nonlinear model.