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Z. Darbenas and M. Oliver,
Uniqueness of solutions for weakly degenerate cordial Volterra
integral equations,
J. Integral Equ. Appl. 31 (2019), 307-327,
doi:10.1216/JIE-2019-31-3-307.
Abstract:
We study the question of uniqueness of solutions to cordial Volterra
integral equations in the sense of Vainikko (Numer. Funct. Anal.
Optim. 30, 2009, pp. 1145–1172) in the case where the kernel (or
core) 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 non-increasing 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 continuation-of-uniqueness problem where the
conditions on the kernel need only be satisfied in some neighborhood
of the diagonal. We give examples illustrating 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.
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