On the regularity of solutions to a class of nonlinear Volterra integral equations with singularities (Q6593407)

The authors study the regularity of the solutions to the nonlinear Volterra integral equation \[ y(t) = f(t) + \int_0^t K(t,s,y(s))\, ds, \qquad 0 \leq t \leq b, \] and extend the results in [\textit{H. Liang} and \textit{M. Stynes}, Discrete Contin. Dyn. Syst., Ser. B 28, No. 3, 2211--2223 (2023; Zbl 1506.45002)] to the case where the kernel \(K(t, s, u)\) may have, in addition to a singularity as \(s \to t\), also a singularity as \(s\to 0\). In particular the authors assume that for some non-negative integer \(m\) and some numbers \(\kappa\) and \(\lambda\) and such that \(\kappa>0\), \(\lambda\geq 0\) and \(\kappa+\lambda<1\) one has \[ \left |\left(\frac \partial {\partial t}\right )^i \left(\frac \partial {\partial t}+\frac \partial {\partial s}\right )^j \left(\frac \partial {\partial u}\right )^k K(t,s,u)\right | \leq b_1(|u|)\,(t-s)^{-\kappa-i} \,s^{-\lambda-j}, \] and\N\[\N\bigg|\left(\frac \partial {\partial t}\right )^i \left(\frac \partial {\partial t}+\frac \partial {\partial s}\right )^j \left(\frac \partial {\partial u}\right )^k K(t,s,u)\N- \left(\frac \partial {\partial t}\right )^i \left(\frac \partial {\partial t}+\frac \partial {\partial s}\right )^j \left(\frac \partial {\partial v}\right )^k K(t,s,v)\bigg| \]\N\[\leq b_2(\max\{|u|,|v|\})\,|u-v|\, (t-s)^{-\kappa-i}\, s^{-\lambda-j},\]\Nwhere \(0<s<t\leq b\), \(i,j,k\geq 0\) and \(i+j+k\leq m\). The conclusion in the main theorem is that if \(f\in C^{m,\kappa+\lambda}(0,b]\) and \(y\) is a bounded solution to the equation under study, then \(y\) is unique and \(y\in C^{m,\kappa+\lambda}(0,b]\).