Regularity of the solution of a nonlinear Volterra integral equation of the second kind
The following nonlinear Volterra integral equation of Urysohn type is discussed: \[ y(t)=g(t)+\int\limits_{0}^{t}K(t,s,y(s))ds, \quad 0\leq t\leq T. \] Here the kernel \(K\) may be smooth or have an integrable singularity and \(g\in C^{m,\beta}(0,T]\) for some positive integer \(m\). The authors study the regularity of the solution \(y\) of this equation. A well-known approach to analyze the smoothness of solutions of such equations consists in reducing them to Fredholm equations and using the corresponding results, see, e.g., [\textit{G. Vainikko}, Multidimensional weakly singular integral equations. Berlin: Springer-Verlag (1993; Zbl 0789.65097)]. Within the framework of Fredholm equations, an alternative method of analysis is proposed.
scientific article; zbMATH DE number 7622300