The smoothness of solutions to nonlinear weakly singular integral equations
The authors consider the smoothness of nonlinear weakly singular Urysohn integral equations \(u(x) = \int_ GK(x,y,u(y))dy + f(x)\) \((x \in G)\). They establish \(u \in C^{m,\nu} (G)\) if \(u \in L^ \infty(G)\) and \(f \in C^{m,\nu}(G)\) and the Urysohn kernel \(K(x,y,u)\) satisfy the (sufficiently simple general) so-called weakly singularity conditions, where \(C^{m,\nu}(G)\) is the Banach space of all \(m\) times continuously differentiable functions \(u\) on an open bounded set \(G\) with the weight \[ w_ \lambda (x) = \begin{cases} 1 & \text{if } \lambda < 0 \\ \biggl(1 + \bigl | \log \rho (x)\bigr | \biggr)^{-1} \quad & \text{if } \lambda = 0\;(x \in G, \lambda \in \mathbb{R})\\ \rho(x)^ \lambda & \text{if } \lambda > 0 \end{cases} \] and \(\rho (x) = \text{dist} (x,\partial G)\). This theorem was formulated and partly (for \(m=2)\) proved by \textit{G. Vainikko} [Multidimensional weakly singular integral equations, Lect. Notes Math. 1549 (1993; Zbl 0789.65097)]. In this article the authors give a full proof by extending some new idea in this book. This theorem is more general even in the case of nonlinear Hammerstein equations than the results of \textit{H. Kaneko}, \textit{R. Noren} and \textit{Y. Xu} [Integral Equations Oper. Theory 13, No. 5, 660-670 (1990; Zbl 0706.45005)]. It is remarkable that all techniques for getting the smoothness of solutions in the linear case excepting the idea of G. Vainikko in his book do not work for the nonlinear case. The idea of G. Vanikko is based on the ``smallness'' of \(\int_ \Omega K(x,y,u(y))dy\), where \(\Omega \subset G\) is a small subregion; the integral over \(G \backslash \Omega\) is treated as a part of the inhomogeneity.
nonlinear weakly singular Urysohn integral equations
scientific article; zbMATH DE number 627647