Stability of Cauchy-type singular integral equations over an interval (Q1774560)

The author considers singular integral equations of Cauchy-type, \[ a(x)\varphi(x) +\frac{b(x)}{\pi}\int_{-1}^1\frac{\varphi(\tau)}{\tau-x} \,d\tau+\lambda \int_{-1}^1k(x,\tau)\varphi(\tau)\,d\tau = f(x) \] with Hölder continuous coefficients \(a\), \(b\), kernel \(k\) and right hand side \(f\). The solution \(\varphi\) is required to be Hölder continuous in the open interval \((-1,1)\) with integral singularity at the endpoints of the interval and bounded in a (possibly empty) subset \(S\) of \(\{-1,1\}\). The index of the equation can be either positive, zero or negative. This paper is a continuation of the author's work [Integral Equations Oper. Theory 46, No.~1, 1--10 (2003; Zbl 1033.45001)], where the case of negative index of the equation was considered. The main results of the paper are: a unified normalization form for the equation, according to the index being negative or non-negative (Theorem 2.1), a stability result for the solution of the normalized equation under small perturbations of the data: coefficients, kernel and right hand side (Theorem 2.2), and a pointwise error estimate, between the solution of the original and of the perturbed equation, for the different choices of the subset \(S\) (Theorem 2.3). Some applications of this results to estimate the rates of convergence of some numerical schemes for the equation are given at the end of the paper.