On the theory of convolution equations of the third kind (Q878500)

This paper deals with a general class of autoconvolution equations of the third kind of the form \[ k(x)y(x)=\int_{0}^{t}y(\xi)y(x-\xi)\,d\xi+p(x), \quad x\in (0,1), \] with coefficient \(k(x) \thicksim Ax^{\alpha}\) for arbitrary positive \(\alpha\) as \(x\) goes to zero. For these equations two existence theorems and a uniqueness theorem are derived which generalize corresponding results in some previous papers by the author with \textit{L. Berg} [Z. Anal. Anwend. 24, No. 2, 217--250 (2005; Zbl 1104.45001)] and \textit{J. Janno} [Z. Anal. Anwend. 24, No. 3, 523--543 (2005; Zbl 1094.45002)] for \(\alpha=1\) and with \textit{B. Hofmann} [Numer. Funct. Anal. Optim. 27, 357--375 (2006; Zbl 1099.47049)] for \(\alpha=n\in \mathbb N\). The method of weighted norms is applied using a result by \textit{J. Janno} [Z. Anal. Anwend. 18, No. 2, 287--295 (1999; Zbl 0937.47061)].