On autoconvolution and regularization

We consider the ill-posed nonlinear integral equation x*x=y of autoconvolution defined on the interval (0,1). We discuss conditions for the compactness, injectivity and weak closedness of the associated integral operator. The general theory of Tikhonov regularization for nonlinear ill-posed problems can be applied, and provides an approach to define different levels and degrees of ill-posedness in Hilbert spaces. For the autoconvolution problem we observe a varying degree of ill-posedness depending on the smoothness of solutions and on the behaviour of solutions and their derivatives for small arguments.