Pseudosolution of an integral convolution equation of the first kind

We consider the equation \(\int_{0}^{r}T(|x-t|)f(t)\,dt =g(x)\), where \(r<\infty\) and the functions \(T\) and \(g\) and their first derivatives are absolutely continuous on \([0,r]\) and \(T'(0)\ne 0\). An arbitrary term \(ax+b\) is added to the right-hand side of the equation. The obtained family of equations is reduced by double differentiation to an equation of the second kind. In the case of its unique solvability in \(L_{1}(0,r)\), the solution \(\tilde{f}\) is called a \(D^{2}\)-pseudosolution of the original equation. We introduce the partial regularization of the equation and present some cases of the existence of a \(D^{2}\)-pseudosolution. We propose a criterion for the suitability of \(\tilde{f}\) as an approximate solution. The problem of constructing a pseudosolution of an equation on the half-line is discussed.