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.