Abstract:
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.
Keywords:convolution equations of the second kind, absolute continuity, additional linear term, double differentiation, pseudosolution, partial regularization, total monotonicity, equation on the half-line.
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