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On a class of generalized autoconvolution equations of the third kind. (English) Zbl 1104.45001
The authors investigate Volterra-type integral equations of type
\[ k(x)y(x)= \int^x_0 a(s) y(s) y(x- s)\,ds,\quad x> 0, \]
i.e., so-called autoconvolution equations of the third kind. By using a result by J. Janno [Z. Anal. Anwend. 18, No. 2, 287–295 (1999; Zbl 0937.47061)] existence theorems for continuous solutions are deduced. Asymptotics at infinity is also considered. Under certain smoothness assumptions on the given functions \(k\) and \(a\), the existence of first and second derivatives of the solution are deduced. In addition, holomorphic solutions \(u\) (with \(a(x)\) identically 1) are investigated. Finally, a procedure for numerically solving the equation is presented.

45G10 Other nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
65R20 Numerical methods for integral equations