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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.

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