Solution of Schlömilch integral equation. (Ukrainian) Zbl 0932.45008

Golovach, G. P.

Visn., Ser. Fiz.-Mat. Nauky, Kyïv. Univ. Im. Tarasa Shevchenka 1996, No. 2, 7-10 (1996).

The Schlömilch integral equation \[ f(x) = \frac {2}{\pi}\int _0^{\pi /2}\varphi (x\sin \theta) d\theta \] is considered. An effective method of solution of this equation and its generalizations are proposed. It is proved that if a continuous solution \(\varphi \) of the equation \((1)\) exists and the function \(f\) is continuous in the corresponding domain, then the solution of equation (1) is given by the formula \[ \varphi (t) = \text{sgn}\{t\}\frac {d}{dt}\int_0^t \frac {xf(x)dx}{\sqrt {t^2-x^2}}; \qquad t\not = 0. \] The corresponding formula for the equation \[ f(x) = \frac {2}{\pi}\int _0^{\pi /2} \varphi (x\sin ^\alpha \theta)d\theta, \qquad \alpha >0, \] is obtained, too.

Reviewer: O.A.Voina (Kyïv)

MSC:

45G10	Other nonlinear integral equations
45H05	Integral equations with miscellaneous special kernels

Keywords:

explicit solution formula; Schlömilch integral equation; continuous solution