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Solution of Schlömilch integral equation. (Ukrainian) Zbl 0932.45008
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