Solution of multiple generalized integral equations of Schlömilch’s type. (Ukrainian) Zbl 0924.45007

Golovach, G. P.

Visn., Ser. Fiz.-Mat. Nauky, Kyïv. Univ. Im. Tarasa Shevchenka 1997, No. 3, 16-18 (1997).

The author solves the integral equation \[ \left({2\over \pi}\right)^{n} \int\limits_{0}^{+\infty}\ldots\int\limits_{0}^{+\infty}\phi(x_1 \cosh^{\alpha_1}\theta_1, x_2\cosh^{\alpha_2}\theta_2,\ldots,x_{n}\cosh^{\alpha_{n}} \theta_{n})d\theta_1\ldots d\theta_{n}=f(x_1,\ldots, x_{n}), \] \(\alpha_{i}>0\), \(x_{i}>0\), \(i=1,\dots, n\), where \(f,\phi\) are continuous functions; \(f\) is a known function; \(\phi\) is an unknown function. The solution of the given integral equation has the form \[ \phi(s_1,\ldots,\phi_{n})=(-1)^{n}\prod\limits_{i=1}^{n}s_{i} ^{1-1/\alpha_{i}}{\partial^{n}\over\partial s_1\ldots\partial s_{n}}\int\limits_{s_1}^{+\infty}\ldots \int\limits_{s_{n}}^{+\infty}Afdx_1\ldots dx_{n}, \] where \[ A=\prod\limits_{i=1}^{n}x_{i}^{2/\alpha_{i}-1}/\prod\limits_{i= 1}^{n} (x_{i}^{2/\alpha_{i}}-s_{i}^{2/\alpha_{i}})^{1/2}, x_{i}>s_{i}. \]

Reviewer: A.D.Borisenko (Kyïv)

MSC:

45E10

Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

Keywords:

multiple generalized integral equations; Schlömilch’s type; closed form solution