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On the convolution equation with positive kernel expressed via an alternating measure. (English. Russian original) Zbl 1153.45003
Math. Notes 81, No. 5, 620-627 (2007); translation from Mat. Zametki 81, No. 5, 693-702 (2007).
The author considers finite interval convolution operators \[ f(x)=g(x)+\int_0^rK(x-t)f(t)\,dt, \qquad r<+\infty\tag{1} \] with a positive kernel
\[ K(x):=\int_a^be^{-| x| s}\,d\sigma(s)>0. \qquad x\in\mathbb{R}, \] with an alternating measure \(\sigma(t)\) such that
\[ \int_a^b\frac1\sigma | d\sigma(s)| <\infty, \qquad \int_{-\infty}^\infty K(t)dt=2\int_a^b\frac1\sigma \,d\sigma(s)\leq1. \] The solvability of the nonlinear Ambartsumyan equation
\[ \varphi(s)=1+\varphi(s)\int_a^b\frac{\varphi(p)d\sigma(p)}{s+p}, \qquad r<+\infty \] is proved by iteration and applies the solution to solve the above convolution equation (1).
MSC:
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
45G05 Singular nonlinear integral equations
45L05 Theoretical approximation of solutions to integral equations
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