Mathematical Notes

, Volume 81, Issue 5–6, pp 620–627 | Cite as

On the convolution equation with positive kernel expressed via an alternating measure

  • B. N. Engibaryan


We consider the integral convolution equation on the half-line or on a finite interval with kernel
$$K(x - t) = \int_a^b {e^{ - \left| {x - t} \right|s} d\sigma (s)} $$
with an alternating measure under the conditions
$$K(x) > 0, \int_a^b {\frac{1}{s}\left| {d\sigma (s)} \right| < + \infty } , \int_{ - \infty }^\infty {K(x)dx = 2} \int_a^b {\frac{1}{s}d\sigma (s) \leqslant 1} .$$
The solution of the nonlinear Ambartsumyan equation
$$\varphi (s) = 1 + \varphi (s) \int_a^b {\frac{{\varphi (p)}}{{s + p}}d\sigma (p)} ,$$
is constructed; it can be effectively used for solving the original convolution equation.

Key words

integral convolution equation nonlinear Ambartsumyan equation alternating measure Wiener-Hopf operator nonlinear factorization equation Volterra equation 


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Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • B. N. Engibaryan
    • 1
  1. 1.Mathematics InstituteNational Academy of Sciences of ArmeniaYerevanRussia

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