Integro-differential equation of non-local wave interaction

The integro-differential equation
$$ \frac{d^2f}{dx^2}+Af=\int^\infty_0K(x-t)f(t)\,dt+g(x) $$
with kernel
$$ K(x)=\lambda\int^\infty_ae^{-|x|p}G(p)\,dp, \qquad a\geqslant0, $$
is considered, in which
$$ A>0,\qquad \lambda\in(-\infty,\infty), \qquad G(p)\geqslant0, \qquad 2\int^\infty_a\frac1p\,G(p)\,dp=1. $$
These equations arise, in particular, in the theory of non-local wave interaction. A factorization method of their analysis and solution is developed.
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