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On the Cauchy problem for a class of integro-differential equations on the half-line with a difference kernel. (English. Russian original) Zbl 0859.45004
Russ. Acad. Sci., Dokl., Math. 50, No. 2, 215-219 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 338, No. 2, 162-164 (1994).
We consider an equation of the form $P \left(i{d\over dx} \right) y(x)+ \int^\infty_0 K(x-x')Q \left(i{d\over dx'} \right)y(x')dx'=0,\;x>0, \tag{1}$ together with the initial conditions $i^{k+1} y^{(k)} (0)= \sigma_k, \quad k=0, 1, \dots, m-1. \tag{2}$ Here $$P$$ and $$Q$$ are polynomials, and the unknown function $$y(x)$$ and its derivatives of order up to $$m$$ are assumed to be square-integrable on $$(-\infty, \infty)$$.
We pose the problem of finding all tuples $$\{\sigma_\alpha\}_0^{m-1}$$ for which the Cauchy problem (1), (2) is solvable, and present necessary and sufficient conditions for its solvability.
MSC:
 45J05 Integro-ordinary differential equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)