zbMATH — the first resource for mathematics

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.
45J05 Integro-ordinary differential equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)