On the Method of Two-Sided Continuation of Solutions of the Integral Convolution Equation on a Finite Interval

A. G. Barseghyan

Matematičeskie zametki, Tome 97 (2015) no. 3, pp. 323-335

Résumé

The paper is devoted to the development of the method of two-sided continuation of the solution of the integral convolution equation $$ S(x)=g(x)+\int _{0}^{r} K(x-t)S(t)\,dt,\qquad 0,\quad r \infty, $$ with an even kernel function $K\in L_{1} (-r,r)$. Two continuations of the solution $S$ are considered: to $(-\infty, 0]$ and to $[r,\infty)$. A Wiener–Hopf-type factorization is used. Under invertibility conditions for some operators, the problem can be reduced to two equations with sum kernels: $$ H^{\pm } (x)=q_{0}^{\pm } (x) \mp \int _{0}^{\infty } U(x+t+r)H^{\pm } (t)\,dt,\qquad x>0,\quad U\in L^{+} . $$ Applied aspects of the realization of the method are discussed.

Détail

Keywords: integral convolution equation, two-sided continuation of a solution, kernel function, Wiener–Hopf-type factorization, Baxter–Hirschman method.