On the method of two-sided continuation of solutions of the integral convolution equation on a finite Interval

Abstract

The paper is devoted to the development of the method of two-sided continuation of the solution of the integral convolution equation

$$\begin{array}{*{20}c} {S(x) = g(x) + \int_0^r {K(x - t)S(t)dt,} } & {0 < x < r,} & {r < \infty ,} \\ \end{array}$$

with an even kernel function K ∈ L ₁(−r, r). Two continuations of the solution S are considered: to (−∞, 0] and to [r,∞). A Wiener–Hopf-type factorization is used. Under invertibility conditions for some operators, the problem can be reduced to two equations with sum kernels:

$$\begin{array}{*{20}c} {H^ \pm (x) = q_0^ \pm (x) \mp \int_0^\infty {U(x + t + r)H^ \pm (t)dt,} } & {x > 0,} & {U \in L^ + .} \\ \end{array}$$

Applied aspects of the realization of the method are discussed.