Integral Equation f(x)-c/L(x) Integral from x=0 to x=L(x) of (f(y))dy=g(x), WhereL(x)=min(ax,1), a greater than or = 1

摘要

In the present paper the authors consider a function f which is a solution of theintegral equation f(x)-c/L(x)(the integral from 0 to L(x), of f(y)dy) = g(x). Here g is a given, smooth function defined in the interval (left bracket) 0, 1 (right bracket) c (epsilon) (0,1) is a constant, and L is a continuous piecewise linear function through the points (0,0), (a(sup -1),1), (1,1), where also a > 1 is a constant. The authors mainly focus their attention on the regularity properties of f. Away from the origin the regularity is analyzed by applying the Banach fixed point theorem, while near the origin the authors get a singular expansion for f by using the Melling transform techniques. (Copyright (c) 1995 by Helsinki University of Technology.)