Existence results for singular integral equations of fredholm type

1. I N T R O D U C T I O N This paper discusses the existence of one (or indeed more) solutions to singular integral equations of the form y(t) = O(t) + k(t, s)[g(y(s)) + h(y(s))] ds, for t ~ [0, 1]; (1.1) here our nonlinearity g + h may be singular at y = 0. In [1], we established the existence of one nonnegative solution to (1.1) using the Leray-Schauder alternative. However, in [1], we had to assume a rather strong lower-growth type assumption, namely that there exists a ¢ > 0 continuous on [0, 11 with g(u) + h(u) > ¢(t) on (0, ~ ) and there exists a subset I of [0, 1] of measure zero with 8(t) + (1.2) fo k(t, s)¢(s) ds > 0 for t e [0, 1]\I. So for example if g ( u ) + h ( u ) = u ~ + u a + A , ~ > 0 , B_>0, we need to assume usually that A > 0. The case A -0 was not discussed in [1], and this is the situation that occurs most frequently in applications (for example, in fluid dynamics [2,3]). In this paper, we present results for the case A = 0 (in particular, we will remove assumption (1.2) and replace it with a concavity type assumption). To do so, we will use Krasnoselski's fixed-point theorem in a cone. This paper has two main sections. Section 2 presents results for singular problems, whereas Section 3 discusses nonsingular problems. For the remainder of this section, we present some results from the literature which will be needed in Sections 2 and 3. First we state Krasnoselski's fixed-point theorem in a cone. 0893-9659/00/$ see front matter (~) 2000 Elsevier Science Ltd. All rights reserved. Typeset by .Afl~S-~_X PII: S0893-9659(99)00161-5