Introduction

This paper presents three new existence results for semipositone Fredholm integral equations of the form

$$y(t)=\mu\,\int^1_0 \,k(t,s)\,f(s,y(s))\,ds\,\,\,\hbox{ for }\,\,t\in [0,1], \leqno{(1.1)}$$

where $\,\mu>0\,$ is a constant. Existence in both $\,C[0,1]\,$ and $\,L^p[0,1]\,$ will be discussed. Throughout this paper $\,k\,$ is nonnegative but our nonlinearity $\,f\,$ may take negative values. Problems of this type are referred to as semipositone problems in the literature and they arise naturally in chemical reactor theory [4]. The constant $\,\mu\,$ is called the Thiele modulus and of physical interest is the existence of positive solutions to $(1.1)$ when $\,\mu>0\,$ is small. The literature on positive solutions to Fredholm integral equations (see [3]-[8]] and the references therein) is almost totally devoted to $(1.1)$ when $\,f\,$ takes nonnegative values (i.e. positone problems). Only a few results (see [1, Chapter 4]) are available for the semipositone problem.

Existence in this paper will be established using Krasnoselskii's fixed point theorem in a cone, which we state here for the convenience of the reader.

Theorem 1.1. Let $\,E=(E, \Vert\,.\,\Vert)\,$ be a Banach space and let $\,K \subset E\,$ be a cone in $\,E$. Assume $\,\Omega_1\,$ and $\,\Omega_2\,$ are open subsets of $\,E\,$ with $\,0\in \Omega_1\,$ and $\,\overline{\Omega_1} \subset \Omega_2\,$ and let $\,A:K \cap \left( \overline{\Omega_2} \backslash \Omega_1 \right) \to K\,$ be continuous and completely continuous. In addition suppose either

$$\Vert A\,u\Vert \leq \Vert u\Vert\,\,\hbox{ for }\,\,u\in K ... ... \Vert u\Vert\,\,\hbox{ for }\,\,u\in K \cap \partial \Omega_2$$

or

$$\Vert A\,u\Vert \geq \Vert u\Vert\,\,\hbox{ for }\,\,u\in K ... ... \Vert u\Vert\,\,\hbox{ for }\,\,u\in K \cap \partial \Omega_2$$

hold. Then $\,A\,$ has a fixed point in $\,K \cap \left( \overline{\Omega_2} \backslash \Omega_1 \right)$.