System of Fredholm Integral Equations: Existence of a Constant-Sign L ^p Solution

Abstract

In this chapter we shall consider two systems of integral equations, one is on a finite interval

$$\displaystyle{ u_{i}(t) =\int _{ 0}^{1}g_{ i}(t,s)f(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,1],\ 1 \leq i \leq n }$$

(5.1.1)

and the other is on the half-line [0, ∞)

$$\displaystyle{ u_{i}(t) =\int _{ 0}^{\infty }g_{ i}(t,s)f(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,\infty ),\ 1 \leq i \leq n. }$$

(5.1.2)

In both (5.1.1) and (5.1.2), we shall include both cases when the function f is “nonnegative” as well as when f may take “negative” values.