System of Fredholm Integral Equations: Semipositone and Singular Case

Abstract

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

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

(6.1.1)

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

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

(6.1.2)

In both (6.1.1) and (6.1.2), μ is a positive number, the function f may take negative values, and $f(\cdot,u_{1},u_{2},\cdots \,,u_{n})$ may be singular at u _j = 0, $j \in \{ 1,2,\cdots \,,n\}.$