Systems of Fredholm and Volterra Integral Equations: Integrable Singularities

Abstract

In this chapter we consider three systems of singular integral equations. Specifically we are interested in the following systems of Fredholm integral equations $$\displaystyle{ u_{i}(t) =\int _{ 0}^{1}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,1],\ 1 \leq i \leq n }$$ (7.1.1) $$\displaystyle{ u_{i}(t) =\int _{ 0}^{\infty }g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,\infty ),\ 1 \leq i \leq n }$$ (7.1.2) and the system of Volterra integral equations $$\displaystyle{ u_{i}(t) =\int _{ 0}^{t}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,T],\ 1 \leq i \leq n }$$ (7.1.3) where T > 0 is fixed. The nonlinearities \(f_{i},\ 1 \leq i \leq n\) in the above systems may be singular in the independent variable and may also be singular at \(u_{j} = 0,\ j \in \{ 1,2,\cdots \,,n\}.\)