Constant-sign solutions for systems of Fredholm and Volterra integral equations: The singular case (Q959988)

The authors provide sufficient conditions for the existence of constant-sign solutions in \((C[0,T])^{n}\) space of the following system of nonlinear Fredholm integral equations \[ \begin{aligned} u_{i}(t)=\int_{0}^{T}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\dots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\dots,u_{n}(s))]ds,\\ t\in [0,T], \quad 1\leq i\leq n.\end{aligned} \] and of the system of nonlinear Volterra integral equations \[ \begin{aligned} u_{i}(t)=\int_{0}^{t}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\dots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\dots,u_{n}(s))]ds,\\ t\in [0,T], \;1\leq i\leq n,\end{aligned} \] where \(T>0\) is fixed and the nonlinearities \(h_{i}(t,u_{1},u_{2},\dots,u_{n})\) can be singular at \(t=0\) and \(u_{j}=0, \;j=1,2,\dots,n.\) Several examples are presented. These results are obtained via the Leray-Schauder alternative.