Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case

Abstract

We consider the system of Fredholm integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{T}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$ and also the system of Volterra integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{t}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$ where T>0 is fixed and the nonlinearities h i (t,u 1,u 2,…,u n ) can be singular at t=0 and u j =0 where j∈{1,2,…,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ i u i (t)≥0 for t∈[0,1] and 1≤i≤n, where θ i ∈{1,−1} is fixed. We also include examples to illustrate the usefulness of the results obtained.