Constant-sign solutions for systems of fredholm and volterra integral equations: The singular case

We consider the system of Fredholm integral equations u_i(t) = ∫₀}^T} g_i(t,s)[h_i(s,u ₁(s),u₂(s),...,u_n(s))+k_i(s,u ₁(s),u₂(s),...,u_n(s))]ds, t ∈ [0,T], 1≤ i ≤ n and also the system of Volterra integral equations u_i(t) = ∫₀^t g_i(t,s)[h_i(s,u ₁(s),u₂(s),...,u_n(s)+k_i(s,u ₁(s),u₂(s),...,u_n(s))]ds, t ∈ [0,T], 1 ≤ i≤ n where T>0 is fixed and the nonlinearities h_i (t,u₁,u₂,...,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.