Eigenvalues of a system of Fredholm integral equations

We consider the following system of Fredholm integral equations: ui=λ∫01gi(t,s)Pi(s,u1(s),u2(s),…,un(s))ds, tϵ[0,1], 1≤i≤n, where λ > 0. Our aim is to determine those values of λ such that the above system has a constant-sign solution. In addition, explicit intervals for λ will be presented. The generality of the results obtained is illustrated through applications to several well-known boundary value problems. We also extend the above problem to that on the half-line [0, ∞) ui=λ∫0∞gi(t,s)Pi(s,u1(s),u2(s),…,un(s))ds, tϵ[0,∞], 1≤i≤n, Finally, both the systems above are extended to the general case when λ is replaced by λi.