Constant‐sign solutions of a system of integral equations: The semipositone and singular case

Abstract

We consider the following system of integral equations u_i(t)=μ∫₀¹g_i(t,s)f(s,u₁(s),u₂(s),…,u_n(s)) ds, t∈[0,1], 1≤i≤n, where μ>0, the function f may take negative values and f(·,u₁,u₂,…,u_n) may be singular at u_j=0, j∈{1,2,…,n}. Our aim is to establish criteria such that the above system has a constant‐sign solution. To illustrate the generality of the results obtained, application to a well known boundary value problem is included. We also extend the above problem to that on the half‐line [0,∞) u_i(t)=μ∫₀^∞g_i(t,s)f(s,u₁(s),u₂(s),…,u_n(s)) ds, t∈[0,∞), 1≤i≤n.