Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations ui(t)=∫0Tgi(t,s)fi(s,u1(s)+ρ1(s),u2(s)+ρ2(s),…,un(s)+ρn(s))ds,t∈[0,T],1≤i≤n where T>0T>0 is fixed, ρiρi’s are given functions and the nonlinearities fi(t,x1,x2,…,xn)fi(t,x1,x2,…,xn) can be singular   at t=0t=0 and xj=0xj=0 where j∈{1,2,⋯,n}j∈{1,2,⋯,n}. Criteria are offered for the existence of constant-sign solutions  , i.e., θiui(t)≥0θiui(t)≥0 for t∈[0,T]t∈[0,T] and 1≤i≤n1≤i≤n, where θi∈{1,−1}θi∈{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.