Constant-sign solutions of a system of Volterra integral equations

We consider the following system of Volterra integral equations: ui(t)=∫0tgi(t,s)[fi(s,u1(s),u2(s),…,un(s))+hi(s,u1(s),u2(s),…,un(s))]ds,t∈[0,T],1≤i≤n and some of its particular cases that arise from physical problems. Criteria are offered for the existence of one and more constant-sign   solutions u=(u1,u2,…,un)u=(u1,u2,…,un) of the system in (C[0,T])n(C[0,T])n. We say uu is of constant sign   if for each 1≤i≤n,θiui(t)≥01≤i≤n,θiui(t)≥0 for all t∈[0,T]t∈[0,T], where θi∈{1,−1}θi∈{1,−1} is fixed. Examples are also included to illustrate the results obtained.