Solutions for singular Volterra integral equations

Abstract

We consider the system of Volterra integral equations ui(t)=∫t0gi(t,s)[Pi(s,u1(s),u2(s),⋯,un(s))+Qi(s,u1(s),u2(s),⋯,un(s))]ds,t∈[0,T],1≤i≤n where T>0 is fixed and the nonlinearities Pi(t,u1,u2,⋯,un) can be singular at t=0 and uj=0 where j∈{1,2,⋯,n}. Criteria are offered for the existence of fixed-sign solutions (u∗1,u∗2,⋯,u∗n) to the system of Volterra integral equations, i.e., θiu∗i(t)≥0 for t∈[0,1] and 1≤i≤n, where θi∈{1,−1} is fixed. We also include an example to illustrate the usefulness of the results obtained.