Solutions for singular Volterra integral equations

Abstract

We consider the system of Volterra integral equations $$ \begin{array}{l} u_i(t)=\int_{0}^{t}g_i(t,s)[P_i(s,u_1(s),u_2(s),\cdots, u_n(s)) + Q_i(s,u_1(s),u_2(s),\cdots, u_n(s))]ds, t\in [0,T],1\leq i\leq n \end{array} $$ where $T>0$ is fixed and the nonlinearities $P_i(t,u_1,u_2,\cdots,u_n)$ can be singular at $t=0$ and $u_j=0$ where $j\in\{1,2,\cdots,n\}.$ Criteria are offered for the existence of fixed-sign solutions $(u_1^*,u_2^*,\cdots,u_n^*)$ to the system of Volterra integral equations, i.e., $\theta_iu_i^*(t)\geq 0$ for $t\in [0,1]$ and $1\leq i\leq n,$ where $\theta_i\in \{1,-1\}$ is fixed. We also include an example to illustrate the usefulness of the results obtained.