Existence of solutions to singular integral equations (Q5919923)

The following system of integral equations is investigated: \[ u_i(t)= \int^T_0 g_i(t, s)[a_i(s, u_1(s),\dots, u_n(s))+ b_i(s, u_1(s),\dots, u_n(s))]\,ds,\tag{1} \] for \(t\in [0, T]\) and \(i= 1,2,\dots, n\). The functions \(a_i(t, u_1,\dots, u_n)\) can be singular at \(t= 0\) and \(u_j=0\) for \(j\in \{1,2,\dots, n\}\). Under some assumptions (generally very complicated, but admitting natural realizations) it is proved that (1) has a fixed sign solutions. An example illustrating the theoretical investigations is also included.