Nontrivial Solutions in Abstract Cones for Hammerstein Integral Systems

We establish new criteria for the existence of nonzero solutions of systems of integral equations of the form $$u_i(t)= B_i\left ( \int_{\eta}^{\mu} f_i(s,u_1(s), \dots, u_n(s))\, ds \right )+ \int_0^1 k_i(t,s) f_i(s,u_1(s), \dots, u_n(s))\, ds,$$ where $B_i$ are continuous functions, $[\eta, \mu]\subset [0,1]$, $f_i$ satisfy Carath\'{e}odory conditions and $k_i$ may be discontinuous and change sign. We apply our results to prove the existence of nontrivial solutions of some systems of differential equations with nonlinear boundary conditions