Positive and nontrivial solutions to a system of first-order impulsive nonlocal boundary value problems with sign changing nonlinearities (Q830281)

The authors consider a system of two impulsive differential equations with non-local boundary value conditions: \[ \begin{aligned} u'_i(t) + a_i(t)u_i(t) &= f_i(t,u_1(t),u_2(t)), \quad &&\text{for a.e. } t \in [0,1],\\ \triangle u_i(t_k) &= I_i^k(u_1(t_k),u_2(t_k)), && k = 1,\dots,p,\\ \gamma u_i(0) - \delta u_i(1) &= \int_{\alpha_i}^{\beta_i} g_i(s)u_i(s)\,\mathrm{d}s, && i=1,2, \end{aligned} \] where \(a_i,g_i \in L([0,1])\), \(f_i\) are \(L^1\)-Carathéodory function, \(I_i^k\) are continuous and \(\alpha_i\), \(\beta_i\), \(\delta\), \(\gamma\) are real numbers, \(0 < t_1 < t_2 < \ldots < t_p < 1\). Using a fixed point theorem in cones sufficient conditions for the existence of at least two nonnegative solutions of the problem are obtained.