On a class of second-order impulsive boundary value problem at resonance (Q2468984)

The authors consider the impulsive boundary value problem for ordinary differential equation of second order \[ \begin{aligned} &x''(t) = f(t,x(t),x'(t)),\quad t \in J', \\ &\triangle x(t_i) = I_i(x(t_i),x'(t_i)),\;\triangle x'(t_i) = J_i(x(t_i),x'(t_i)),\;i = 1,\ldots,k, \\ &x(0) = 0,\quad x'(1) = \sum_{j=1}^{m-2} \alpha_j x'(\eta_j), \end{aligned} \] where \(0 < t_1 < \ldots < t_k < 1\), \(J' = [0,1]\setminus\{t_1,\ldots,t_k\}\), \(0 < \eta_1 < \dots < \eta_{m-2} < 1\); the functions \(f\), \(I_i\), \(J_i\) are continuous. Sufficient conditions for the existence of at least one solution to this problem are derived. An example illustrating the application of this result is presented, too.