Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary-value problems - Domoshnitsky, A.; Mizgireva, Iu.; Raichik, V. (Ukrains’kyi Matematychnyi Zhurnal, 2021)
UDC 517.9We consider the second order impulsive differential equation with delays $$(Lx)(t)\equiv x''(t) + \sum\limits _{j = 1}^{p} a_j(t) x'(t-\tau_j(t)) + \sum\limits _{j = 1}^{p} b_j(t) x(t-\theta_j(t)) = f(t),  \quad t \in [0, \omega], $$ $$x(t_k) = \gamma_k x(t_k-0), \quad x'(t_k) = \delta_k x'(t_k-0),\quad  k = 1, 2, \ldots , r,$$where $\gamma_k > 0,$ $\delta_k >0$ for $k = 1, 2, \ldots , r.$ In this paper, we obtain the conditions of semi-nonoscillation for the corresponding homogeneous equation on the interval $[0, \omega].$  Using these results, we formulate theorems on sign-constancy of Green's functions for two-point impulsive boundary-value problems in terms of differential inequalities.