MMN-2768
Sign-constancy of Green's functions for two-point impulsive boundary value problems
A. Domoshnitsky; Iu. Mizgireva;Abstract
We consider the following second order impulsive differential equation with delays
\begin {equation}
\begin{cases}
(Lx)(t)\equiv x''(t)+\sum_{j=1}^{p} a_j(t) x'(t-\tau_j(t)) + \sum_{j=1}^{p} b_j(t) x(t-\theta_j(t)) = f(t), \notag \\
\quad t \in [0,\omega], \notag \\
x(t_k)=\gamma_k x(t_k-0), \quad x'(t_k)=\delta_k x'(t_k-0), \quad k=1,2,...,r. \notag
\end{cases}
\end {equation}
In this paper we obtain sufficient conditions of nonpositivity of Green's functions for impulsive differential equation.
All results are formulated in the form of theorems about differential inequalities.
It should be noted that the sign-constancy of the coefficients $b_j(t)$ was assumed in all the literature devoted to impulsive functional differential equations.
One of the main purposes of this work is to propose a technique allowing us to avoid these assumptions.
Vol. 20 (2019), No. 1, pp. 193-208
DOI: 10.18514/MMN.2019.2768