TY  - JOUR
ID  - Domoshnitsky2014
LB  - Domoshnitsky2014
AU  - Domoshnitsky, Alexander
AU  - Landsman, Guy
AU  - Yanetz, Shlomo
TI  - About sign-constancy of Green's functions for impulsive second order delay equations
ST  - About sign-constancy of Green's functions for impulsive second order delay equations
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2014
VL  - 34
IS  - 2
SP  - 339
EP  - 362
DO  - http://dx.doi.org/10.7494/OpMath.2014.34.2.339
UR  - http://dx.doi.org/10.7494/OpMath.2014.34.2.339
AB  - We consider the following second order differential equation with delay \[\begin{cases} (Lx)(t)\equiv{x''(t)+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), \quad t\in[0,\omega],\\ x(t_j)=\gamma_{j}x(t_j-0), x'(t_j)=\delta_{j}x'(t_j-0), \quad j=1,2,\ldots,r. \end{cases}\] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality \(\sum_{i=1}^p{b_i(t)\left(\frac{1}{4}+r\right)}\lt \frac{2}{\omega^2}\) is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case \(0\lt \gamma_i\leq{1}\), \(0\lt \delta_i\leq{1}\) for \(i=1,\ldots ,p\).
KW  - impulsive equations
KW  - Green's functions
KW  - positivity/negativity of Green's functions
KW  - boundary value problem
KW  - second order
SN  - 1232-9274
ER  -