About sign-constancy of Green’s function of a two-point problem for impulsive second order delay equations
We consider the following second order differential equation with delay $$ \begin{cases} \begin{split} (Lx)(t)&\equiv{x''(t)+\sum_{j=1}^p {a_{j}(t)x'(t-\tau_{j}(t))}+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), & t\in[0,\omega] \end{split} \\ x(t_k)=\gamma_{k}x(t_k-0), \...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Szeged
2016-08-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=4268 |
| Summary: | We consider the following second order differential equation with delay
$$
\begin{cases}
\begin{split}
(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), & t\in[0,\omega]
\end{split} \\
x(t_k)=\gamma_{k}x(t_k-0), \ x'(t_k)=\delta_{k}x'(t_k-0), \quad k=1,2,\dots,r.
\end{cases}
$$
In this paper we find sufficient conditions of positivity of Green's functions for this impulsive equation coupled with two-point boundary conditions in the form of theorems about differential inequalities.
Choosing the test function in these theorems, we obtain simple sufficient conditions. |
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| ISSN: | 1417-3875 1417-3875 |