Opuscula Math. 34, no. 2 (2014), 339-362
http://dx.doi.org/10.7494/OpMath.2014.34.2.339

 
Opuscula Mathematica

About sign-constancy of Green's functions for impulsive second order delay equations

Alexander Domoshnitsky
Guy Landsman
Shlomo Yanetz

Abstract. 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 \(i=1,\ldots ,p\).

Keywords: impulsive equations, Green's functions, positivity/negativity of Green's functions, boundary value problem, second order.

Mathematics Subject Classification: 34K10, 34B37, 34A40, 34A37, 34K48.

Full text (pdf)

Opuscula Mathematica - cover

Cite this article as:
Alexander Domoshnitsky, Guy Landsman, Shlomo Yanetz, About sign-constancy of Green's functions for impulsive second order delay equations, Opuscula Math. 34, no. 2 (2014), 339-362, http://dx.doi.org/10.7494/OpMath.2014.34.2.339

Download this article's citation as:
a .bib file (BibTeX),
a .ris file (RefMan),
a .enw file (EndNote)
or export to RefWorks.

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.