About sign-constancy of Green's functions for impulsive second order delay equations
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 thi...
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| Format: | Article |
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AGH Univeristy of Science and Technology Press
2014-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | http://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3421.pdf |
| _version_ | 1819174102013837312 |
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| author | Alexander Domoshnitsky Guy Landsman Shlomo Yanetz |
| author_facet | Alexander Domoshnitsky Guy Landsman Shlomo Yanetz |
| author_sort | Alexander Domoshnitsky |
| collection | DOAJ |
| description | 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 \(i=1,\ldots ,p\). |
| first_indexed | 2024-12-22T20:33:38Z |
| format | Article |
| id | doaj.art-186b4ffc7ac44135b6e6f1fea6204176 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-22T20:33:38Z |
| publishDate | 2014-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-186b4ffc7ac44135b6e6f1fea62041762022-12-21T18:13:31ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742014-01-01342339362http://dx.doi.org/10.7494/OpMath.2014.34.2.3393421About sign-constancy of Green's functions for impulsive second order delay equationsAlexander Domoshnitsky0Guy Landsman1Shlomo Yanetz2Ariel University, Department of Computer Science and Mathematics, 44837 Ariel, IsraelBar Ilan University, Department of Mathematics, 52990 Ramat-Gan, IsraelBar Ilan University, Department of Mathematics, 52990 Ramat-Gan, IsraelWe 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 \(i=1,\ldots ,p\).http://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3421.pdfimpulsive equationsGreen's functionspositivity/negativity of Green's functionsboundary value problemsecond order |
| spellingShingle | Alexander Domoshnitsky Guy Landsman Shlomo Yanetz About sign-constancy of Green's functions for impulsive second order delay equations Opuscula Mathematica impulsive equations Green's functions positivity/negativity of Green's functions boundary value problem second order |
| title | About sign-constancy of Green's functions for impulsive second order delay equations |
| title_full | About sign-constancy of Green's functions for impulsive second order delay equations |
| title_fullStr | About sign-constancy of Green's functions for impulsive second order delay equations |
| title_full_unstemmed | About sign-constancy of Green's functions for impulsive second order delay equations |
| title_short | About sign-constancy of Green's functions for impulsive second order delay equations |
| title_sort | about sign constancy of green s functions for impulsive second order delay equations |
| topic | impulsive equations Green's functions positivity/negativity of Green's functions boundary value problem second order |
| url | http://www.opuscula.agh.edu.pl/vol34/2/art/opuscula_math_3421.pdf |
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