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Title: Positivity of Green's matrix of nonlocal boundary value problems (English)
Author: Domoshnitsky, Alexander
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959
Volume: 139
Issue: 4
Year: 2014
Pages: 621-638
Summary lang: English
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Category: math
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Summary: We propose an approach for studying positivity of Green's operators of a nonlocal boundary value problem for the system of $n$ linear functional differential equations with the boundary conditions $n_{i}x_{i}-\sum \nolimits _{j=1}^{n}m_{ij}x_{j}=\beta _{i}$, $i=1,\dots ,n$, where $n_{i}$ and $m_{ij}$ are linear bounded “local” and “nonlocal“ functionals, respectively, from the space of absolutely continuous functions. For instance, $n_{i}x_{i}=x_{i}(\omega )$ or $n_{i}x_{i}=x_{i}(0)-x_{i}(\omega )$ and $m_{ij}x_{j}=\int _{0}^{\omega }k(s)x_{j}(s) {\rm d} s +\sum \nolimits _{r=1}^{n_{ij}}c_{ijr}x_{j}(t_{ijr})$ can be considered. It is demonstrated that the positivity of Green's operator of nonlocal problem follows from the positivity of Green's operator for auxiliary “local” problem which consists of a “close” equation and the local conditions $n_{i}x_{i}=\alpha _{i}$, $i=1,\dots ,n.$ (English)
Keyword: functional differential equation
Keyword: nonlocal boundary value problem
Keyword: positivity of Green's operator
Keyword: fundamental matrix
Keyword: differential inequalities
MSC: 34B27
MSC: 34B40
MSC: 34K06
MSC: 34K10
idZBL: Zbl 06433686
idMR: MR3306852
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Date available: 2015-02-04T09:19:50Z
Last updated: 2016-01-04
Stable URL: http://hdl.handle.net/10338.dmlcz/144139
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