# Article

 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 . Category: math . 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 . Date available: 2015-02-04T09:19:50Z Last updated: 2016-01-04 Stable URL: http://hdl.handle.net/10338.dmlcz/144139 . Reference: [1] Agarwal, R. P., Berezansky, L., Braverman, E., Domoshnitsky, A.: Nonoscillation Theory of Functional Differential Equations with Applications.Springer, New York (2012). Zbl 1253.34002, MR 2908263 Reference: [2] Agarwal, R. 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