Positivity of Green's matrix of nonlocal boundary value problems

Alexander Domoshnitsky

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 4, page 621-638
  • ISSN: 0862-7959

Abstract

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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 - ∑ j = 1 n m i j x j = β i , i = 1 , ⋯ , n , where n i and m i j are linear bounded “local” and “nonlocal“ functionals, respectively, from the space of absolutely continuous functions. For instance, n i x i = x i ( ω ) or n i x i = x i ( 0 ) - x i ( ω ) and m i j x j = ∫ 0 ω k ( s ) x j ( s ) d s + ∑ r = 1 n i j c i j r x j ( t i j r ) 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 = α i , i = 1 , ⋯ , n .

How to cite

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Domoshnitsky, Alexander. "Positivity of Green's matrix of nonlocal boundary value problems." Mathematica Bohemica 139.4 (2014): 621-638. <http://eudml.org/doc/269830>.

@article{Domoshnitsky2014,
abstract = {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 $n_\{i\}x_\{i\}=\alpha _\{i\}$, $i=1,\dots ,n.$},
author = {Domoshnitsky, Alexander},
journal = {Mathematica Bohemica},
keywords = {functional differential equation; nonlocal boundary value problem; positivity of Green's operator; fundamental matrix; differential inequalities; functional differential equation; nonlocal boundary value problem; positivity of Green's operator; fundamental matrix; differential inequalities},
language = {eng},
number = {4},
pages = {621-638},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Positivity of Green's matrix of nonlocal boundary value problems},
url = {http://eudml.org/doc/269830},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Domoshnitsky, Alexander
TI - Positivity of Green's matrix of nonlocal boundary value problems
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 4
SP - 621
EP - 638
AB - 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 $n_{i}x_{i}=\alpha _{i}$, $i=1,\dots ,n.$
LA - eng
KW - functional differential equation; nonlocal boundary value problem; positivity of Green's operator; fundamental matrix; differential inequalities; functional differential equation; nonlocal boundary value problem; positivity of Green's operator; fundamental matrix; differential inequalities
UR - http://eudml.org/doc/269830
ER -

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