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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
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 $n_{i}x_{i}=\alpha _{i}$, $i=1,\dots ,n.$
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