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Positivity of Green’s matrix of nonlocal boundary value problems. (English) Zbl 1349.34255
Summary: We propose an approach for studying the 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 \(m_{ij}x_{j}=\int_{0}^{\omega}k(s)x_{j}(s)\,ds +\sum \nolimits_{r=1}^{n_{ij}}c_{ijr}x_{j}(t_{ijr})\) 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 \(n_{i}x_{i}=\alpha_{i}\), \(i=1,\dots ,n\).

34K10 Boundary value problems for functional-differential equations
34K06 Linear functional-differential equations
34B27 Green’s functions for ordinary differential equations
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