MATHEMATICA BOHEMICA, Vol. 139, No. 4, pp. 621-638, 2014

Positivity of Green's matrix of nonlocal boundary value problems

Alexander Domoshnitsky

Alexander Domoshnitsky, Ariel University, Qiryat University, 40700 Ariel, Israel, e-mail: adom@ariel.ac.il

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 $n_ix_i-\sum\nolimits_{j=1}^nm_{ij}x_j=\beta_i$, $i=1,\dots,n$, where $n_i$ and $m_{ij}$ are linear bounded "local" and "nonlocal" functionals, respectively, from the space of absolutely continuous functions. For instance, $n_ix_i=x_i(\omega)$ or $n_ix_i=x_i(0)-x_i(\omega)$ and $m_{ij}x_j=\int_0^{\omega}k(s)x_j(s)  d s +\sum\nolimits_{r=1}^{n_{ij}}c_{ijr}x_j(t_{ijr})$ 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 $n_ix_i=\alpha_i$, $i=1,\dots,n.$

Keywords: functional differential equation; nonlocal boundary value problem; positivity of Green's operator; fundamental matrix; differential inequalities

Classification (MSC 2010): 34K10, 34K06, 34B27, 34B40


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