On the sign of Green's function for an impulsive differential equation with periodic boundary conditions
type: Item
instance of: scholarly article
title: On the sign of Green's function for an impulsive differential equation with periodic boundary conditions
zbMATH Open document ID: 1170.34018
author: Aydin Huseynov
published in: Applied Mathematics and Computation
publication date: 2009-03-02T00:00:00Z
review text: According to the abstract of the paper, the author investigates ``Green's function of second order linear differential equations subject to linear impulse conditions and periodic boundary conditions. Sufficient conditions that ensure the positiveness of the Green's function are established.'' To be more precise, the author considers the following linear boundary value problem with impulse (BVPI): \[ (1):\quad -[p(x)y'(x)]'+q(x)y(x)=h(x), \quad x\in[a,c)\cup(c,b], \] \[ (2):\quad y(c^{-})=d_1y(c^{+}), \quad y^{[1]}(c^{-})=d_2y^{[1]}(c^{+}), \] \[ (3):\quad y(a)=y(b), \quad y^{[1]}(a)=y^{[1]}(b), \] where \(a, b, c\) are fixed real numbers, \(a<c<b,\) \(d_1, d_2\) are nonzero real numbers, \(p(x), q(x)\) and \(h(x)\) are real-valued measurable functions on \([a,b]\) such that \[ \int_{a}^{b}\frac{1}{|p(x)|}dx<\infty,\quad \int_{a}^{b}|q(x)|dx<\infty,\quad \int_{a}^{b}|h(x)|dx<\infty, \] and \(y^{[1]}=py'\) denotes the quasi-derivative of the solution \(y.\) When \(h=0,\) let \(u, v\) the solutions of the homogeneous problem (1)-(2) associated to (BVPI) satisfying the initial conditions \(u(a)=1, u^{[1]}(a)=0\) and \(v(a)=0, v^{[1]}(a)=1,\) respectively. Consider the value \(D=u(b)+v^{[1]}(b)-\frac{1}{d_1 d_2}-1.\) If \(D\neq 0,\) the author shows that the non-homogeneous problem (1)-(2)-(3) has a unique solution \(y(x)\) which can be represented (in terms of a Green function whose description is too involved to be described here), as \(y(x)=\int_{a}^{b}G(x,s)h(s)ds,\) for \(x\in[a,c)\cup(c,b].\) Moreover, if the following conditions are assumed, \((H1): p(x)>0, q(x)\geq 0, q(x)\neq 0 \) almost everywhere, and \((H2):\quad d_1>0, d_2>0, d_1+d_2\geq 1+d_1d_2,\) then the Green function is positive for \(x,s\in [a,c)\cup(c,b],\) in fact, \(\inf_{x,s}G(x,s)>0.\)
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 34B37
zbMATH DE Number: 5522174
zbMATH Keywords: Impulsive differential equations
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: Green's function
zbMATH Keywords: quasi-derivative
zbMATH Keywords: wronskian
zbMATH Keywords: Volterra integral equation
reviewed by: Antonio Linero Bas
MaRDI profile type: Publication
full work available at URL: j.amc.2008.11.034
OpenAlex ID: W1969790413
Wikidata QID: Q115361730
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DOI: 10.1016/J.AMC.2008.11.034
description: scientific article; zbMATH DE number 5522174