Negativity of Green’s Functions to Focal and Two-Point Boundary Value Problems for Equations of Second Order with Delay and Impulses in Their Derivatives

Alexander Domoshnitsky; Sergey Malev; Vladimir Raichik

doi:10.3390/math10193683

Negativity of Green’s Functions to Focal and Two-Point Boundary Value Problems for Equations of Second Order with Delay and Impulses in Their Derivatives

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Abstract

We consider the second-order impulsive differential equation with impulses in derivative and without the damping term. Sufficient conditions that a nontrivial solution of the homogeneous equation having a zero of its derivative does not have a zero itself are obtained. On the basis of the obtained results on differential inequalities, which can be considered as analogues of the Vallee–Poussin theorems, new sufficient conditions on the negativity of Green’s functions are obtained.

Original language	English
Article number	3683
Journal	Mathematics
Volume	10
Issue number	19
DOIs	https://doi.org/10.3390/math10193683
State	Published - Oct 2022

Keywords

focal intervals
Green’s function
positivity of solutions
second-order impulsive differential equations
semi-nonoscillation intervals
Vallee–Poussin theorem on differential inequality for impulsive equations

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title = "Negativity of Green{\textquoteright}s Functions to Focal and Two-Point Boundary Value Problems for Equations of Second Order with Delay and Impulses in Their Derivatives",

abstract = "We consider the second-order impulsive differential equation with impulses in derivative and without the damping term. Sufficient conditions that a nontrivial solution of the homogeneous equation having a zero of its derivative does not have a zero itself are obtained. On the basis of the obtained results on differential inequalities, which can be considered as analogues of the Vallee–Poussin theorems, new sufficient conditions on the negativity of Green{\textquoteright}s functions are obtained.",

keywords = "focal intervals, Green{\textquoteright}s function, positivity of solutions, second-order impulsive differential equations, semi-nonoscillation intervals, Vallee–Poussin theorem on differential inequality for impulsive equations",

author = "Alexander Domoshnitsky and Sergey Malev and Vladimir Raichik",

note = "Publisher Copyright: {\textcopyright} 2022 by the authors.",

year = "2022",

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doi = "10.3390/math10193683",

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AU - Domoshnitsky, Alexander

AU - Malev, Sergey

AU - Raichik, Vladimir

PY - 2022/10

Y1 - 2022/10

N2 - We consider the second-order impulsive differential equation with impulses in derivative and without the damping term. Sufficient conditions that a nontrivial solution of the homogeneous equation having a zero of its derivative does not have a zero itself are obtained. On the basis of the obtained results on differential inequalities, which can be considered as analogues of the Vallee–Poussin theorems, new sufficient conditions on the negativity of Green’s functions are obtained.

AB - We consider the second-order impulsive differential equation with impulses in derivative and without the damping term. Sufficient conditions that a nontrivial solution of the homogeneous equation having a zero of its derivative does not have a zero itself are obtained. On the basis of the obtained results on differential inequalities, which can be considered as analogues of the Vallee–Poussin theorems, new sufficient conditions on the negativity of Green’s functions are obtained.

KW - focal intervals

KW - Green’s function

KW - positivity of solutions

KW - second-order impulsive differential equations

KW - semi-nonoscillation intervals

KW - Vallee–Poussin theorem on differential inequality for impulsive equations

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ER -

Negativity of Green’s Functions to Focal and Two-Point Boundary Value Problems for Equations of Second Order with Delay and Impulses in Their Derivatives

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Keywords

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