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

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.

[1]  A. Domoshnitsky,et al.  Seminonoscillation Intervals and Sign-Constancy of Green’s Functions of Two-Point Impulsive Boundary-Value Problems , 2021, Ukrainian Mathematical Journal.

[2]  A. Domoshnitsky,et al.  The Sturm Separation Theorem for Impulsive Delay Differential Equations , 2018, Tatra Mountains Mathematical Publications.

[3]  A. Domoshnitsky,et al.  Semi-nonoscillation intervals in the analysis of sign constancy of Green’s functions of Dirichlet, Neumann and focal impulsive problems , 2017, Advances in Difference Equations.

[4]  Antje Baer,et al.  Dynamic Impulse Systems Theory And Applications , 2016 .

[5]  Jonas Gloeckner,et al.  Impulsive Differential Equations , 2016 .

[6]  A. Domoshnitsky,et al.  About differential inequalities for nonlocal boundary value problems with impulsive delay equations , 2015 .

[7]  A. Domoshnitsky,et al.  ABOUT SIGN-CONSTANCY OF GREEN'S FUNCTIONS FOR IMPULSIVE SECOND ORDER DELAY EQUATIONS , 2014 .

[8]  L. Berezansky,et al.  Nonoscillation Theory of Functional Differential Equations with Applications , 2012 .

[9]  M. Feng,et al.  Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations , 2009 .

[10]  Tadeusz Jankowski,et al.  Positive solutions to second order four-point boundary value problems for impulsive differential equations , 2008, Appl. Math. Comput..

[11]  Yonghong Wu,et al.  Positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations , 2008, Appl. Math. Comput..

[12]  D. O’Regan,et al.  Introduction to the theory of functional differential equations : methods and applications , 2007 .

[13]  Daqing Jiang,et al.  Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations , 2006 .

[14]  Eun Kyoung Lee,et al.  Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations , 2004, Appl. Math. Comput..

[15]  I. Mizgireva.  On positivity of Green's Functions of two-point impulsive problems , 2004 .

[16]  A. A. Boichuk,et al.  Generalized Inverse Operators and Fredholm Boundary-Value Problems , 2004 .

[17]  Ravi P. Agarwal,et al.  Multiple nonnegative solutions for second order impulsive differential equations , 2000, Appl. Math. Comput..

[18]  Alexander Domoshnitsky,et al.  Nonoscillation of First Order Impulse Differential Equations with Delay , 1997 .

[19]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

[20]  S. G. Pandit,et al.  Differential systems involving impulses , 1982 .

[21]  M. A. Krasnoselʹskii,et al.  Approximate Solution of Operator Equations , 1972 .