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A Note on the Oscillatory Behavior of Impulsive Differential Equations with Piecewise Constant Arguments via Difference Equations

Ricardo Torres Naranjo, Eugenio Trucco Vera, Özkan Öcal

Abstract

This paper studies the oscillatory behavior of solutions to linear nonautonomous impulsive differential equations with piecewise constant arguments, including both advanced and delayed cases \[ x'(t) = a(t)x(t) + b(t)x([t-k]), \quad k \in \mathbb{Z}. \] By exploiting the hybrid structure of these systems, we reduce the problem to an associated difference equation whose coefficients explicitly incorporate both the continuous dynamics and the impulsive effects. Classical oscillation criteria for difference equations do not account for impulsive phenomena. Through the proposed reduction, we extend these criteria to a class of impulsive and non-impulsive equations (IDEPCA and DEPCA), obtaining explicit sufficient conditions for oscillation in terms of the original system data. An example is provided to illustrate the applicability of the results.

A Note on the Oscillatory Behavior of Impulsive Differential Equations with Piecewise Constant Arguments via Difference Equations

Abstract

This paper studies the oscillatory behavior of solutions to linear nonautonomous impulsive differential equations with piecewise constant arguments, including both advanced and delayed cases \[ x'(t) = a(t)x(t) + b(t)x([t-k]), \quad k \in \mathbb{Z}. \] By exploiting the hybrid structure of these systems, we reduce the problem to an associated difference equation whose coefficients explicitly incorporate both the continuous dynamics and the impulsive effects. Classical oscillation criteria for difference equations do not account for impulsive phenomena. Through the proposed reduction, we extend these criteria to a class of impulsive and non-impulsive equations (IDEPCA and DEPCA), obtaining explicit sufficient conditions for oscillation in terms of the original system data. An example is provided to illustrate the applicability of the results.

Paper Structure

This paper contains 11 sections, 11 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Aim of the work
  3. Recent Advances on Oscillations in Differential Equations with Piecewise Constant Argument
  4. Definition of a solution for (abstract system) in the DEPCA and IDEPCA frameworks
  5. Existence and Uniqueness of Solutions for Eq.
  6. Relationship Between the Oscillatory Behavior of IDEPCA and Difference Equations
  7. Some Useful Oscillatory Criteria from the Theory of Difference Equations
  8. The difference equation zn+1 = an zn + bn zn-k and its simplified form
  9. Difference equations theory: some oscillatory criteria for \ref{['ec_dif_reducida']}
  10. Main Results
  11. Some Examples of Oscillatory Linear DEPCA-IDEPCA Equations

Key Result

Theorem 1

If then every solution of depca_t_menos_1 oscillates.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Definition 3: DEPCA solution of \ref{['sistema_idepcag_general_abstract']}
  • Definition 4: IDEPCA solution of \ref{['sistema_idepcag_general_abstract']}
  • Lemma 1: Existence and Uniqueness
  • proof
  • ...and 12 more