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A variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type

Ricardo Torres, Manuel Pinto

Abstract

In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated arguments investigated before as particular cases. We also give some examples showing the applicability of our results.

A variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type

Abstract

In this work, we give a variation of parameters formula for nonautonomous linear impulsive differential equations with piecewise constant arguments of generalized type. We cover several cases of differential equations with deviated arguments investigated before as particular cases. We also give some examples showing the applicability of our results.
Paper Structure (19 sections, 6 theorems, 97 equations, 6 figures, 1 table)

This paper contains 19 sections, 6 theorems, 97 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Why study IDEPCAG?: impulses in action
  3. Fundamental matrices and variation of parameters formulas: an overview
  4. The fundamental matrix of a DEPCA system
  5. Variation of parameters formula for a DEPCA
  6. Variation of parameters formula for an IDEPCA: the impulsive effect applied
  7. Aim of the work
  8. Preliminaires
  9. An Integral equation associated to \ref{['SISTEMA_GENERICO_IDEPCAG']}
  10. First IDEPCAG Gronwall-Bellman type inequality
  11. Second IDEPCAG Gronwall-Bellman type inequality
  12. Existence and uniqueness for \ref{['SISTEMA_GENERICO_IDEPCAG']}
  13. Variation of parameters formula for IDEPCAG
  14. The fundamental matrix of the linear homogeneous IDEPCAG
  15. The variation of parameter formula for IDEPCAG
  16. ...and 4 more sections

Key Result

Theorem 1

(CTP2019, Lemma 4.2) a function $x(t)=x(t,\tau ,x_0)$, $\tau\in\mathbb{R}^+$ is a solution of SISTEMA_GENERICO_IDEPCAG on $\mathbb{R}^{+}$ if and only if satisfies: where

Figures (6)

  • Figure 1: Solution of \ref{['idepca_parte_entera']} with $\alpha=0.9$, $\beta=1.2$, $x_0=1.8$.
  • Figure 2: solution of \ref{['idepca_parte_entera']} with $\alpha=0.4$, $\beta=-2$, $x_0=2.4$.
  • Figure 3: Solution of \ref{['berek_sistema_avanzado']} with $c_k=-1.1$ and $z(0)=-1.2$
  • Figure 4: Solution of \ref{['SISTEMA_BEREK']} with $\gamma(t)=[t]+7/10$, $D_r=1/r^2$, $A(t)=1/(t+1),$$f(t)=\exp(-t)$ and $y(0)=y_0=-1.$
  • Figure 5: Solution of \ref{['SISTEMA_BEREK']} with $D_k=1/k^2,\,\,f(t)=0$ and $z(0)=1.$
  • ...and 1 more figures

Theorems & Definitions (14)

  • Remark 1
  • Definition 1: DEPCAG solution
  • Definition 2: IDEPCAG solution
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Lemma 3
  • Theorem 3
  • Remark 2
  • ...and 4 more