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Bifurcation of periodic solutions to nonlinear measure differential equations

Maria Carolina Mesquita Macena Stefani, Milan Tvrdý

Abstract

This paper is devoted to bifurcations of periodic solutions of nonlinear measure differential equations with a parameter. Main tools are nonlinear generalized differential equations (in the sense of Kurzweil) and the Kurzweil gauge type generalized integral. We continue the research started by the first author under the supervision of the second one.

Bifurcation of periodic solutions to nonlinear measure differential equations

Abstract

This paper is devoted to bifurcations of periodic solutions of nonlinear measure differential equations with a parameter. Main tools are nonlinear generalized differential equations (in the sense of Kurzweil) and the Kurzweil gauge type generalized integral. We continue the research started by the first author under the supervision of the second one.
Paper Structure (5 sections, 27 theorems, 148 equations)

This paper contains 5 sections, 27 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Preliminaries (Kurzweil integral and generalized ODEs)
  3. Bifurcation theory for generalized ODEs
  4. Measure Differential Equations
  5. Bifurcation theory for Measure Differential Equations

Key Result

Lemma 2.3

Let $U:[a,b]\times[a,b]\to X$ be Kurzweil integrable and regulated in the second variable on $[a,b]$ and Then $v$ is regulated on $[a,b],$ Moreover, if there are functions $f\,{:}\,[a,b]\,{\to}\,\mathbb{R}$ regulated on $[a,b]$ and $g\,{:}\,[a,b]\,{\to}\,\mathbb{R}$ nondecreasing on $[a,b]$ and such that then

Theorems & Definitions (58)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Remark 2.8
  • Theorem 2.9
  • Definition 3.1
  • ...and 48 more