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On the Liénard's type equation: an icon of the Nonlinear Analysis

Juan E. Nápoles Valdés

TL;DR

This paper surveys qualitative results for the Liénard equation under non-conformable, generalized, and Caputo fractional operators. It develops a unified framework of a generalized integral operator and N-derivatives, then derives oscillation, boundedness, and stability results across three operator regimes. The findings extend classical Liénard theory to broader contexts, showing that core qualitative behaviors persist under these generalized derivatives and providing criteria for oscillations, continuability, and stability. Together, these contributions offer a concise compendium for researchers studying nonlinear oscillations and their applications in physics and engineering.

Abstract

In this note, we review the latest qualitative results, referring to the Liénard Equation, in the framework of non-conformable, generalized and fractional differential operators.

On the Liénard's type equation: an icon of the Nonlinear Analysis

TL;DR

This paper surveys qualitative results for the Liénard equation under non-conformable, generalized, and Caputo fractional operators. It develops a unified framework of a generalized integral operator and N-derivatives, then derives oscillation, boundedness, and stability results across three operator regimes. The findings extend classical Liénard theory to broader contexts, showing that core qualitative behaviors persist under these generalized derivatives and providing criteria for oscillations, continuability, and stability. Together, these contributions offer a concise compendium for researchers studying nonlinear oscillations and their applications in physics and engineering.

Abstract

In this note, we review the latest qualitative results, referring to the Liénard Equation, in the framework of non-conformable, generalized and fractional differential operators.

Paper Structure

This paper contains 9 sections, 13 theorems, 28 equations.

Table of Contents

  1. Preliminars
  2. Historic context
  3. The Liénard Equation
  4. A general integral operator
  5. Results
  6. The non-conformable case
  7. The generalized case
  8. The fractional case
  9. Conclusions

Key Result

Proposition 5

Let $I$ be an interval $I \subseteq \mathbb{R}$, $a \in I$, $0<\alpha \le 1$ and $f$ a $\alpha$-differentiable function on $I$ such that $f'$ is a locally integrable function on $I$. Then, we have for all $x \in I$

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Remark 3
  • Remark 4
  • Proposition 5
  • Proposition 6
  • Theorem 7
  • Theorem 8
  • Remark 9
  • Theorem 10
  • ...and 11 more