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A simple oscillation criteria for second-order differential equations with piecewise constant argument of generalized type

Ricardo Torres Naranjo

TL;DR

The paper addresses the problem of establishing simple, verifiable oscillation criteria for second-order differential equations with a generalized piecewise constant argument (DEPCAG). It develops two Leighton-Wintner-type oscillation criteria for the DEPCAG $(r(t)x'(t))'+f(t,x(\\gamma(t)))=0$ by leveraging a key lemma that enforces monotone behavior of $x'(t)$ under integral conditions on $r$, and by analyzing the auxiliary quantity $w(t)=\\frac{r(t)x'(t)}{\\phi(x(\\gamma(t)))}$ across advanced and delayed subintervals. The main contributions are two criteria: (i) a criterion requiring $\\int_{\\tau}^{\\infty} p(s) ds=\\infty$, and (ii) a second criterion based on mixed integrability conditions involving $\\phi$, $p$, and $r$, with a generalized $\\gamma(t)$ that recovers the classical Wang-Cheng results when $\\gamma(t)=[t]$. These results yield simple, checkable oscillation conditions for DEPCAG models exhibiting hybrid continuous-discrete dynamics, and the paper illustrates applicability with linear and nonlinear examples.

Abstract

This work presents two simple criteria for determining the oscillatory nature of solutions to second-order differential equations with deviated arguments. These criteria extend the (Leighton-Wintner)-type criteria established by G.Q. Wang and S.S. Cheng, considering a generalized piecewise constant argument. Finally, we provide some examples.

A simple oscillation criteria for second-order differential equations with piecewise constant argument of generalized type

TL;DR

The paper addresses the problem of establishing simple, verifiable oscillation criteria for second-order differential equations with a generalized piecewise constant argument (DEPCAG). It develops two Leighton-Wintner-type oscillation criteria for the DEPCAG by leveraging a key lemma that enforces monotone behavior of under integral conditions on , and by analyzing the auxiliary quantity across advanced and delayed subintervals. The main contributions are two criteria: (i) a criterion requiring , and (ii) a second criterion based on mixed integrability conditions involving , , and , with a generalized that recovers the classical Wang-Cheng results when . These results yield simple, checkable oscillation conditions for DEPCAG models exhibiting hybrid continuous-discrete dynamics, and the paper illustrates applicability with linear and nonlinear examples.

Abstract

This work presents two simple criteria for determining the oscillatory nature of solutions to second-order differential equations with deviated arguments. These criteria extend the (Leighton-Wintner)-type criteria established by G.Q. Wang and S.S. Cheng, considering a generalized piecewise constant argument. Finally, we provide some examples.

Paper Structure

This paper contains 6 sections, 3 theorems, 80 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Recent Developments in Second-Order Differential Equations with Deviating Arguments
  3. Aim of the work
  4. Auxiliary Results
  5. Main Results
  6. Examples

Key Result

Lemma 1

Let $x(t)$ be a solution of osc_DEPCAG such that there is some $M\geq 0$ and $x(t)\geq 0$ for $t\geq M$, $\gamma(t)$ be a generalized piecewise constant argument. If then $x'(t)\geq 0$ for all $\{t_k\}_{k\geq N},$ for some $N$ sufficiently large such that $t_k\geq M$.

Figures (2)

  • Figure 1: $f(t)=2\left[\frac{t}{2}\right]+1$. An example of the previous piecewise constant argument, with $m=2$ and $\alpha=0.5$.
  • Figure 2: Discrete solution $x(n)$ of \ref{['ejemplo1']}, with $x(0)=1$ and $x'(0)=0$.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Remark 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 2
  • ...and 2 more