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Oscillation results for first order neutral delay differential equations with several positive and negative coefficients

Ábel Garab, Gergő Tóth

TL;DR

The paper addresses oscillation of linear neutral delay differential equations with several variable delays and a mix of positive and negative coefficients. It develops two main results: a general oscillation criterion that reduces the problem to a first-order non-neutral DDE (extending Gy\'ori–Ladas) and a comparison-based criterion for multiple constant delays (extending Agarwal et al.), with sharp improvements when coefficients are slowly varying. A key construction uses differentiable shift functions $c_i(t)$ solving $c_i(t)=\tau_i(t)-\delta_i(t-c_i(t))$ to form transformed coefficients $\bar{P}_i(t)=P_i(t)-Q_i(t-c_i(t))(1-c_i'(t))$, enabling a bridge to retarded inequalities. The results are illustrated by examples, underscoring applicability to physical, biological, and engineering systems where NDDEs arise and delay structures are complex and variable.

Abstract

We provide sufficient criteria for the oscillation of all solutions of neutral delay differential equations of the form \[ \left[x(t) - \sum_{i=1}^{N_r}R_i(t)x(t - r_i(t)) \right]' + \sum_{i=1}^{N_p}P_i(t)x(t - τ_i(t)) - \sum_{i=1}^{N_q}Q_i(t)x(t - δ_i(t))=0, \] with both positive and negative terms and time-variable delays. Our results improve and generalize several existing criteria available in the literature that address restricted cases, such as constant delays or the absence of negative coefficients. Under additional assumptions on slowly varying parameters, we derive sharper oscillation conditions. We demonstrate the applicability of our findings through illustrative examples.

Oscillation results for first order neutral delay differential equations with several positive and negative coefficients

TL;DR

The paper addresses oscillation of linear neutral delay differential equations with several variable delays and a mix of positive and negative coefficients. It develops two main results: a general oscillation criterion that reduces the problem to a first-order non-neutral DDE (extending Gy\'ori–Ladas) and a comparison-based criterion for multiple constant delays (extending Agarwal et al.), with sharp improvements when coefficients are slowly varying. A key construction uses differentiable shift functions solving to form transformed coefficients , enabling a bridge to retarded inequalities. The results are illustrated by examples, underscoring applicability to physical, biological, and engineering systems where NDDEs arise and delay structures are complex and variable.

Abstract

We provide sufficient criteria for the oscillation of all solutions of neutral delay differential equations of the form \[ \left[x(t) - \sum_{i=1}^{N_r}R_i(t)x(t - r_i(t)) \right]' + \sum_{i=1}^{N_p}P_i(t)x(t - τ_i(t)) - \sum_{i=1}^{N_q}Q_i(t)x(t - δ_i(t))=0, \] with both positive and negative terms and time-variable delays. Our results improve and generalize several existing criteria available in the literature that address restricted cases, such as constant delays or the absence of negative coefficients. Under additional assumptions on slowly varying parameters, we derive sharper oscillation conditions. We demonstrate the applicability of our findings through illustrative examples.
Paper Structure (5 sections, 10 theorems, 84 equations)

This paper contains 5 sections, 10 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results -- variable delays
  4. Results -- constant delays
  5. Examples

Key Result

Proposition 2.1

Assume that $\tau$ and $\delta$ are both continuously differentiable functions on $[t_0,\infty)$ with $0 \leq \delta(t)\leq \tau(t)\leq D$ for some positive constant $D$, such that $\delta'(t)<1$ for all $t\geq t_0$. Then for any $t\geq t_0$, there exists a unique $c(t)\in [0,D]$ such that Moreover, the function $t\mapsto c(t)$ is differentiable and fulfills

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 14 more