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

TL;DR

This paper establishes new sufficient conditions for the oscillation of solutions to first order neutral delay differential equations with multiple positive and negative coefficients and variable delays, extending previous results.

Contribution

It provides generalized oscillation criteria for complex neutral delay differential equations with variable delays and mixed coefficient signs, improving upon existing literature.

Findings

New oscillation criteria applicable to equations with variable delays

Criteria that handle both positive and negative coefficients

Illustrative examples demonstrating the criteria's effectiveness

Abstract

We provide sufficient criteria for the oscillation of all solutions of neutral delay differential equations of the form \[ \left[x(t) - \sum_{i=1}^{N_r}R_i(t)x(t - r_i(t)) \right]' + \sum_{i=1}^{N_p}P_i(t)x(t - \tau_i(t)) - \sum_{i=1}^{N_q}Q_i(t)x(t - \delta_i(t))=0, \] 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.