The equivalence of the oscillation of delay and ordinary differential equations with applications (Q2753278)

The authors consider the linear first-order differential equation with constant delays NEWLINE\[NEWLINEx'(t)+\sum_{i=1}^n p_i(t)x(t-\tau_i)=0,NEWLINE\]NEWLINE with \(p_i\in C([t_0,\infty),\mathbb{R}^+)\), \(\tau_i>0\) and \(\lim_{t\rightarrow\infty}\inf p_i(t)=p_i\). They prove that in the critical case the oscillation (nonoscillation) of this equation is equivalent to one of the following second-order differential equation without delay NEWLINE\[NEWLINEy''+\left[2\left(\sum_{i=1}^n\tau_i^2p_ie^{-\lambda_0\tau_i}\right)^{-1}\sum_{i=1}^ne^{-\lambda_0\tau_i}(p_i(t)-p_i)\right]=0,NEWLINE\]NEWLINE where \(\lambda_0\) is a simple real root of the characteristic equation NEWLINE\[NEWLINE\lambda+\sum_{i=1}^n p_ie^{-\lambda\tau_i}=0.NEWLINE\]NEWLINE New explicit oscillation and nonoscillation conditions are obtained.