New product-type oscillation criteria for first-order linear differential equations with several nonmonotone arguments

Abstract

AbstractWe use an improved technique to establish new sufficient criteria of product type for the oscillation of the delay differential equation $$\begin{aligned} x'(t)+\sum_{l=1}^{m} b_{l}(t)x\bigl(\sigma _{l}(t)\bigr)= 0,\quad t\geq t_{0}, \end{aligned}$$ x ′ ( t ) + ∑ l = 1 m b l ( t ) x ( σ l ( t ) ) = 0 , t ≥ t 0 , with $b_{l},\sigma _{l}\in C([t_{0},\infty ),[0,\infty ))$ b l , σ l ∈ C ( [ t 0 , ∞ ) , [ 0 , ∞ ) ) such that $\sigma _{l}(t)\leq t$ σ l ( t ) ≤ t and $\lim_{t \rightarrow \infty} \sigma _{l}(t)=\infty $ lim t → ∞ σ l ( t ) = ∞ , $l=1,2,\ldots,m$ l = 1 , 2 , … , m . The obtained results are applicable for the nonmonotone delay case. Their strength is supported by a detailed practical example.