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

Abstract

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

Keywords

• Oscillation
• Delay differential equations
• Several nonmonotone delays