Refined oscillation criteria for second-order mixed neutral equations: A finite-sum iterative approach

This paper establishes refined oscillation criteria for second-order mixed neutral differential equations. For equations with variable delays, we construct a bridge function to prove a comparison theorem, which reduces the study of the second-order equation to a first-order delay differential equation. For equations with constant delays, we develop a finite-sum iterative method. By applying an integral operator prior to the iteration steps and iterating the neutral feedback inside this integral bound, we generate increasing sequences that yield sharper oscillation conditions. Finally, we provide two illustrative examples. The first validates the criteria for variable delays, while the second shows that the iterative process detects oscillation even when classical results and the initial iteration step fail.