2025, Vol. 6, Issue 2, Part E

The higher-order delay differential equations are used in the describing of many natural phenomena. This paper presents a study on the oscillatory behavior of solutions of even order neutral differential equations with several delays of the form

(a(t)〖〖((m(t)x(t)+p(t)x(τ(t)))〗^(n-1))〗^γ )^'+r(t) 〖〖((m(t)x(t)+p(t)x(τ(t)))〗^(n-1))〗^γ+∑_(i=1)^n▒〖q_i (t) x^(γ_i ) (σ_i (t)) 〗=0 (1)

is considered. In this research, we applied three techniques – the comparison technique, the Riccati technique, and the integral averages technique to analyze and establish the sufficient conditions for the oscillatory behaviour of the equations under the condition that R(t)=∫_(t_0)^∞▒〖[1/(a(s)) exp⁡(-∫_(t_0)^s▒(r(u))/(a(u)) du)ds]^(1/γ)=∞〗 as t→∞. Also, the results are an extension and simplification as well as improvement of the previous results.