Oscillation criteria for even-order neutral differential equations with distributed deviating arguments

In this work, new conditions are obtained for the oscillation of solutions of the even-order equation ( r ( ζ ) z ( n − 1 ) ( ζ ) ) ′ + ∫ a b q ( ζ , s ) f ( x ( g ( ζ , s ) ) ) d s = 0 , ζ ≥ ζ 0 , $$ \bigl( r ( \zeta ) z^{ ( n-1 ) } ( \zeta ) \bigr) ^{\prime }+ \int _{a}^{b}q ( \zeta ,s ) f \bigl( x \bigl( g ( \zeta ,s ) \bigr) \bigr) \,\mathrm{d}s=0, \quad \zeta \geq \zeta _{0}, $$ where n ≥ 2 $n\geq 2$ is an even integer and z ( ζ ) = x α ( ζ ) + p ( ζ ) x ( σ ( ζ ) ) $z ( \zeta ) =x ^{\alpha } ( \zeta ) +p ( \zeta ) x ( \sigma ( \zeta ) ) $ . By using the theory of comparison with first-order delay equations and the technique of Riccati transformation, we get two various conditions to ensure oscillation of solutions of this equation. Moreover, the importance of the obtained conditions is illustrated via some examples.