Asymptotic behavior of even-order noncanonical neutral differential equations

Abstract In this article, we study the asymptotic behavior of even-order neutral delay differential equation ( a ⋅ u + ρ ∘ τ stretchy="false">) n − 1 accent="true">′ ℓ h g = 0 , ≥ {(a\cdot {(u+\rho \cdot u\circ \tau )}^{(n-1)})}^{^{\prime} }(\ell )+h(\ell )u(g(\ell ))=0,\hspace{1.0em}\ell \ge {\ell }_{0}, where xmlns:m="http://www.w3.org/1998/Math/MathML"> 4 n\ge 4 , and in noncanonical case, that is, ∫ ∞ ( s ) d < . \mathop{\int }\limits^{\infty }{a}^{-1}\left(s){\rm{d}}s\lt \infty . To best our knowledge, most previous studies were concerned only with n -order equations canonical case. By using comparison principle Riccati transformation technique, obtain new criteria which ensure every solution studied is either oscillatory or converges to zero. Examples are presented illustrate results.