Asymptotic behavior of even-order noncanonical neutral differential equations

Abstract

Abstract In this article, we study the asymptotic behavior of even-order neutral delay differential equation ( a ⋅ ( u + ρ ⋅ u ∘ τ ) ( n − 1 ) ) ′ ( ℓ ) + h ( ℓ ) u ( g ( ℓ ) ) = 0 , ℓ ≥ ℓ 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 n ≥ 4 n\ge 4 , and in noncanonical case, that is, ∫ ∞ a − 1 ( s ) d s < ∞ . \mathop{\int }\limits^{\infty }{a}^{-1}\left(s){\rm{d}}s\lt \infty . To the best of our knowledge, most of the previous studies were concerned only with the study of n n -order neutral equations in canonical case. By using comparison principle and Riccati transformation technique, we obtain new criteria which ensure that every solution of the studied equation is either oscillatory or converges to zero. Examples are presented to illustrate our new results.