Existence and bounds for Kneser-type solutions to noncanonical third-order neutral differential equations

This article focuses on the existence and asymptotic behavior of Kneser-type solutions to third-order noncanonical differential equations with a delay or advanced argument in the neutral term (r₂ (t) (r₁ (t) z' (t) ) ') '+g (t) x (t) =0, where \ (z (t) =x (t) +p (t) x ( (t) ) \). This equation is transformed into a canonical equation, which reduces the number of classes of positive solutions from 4 to 2. This is done without assuming extra conditions, and greatly simplifies the process of obtaining conditions for the existence of Kneser-type solutions. Also we obtain lower and upper bounds for these solutions, and obtain their rate of convergence to zero. Two examples are provided to illustrate our main results, one with a delay neutral term, and one with an advanced neutral term. For more information see https: //ejde. math. txstate. edu/Volumes/2024/55/abstr. html