Properties of Kneser’s solution for half-linear third order neutral differential equations

We examine properties of positive solutions of the third order differential equation with advanced argument in neutral term of the form $$(a(t)(b(t)(z'(t))^{\alpha})')'+q(t)y^{\alpha}(t)=0,$$ where $${z(t)=y(t)+p(t)y(\sigma(t))}$$ . They show that the positive solutions are in fact Kneser type solutions and then provide lower and upper estimate that yield the rate of convergence to zero for such solutions. An example is provided to illustrate the importance of the main result.