Existence and nonexistence of eventually positive solutions for nonlinear neutral differential equations (Q1888527)

The authors consider the existence and nonexistence of eventually positive solutions to nonlinear neutral differential equations of the form \[ {[x(t)-p(t) x(t-\tau)]'}^\alpha+q(t)\left(\prod_{i=1}^m x^{\alpha_i}(t-\sigma_i)\right)^\alpha=0. \] By employing a Riccati technique, necessary and sufficient conditions are obtained. The presented results are also applied to discuss the existence and nonexistence of eventually positive solutions to a class of impulsive neutral differential equations.