Asymptotic performance of nonoscillatory solutions of functional differential equations involving a delayed damping term

This work investigated the asymptotic performance of nonoscillatory solutions to the functional differential equation (FDE) ${\left(\alpha \left(u\right){\left|y^{\mathrm{\prime }}\left(u\right)\right|}^{\kappa -1}{y}^{\mathrm{\prime }}\left(u\right)\right)}^{\mathrm{\prime }}+p\left(u\right)F\left(y^{\mathrm{\prime }}\left(\delta \left(u\right)\right)\right)+q\left(u\right)G\left(y\left(\rho \left(u\right)\right)\right)=0$, which involves a delayed damping term. Using Riccati and comparison methods, we extended the previous results to the nonlinear case of the considered equation. Furthermore, the new criteria improved upon the previous ones by removing some constraints on the delay functions. Then, for the linear case, we derived new criteria that take into account all parameters of the equation. The examples and comparisons provided illustrate the importance and novelty of our results.