Lyapunov type inequalities for even order differential equations with mixed nonlinearities

In the case of oscillatory potentials, we present Lyapunov and Hartman type inequalities for even order differential equations with mixed nonlinearities: x(2n)(t)+(−1)n−1∑i=1mqi(t)|x(t)|αi−1x(t)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x^{(2n)}(t)+(-1)^{n-1}\sum_{i=1}^{m}q_{i}(t)|x(t)|^{\alpha_{i}-1}x(t)=0$\end{document}, where n,m∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n,m\in{\mathbb{N}}$\end{document} and the nonlinearities satisfy 0<α1<⋯<αj<1<αj+1<⋯<αm<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\alpha_{1}<\cdots<\alpha_{j}<1<\alpha_{j+1}<\cdots<\alpha_{m}<2$\end{document}.