Lyapunov type inequalities for the Riemann-Liouville fractional differential equations of higher order

In this paper, some new Lyapunov type inequalities will be presented for Riemann-Liouville fractional differential equations of the form (Daαx)(t)+p(t)|x(t)|μ−1x(t)+q(t)|x(t)|γ−1(t)x(t)=f(t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bigl(D^{\alpha}_{a}x\bigr) (t)+p(t)\big| x(t)\big|^{\mu-1}x (t)+q(t)\big| x(t)\big|^{\gamma -1}(t)x(t)=f(t), $$\end{document} where α∈(n−1,n]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha\in(n-1, n]$\end{document} (n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\geq3$\end{document}), p, q, f are real-valued functions and 0<γ<1<μ<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\gamma<1<\mu<n$\end{document}.