Sufficient and necessary conditions for oscillation of linear fractional-order delay differential equations

Abstract

This paper studies the linear fractional-order delay differential equation *D−αCx(t)−px(t−τ)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ {}^{C}D^{\\alpha }_{-}x(t)-px(t-\\tau )= 0, $$\\end{document} where 0<alpha =frac{text{odd integer}}{text{odd integer}}<1, p, tau >0, {}^{C}D_{-}^{alpha }x(t)=-Gamma ^{-1}(1-alpha )int _{t}^{infty }(s-t)^{- alpha }x'(s),ds. We obtain the conclusion that p1/ατ>α/e\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ p^{1/\\alpha } \\tau >\\alpha /e $$\\end{document} is a sufficient and necessary condition of the oscillations for all solutions of Eq. (*). At the same time, some sufficient conditions are obtained for the oscillations of multiple delays linear fractional differential equation. Several examples are given to illustrate our theorems.