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

Advances in Difference Equations

article

Meng, Qiong¹  Jin, Zhen²  Liu, Guirong¹ 

[1] School of Mathematical Sciences, Shanxi University;Complex Systems Research Center, Shanxi University

关键词: Oscillation;    Fractional differential equation;    Autonomous;    Delay;    Linear;   

DOI  :  10.1186/s13662-021-03223-0

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

This paper studies the linear fractional-order delay differential equation* $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ where$0 0$ ,${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ . We obtain the conclusion that$$ p^{1/\alpha } \tau >\alpha /e $$ 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.

【 授权许可】

CC BY