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

Advances in Difference Equations

Zhen Jin¹  Guirong Liu²  Qiong Meng² 

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

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

DOI  :  10.1186/s13662-021-03223-0

来源: DOAJ

【 摘 要 】

Abstract This paper studies the linear fractional-order delay differential equation * D − α C x ( t ) − p x ( t − τ ) = 0 , $$ {}^{C}D^{\alpha }_{-}x(t)-px(t-\tau )= 0, $$ where 0 < α = odd integer odd integer < 1 $0<\alpha =\frac{\text{odd integer}}{\text{odd integer}}<1$ , p , τ > 0 $p, \tau >0$ , D − α C x ( t ) = − Γ − 1 ( 1 − α ) ∫ t ∞ ( s − t ) − α x ′ ( s ) d s ${}^{C}D_{-}^{\alpha }x(t)=-\Gamma ^{-1}(1-\alpha )\int _{t}^{\infty }(s-t)^{- \alpha }x'(s)\,ds$ . We obtain the conclusion that p 1 / α τ > α / e $$ p^{1/\alpha } \tau >\alpha /e $$ 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】

Unknown