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