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.
Qualitative analysis of nonlinear implicit neutral differential equation of fractional order
