In this paper, we investigate the following Kirchhoff type problem involving the fractional
p
x
-Laplacian operator.
a
−
b
∫
Ω
×
Ω
u
x
−
u
y
p
x
,
y
/
p
x
,
y
x
−
y
N
+
s
p
x
,
y
d
x
d
y
L
u
=
λ
u
q
x
−
2
u
+
f
x
,
u
x
∈
Ω
u
=
0
x
∈
∂
Ω
,
, where
Ω
is a bounded domain in
ℝ
N
with Lipschitz boundary,
a
≥
b
>
0
are constants,
p
x
,
y
is a function defined on
Ω
¯
×
Ω
¯
,
s
∈
0
,
1
, and
q
x
>
1
,
L
u
is the fractional
p
x
-Laplacian operator,
N
>
s
p
x
,
y
, for any
x
,
y
∈
Ω
¯
×
Ω
¯
,
p
x
∗
=
p
x
,
x
N
/
N
−
s
p
x
,
x
,
λ
is a given positive parameter, and
f
is a continuous function. By using Ekeland’s variational principle and dual fountain theorem, we obtain some new existence and multiplicity of negative energy solutions for the above problem without the Ambrosetti-Rabinowitz ((AR) for short) condition.