AbstractIn this work we investigate the following fractional p-Laplacian differential equation with Sturm–Liouville boundary value conditions: $$ \textstyle\begin{cases} {}_{t}D_{T}^{\alpha } ( { \frac{1}{{{{ ( {h ( t )} )}^{p - 2}}}}{\phi _{p}} ( {h ( t ){}_{0}^{C}D_{t}^{\alpha }u ( t )} )} ) + a ( t ){\phi _{p}} ( {u ( t )} ) = \lambda f (t,u(t) ),\quad \mbox{a.e. }t \in [ {0,T} ], \\ {\alpha _{1}} {\phi _{p}} ( {u ( 0 )} ) - { \alpha _{2}} {}_{t}D_{T}^{\alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( 0 )} )} ) = 0, \\ {\beta _{1}} { \phi _{p}} ( {u ( T )} ) + {\beta _{2}} {}_{t}D_{T}^{ \alpha - 1} ( {{\phi _{p}} ( {{}_{0}^{C}D_{t}^{\alpha }u ( T )} )} ) = 0, \end{cases} $$
{
D
T
α
t
(
1
(
h
(
t
)
)
p
−
2
ϕ
p
(
h
(
t
)
0
C
D
t
α
u
(
t
)
)
)
+
a
(
t
)
ϕ
p
(
u
(
t
)
)
=
λ
f
(
t
,
u
(
t
)
)
,
a.e.
t
∈
[
0
,
T
]
,
α
1
ϕ
p
(
u
(
0
)
)
−
α
2
t
D
T
α
−
1
(
ϕ
p
(
0
C
D
t
α
u
(
0
)
)
)
=
0
,
β
1
ϕ
p
(
u
(
T
)
)
+
β
2
t
D
T
α
−
1
(
ϕ
p
(
0
C
D
t
α
u
(
T
)
)
)
=
0
,
where ${}_{0}^{C}D_{t}^{\alpha }$
D
t
α
0
C
, ${}_{t}D_{T}^{\alpha }$
D
T
α
t
are the left Caputo and right Riemann–Liouville fractional derivatives of order $\alpha \in ( {\frac{1}{2},1} ]$
α
∈
(
1
2
,
1
]
, respectively. By using variational methods and critical point theory, some new results on the multiplicity of solutions are obtained.
ON WELL POSED IMPULSIVE BOUNDARY VALUE PROBLEMS FOR P(T)-LAPLACIAN'S
