# Positive Solutions for Third-Orderp-Laplacian Functional Dynamic Equations on Time Scales

Authors: Wen Guan; Da-Bin Wang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study the following third-orderp-Laplacian functional dynamic equation on time scales: Φ
p
(
u
Δ
∇
(
t
)
)
∇
+
a
(
t
)
f
(
u
(
t
)
,
u
(
μ
(
t
)
)
)
=
0, t
∈
0
,
T
T,  u
(
t
)
=
φ
(
t
),  t
∈
-
r
,
0
T,  u
Δ
(
0
)
=
u
Δ
∇
(
T
)
=
0, and u
(
T
)
+
B
0
(
u
Δ
(
η
)
)
=
0. By applying the Five-Functional Fixed Point Theorem, the existence criteria of three positive solutions are established.


---

## Body

## 1. Introduction

Recently, much attention has been paid to the existence of positive solutions for the boundary value problems withp-Laplacian operator on time scales; for example, see [1–22] and the references therein. But, to the best of our knowledge, there is not much concerning p-Laplacian functional dynamic equations on time scales [6, 12–14, 19, 21, 22], especially for the third-order p-Laplacian functional dynamic equations on time scales [14, 22].In [14], Song and Gao were concerned with the existence of positive solutions for the p-Laplacian functional dynamic equation on time scales:
(1)
[
Φ
p
(
u
Δ
∇
(
t
)
)
]
∇
+
a
(
t
)
f
(
u
(
t
)
,
u
(
μ
(
t
)
)
)
=
0
,
t
∈
(
0
,
T
)
T
,
u
(
t
)
=
φ
(
t
)
,
t
∈
[
-
r
,
0
]
T
,
u
Δ
(
0
)
=
u
Δ
∇
(
T
)
=
0
,
u
(
T
)
+
B
0
(
u
Δ
(
η
)
)
=
0
,
where η
∈
(
0
,
ρ
(
T
)
)
T and Φ
p
(
s
) is p-Laplacian operator; that is, Φ
p
(
s
)
=
|
s
|
p
-
2
s, p
>
1, (
Φ
p
)
-
1
=
Φ
q, 1
/
p
+
1
/
q
=
1, and(
C
1
)
f
:
(
R
+
)
2
→
R
+ is continuous;
(
C
2
)
a
:
T
→
R
+ is left dense continuous (i.e., a
∈
C
l
d
(
T
,
R
+
)) and does not vanish identically on any closed subinterval of [
0
,
T
], where C
l
d
(
T
,
R
+
) denotes the set of all left dense continuous functions from T to R
+;
(
C
3
)
φ
:
[
-
r
,
0
]
T
→
R
+ is continuous and r
>
0;
(
C
4
)
μ
:
[
0
,
T
]
T
→
[
-
r
,
T
]
T is continuous, μ
(
t
)
≤
0 for all t;
(
C
5
)
B
0
:
R
→
R is continuous and satisfies the condition that there are A
≥
B
≥
0 such that
(2)
B
v
≤
B
0
(
v
)
≤
A
v
,
∀
R
.The existence of two positive solutions to problem (1) was obtained by using a double fixed point theorem due to Avery et al. [23] in a cone.In [22], Wang and Guan considered the existence of positive solutions to problem (1) by applying the well-known Leggett-Williams Fixed Point Theorem.Motivated by [14, 22], we will show that problem (1) has at least three positive solutions by means of the Five-Functional Fixed Point Theorem [24] (which is a generalization of the Leggett-Williams Fixed Point Theorem [25]). It is worth noting that the Five-Functional Fixed Point Theorem is used extensively in yielding three solutions for BVPs of differential equations, difference equations, and/or dynamic equations on time scales; see [6, 26, 27] and references therein.Throughout this work we assume knowledge of time scales and time-scale notation, first introduced by Hilger [28]. For more on time scales, please see the texts by Bohner and Peterson [29, 30].In the remainder of this section, we state the following theorem, which is crucial to our proof.Letγ, β, θ be nonnegative, continuous, and convex functionals on P and let α, ψ be nonnegative, continuous, and concave functionals on P. Then, for nonnegative real numbers h, a, b, d, and c, we define the convex sets
(3)
P
(
γ
,
c
)
=
{
x
∈
P
:
γ
(
x
)
<
c
}
,
P
(
γ
,
α
,
a
,
c
)
=
{
x
∈
P
:
a
≤
α
(
x
)
,
γ
(
x
)
≤
c
}
,
Q
(
γ
,
β
,
d
,
c
)
=
{
x
∈
P
:
β
(
x
)
≤
d
,
γ
(
x
)
≤
c
}
,
P
(
γ
,
θ
,
α
,
a
,
b
,
c
)
=
{
x
∈
P
:
a
≤
α
(
x
)
,
θ
(
x
)
≤
b
,
γ
(
x
)
≤
c
}
,
Q
(
γ
,
β
,
ψ
,
h
,
d
,
c
)
=
{
x
∈
P
:
h
≤
ψ
(
x
)
,
β
(
x
)
≤
d
,
γ
(
x
)
≤
c
}
.Theorem 1 (see [24]).
LetP be a cone in a real Banach space E. Suppose there exist positive numbers c and M; nonnegative, continuous, and concave functionals α and ψ on P; and nonnegative, continuous, and convex functionals γ, β, and θ on P, with
(4)
α
(
x
)
≤
β
(
x
)
,
∥
x
∥
≤
M
γ
(
x
)
for all x
∈
P
(
γ
,
c
)
¯. Suppose
(5)
F
:
P
(
γ
,
c
)
¯
⟶
P
(
γ
,
c
)
¯
is completely continuous and there exist nonnegative numbers h, a, k, b, with 0
<
a
<
b such that(i)
{
x
∈
P
(
γ
,
θ
,
α
,
b
,
k
,
c
)
:
α
(
x
)
>
b
}
≠
⌀ and α
(
F
x
)
>
b for x
∈
P
(
γ
,
θ
,
α
,
b
,
k
,
c
);
(ii)
{
x
∈
Q
(
γ
,
β
,
ψ
,
h
,
a
,
c
)
:
β
(
x
)
<
a
}
≠
⌀ and β
(
F
x
)
<
a for x
∈
Q
(
γ
,
β
,
ψ
,
h
,
a
,
c
);
(iii)
α
(
F
x
)
>
b for x
∈
P
(
γ
,
α
,
b
,
c
) with θ
(
F
x
)
>
k;
(iv)
β
(
F
x
)
<
a for x
∈
Q
(
γ
,
β
,
a
,
c
) with ψ
(
F
x
)
<
h.
ThenF has at least three fixed points x
1
,
x
2
,
x
3
∈
P
(
γ
,
c
)
¯ such that
(6)
β
(
x
1
)
<
a
,
b
<
α
(
x
2
)
,
a
<
β
(
x
3
)
w
i
t
h
α
(
x
3
)
<
b
.


## 2. Existence of Three Positive Solutions

We note thatu
(
t
) is a solution of BVP (1) if and only if
(7)
u
(
t
)
=
{
∫
0
T
‍
(
T
-
s
)
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
i
-
B
0
(
∫
0
η
‍
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
)
i
+
∫
0
t
‍

(
t
-
s
)
Φ
q
(

∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r

)

∇
s
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
t
∈
[
0
,
T
]
T
,
φ
(
t
)
,
kkkkkkkkkkkkkkkkkikkkkkkkkkk
t
∈
[
-
r
,
0
]
T
.LetE
=
C
l
d
(
[
0
,
T
]
T
,
R
) be endowed with ∥
u
∥
=
sup
⁡
t
∈
[
0
,
T
]
T
⁡
|
u
(
t
)
|, so E is a Banach space. Define cone P
⊂
E by
(8)
P
=
{
u
∈
E
:
u
is
concave
and
,
u
Δ
(
0
)
=
0
}
o
ki
k
nonnegative
valued
on
⁡
[
0
,
T
]
T
,
u
Δ
(
0
)
=
0
}
.For eachu
∈
E, extend u
(
t
) to [
-
r
,
T
]
T with u
(
t
)
=
φ
(
t
) for t
∈
[
-
r
,
0
]
T.DefineF
:
P
→
E by
(9)
(
F
u
)
(
t
)
=
∫
0
T
‍
(
T
-
s
)
Φ
q
k
k
K
K
k
×
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
-
B
0
(
∫
0
η
‍
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
)
+
∫
0
t
‍
(
t
-
s
)
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
t
∈
[
0
,
T
]
T
.
We seek a point, u
1, of F in the cone P. Define
(10)
u
(
t
)
=
{
u
1
(
t
)
,
t
∈
[
0
,
T
]
T
,
φ
(
t
)
,
t
∈
[
-
r
,
0
]
T
.Thenu
(
t
) denotes a positive solution of BVP (1).We have the following results.Lemma 2.
Letu
∈
P, and then(1)
F
:
P
→
P is completely continuous;
(2)
u
(
t
)
≥
(
(
T
-
t
)
/
T
)
∥
u
∥ for t
∈
[
0
,
T
]
T;
(3)
u
(
t
) is decreasing [
0
,
T
]
T;
(4)
(
T
-
ς
)
u
(
τ
)
≤
(
T
-
τ
)
u
(
ς
) for 0
<
τ
<
ς
<
T and τ
,
ς
∈
T.Proof.
(1)–(3) are Lemma 3.1 of [14]. It is easy to conclude that (4) is satisfied by the concavity of u.
Letl
∈
T be fixed such that 0
<
l
<
η
<
T, and set
(11)
Y
1
=
{
t
∈
[
0
,
T
]
T
:
μ
(
t
)
<
0
}
;
Y
2
=
{
t
∈
[
0
,
T
]
T
:
μ
(
t
)
≥
0
}
;
Y
3
=
Y
1
∩
[
0
,
l
]
T
.
Throughout this paper, we assumeY
3
≠
⌀ and ∫
Y
3
‍
a
(
r
)
∇
r
>
0.
We define the nonnegative, continuous, and concave functionalsα, ψ and the nonnegative, continuous, and convex functionals β, θ, γ on the cone P, respectively, as
(12)
γ
(
u
)
=
θ
(
u
)
=
max
⁡
t
∈
[
η
,
T
]
T
u
(
t
)
=
u
(
η
)
,
α
(
u
)
=
min
⁡
t
∈
[
0
,
l
]
T
u
(
t
)
=
u
(
l
)
,
β
(
u
)
=
max
⁡
t
∈
[
l
,
T
]
T
u
(
t
)
=
u
(
l
)
,
ψ
(
u
)
=
min
⁡
t
∈
[
0
,
η
]
T
u
(
t
)
=
u
(
η
)
.
We observe thatα
(
u
)
=
β
(
u
) for each u
∈
P.
In addition, by Lemma2, we have γ
(
u
)
=
u
(
η
)
≥
(
(
T
-
η
)
/
T
)
∥
u
∥. Hence ∥
u
∥
≤
(
T
/
(
T
-
η
)
)
γ
(
u
) for all u
∈
P.
For convenience, we define(13)
μ
=
T
(
T
+
η
+
B
)
Φ
q
(
∫
0
T
‍
a
(
r
)
∇
r
)
,
δ
=
A
∫
Y
3
‍
Φ
q
(
∫
0
s
‍
a
(
r
)
∇
r
)
∇
s
,
λ
=
T
(
T
+
l
+
B
)
Φ
q
(
∫
0
T
‍
a
(
r
)
∇
r
)
.We now state growth conditions onf so that BVP (1) has at least three positive solutions.Theorem 3.
Let0
<
a
<
(
(
T
-
l
)
/
T
)
b
<
(
(
T
-
η
)
(
T
-
l
)
/
T
2
)
c, μ
b
<
δ
c, and suppose that f satisfies the following conditions: (
H
1
)
f
(
u
,
φ
(
s
)
)
<
Φ
p
(
c
/
μ
), if 0
≤
u
≤
(
T
/
(
T
-
η
)
)
c, uniformly in s
∈
[
-
r
,
0
]
T, and f
(
u
1
,
u
2
)
<
Φ
p
(
c
/
μ
), if 0
≤
u
i
≤
(
T
/
(
T
-
η
)
)
c, i
=
1,2;
(
H
2
)
f
(
u
,
φ
(
s
)
)
>
Φ
p
(
b
/
δ
), if b
≤
u
≤
(
T
/
(
T
-
η
)
)
2
b, uniformly in s
∈
[
-
r
,
0
]
T;
(
H
3
)
f
(
u
,
φ
(
s
)
)
<
Φ
p
(
a
/
λ
), if 0
≤
u
≤
(
T
/
(
T
-
l
)
)
a, uniformly in s
∈
[
-
r
,
0
]
T, and f
(
u
1
,
u
2
)
<
Φ
p
(
a
/
λ
), if 0
≤
u
i
≤
(
T
/
(
T
-
l
)
)
a, i
=
1,2.
Then BVP (1) has at least three positive solutions of the form
(14)
u
(
t
)
=
{
u
i
(
t
)
,
t
∈
[
0
,
T
]
T
,
i
=
1,2
,
3
,
φ
(
t
)
,
t
∈
[
-
r
,
0
]
T
,
where max
⁡
t
∈
[
l
,
T
]
T
⁡
u
1
(
t
)
<
a, min
⁡
t
∈
[
0
,
l
]
T
⁡
u
2
(
t
)
>
b, and a
<
max
⁡
t
∈
[
l
,
T
]
T
⁡
u
3
(
t
) with min
⁡
t
∈
[
0
,
l
]
T
⁡
u
3
(
t
)
<
b.Proof.
Letu
∈
P
(
γ
,
c
)
¯, and then γ
(
u
)
=
max
⁡
t
∈
[
η
,
T
]
T
⁡
u
(
t
)
=
u
(
η
)
≤
c, and consequently, 0
≤
u
(
t
)
≤
c for t
∈
[
η
,
T
]
T. Since u
(
η
)
≥
(
(
T
-
η
)
/
T
)
u
(
0
), so ∥
u
∥
=
u
(
0
)
≤
(
T
/
(
T
-
η
)
)
u
(
η
)
≤
(
T
/
(
T
-
η
)
)
c, and this implies
(15)
0
≤
u
(
t
)
≤
T
T
-
η
c
,
for
t
∈
[
0
,
T
]
T
.
From(
H
1
), we have
(16)
γ
(
F
u
)
=
(
F
u
)
(
η
)
=
∫
0
T
‍
(
T
-
s
)
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
-
B
0
(
∫
0
η
‍
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
)
+
∫
0
η
‍
(
η
-
s
)
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
≤
∫
0
T
‍
T
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
+
B
∫
0
T
‍
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
+
∫
0
η
‍
T
Φ
q
(
∫
0
T
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
=
T
(
T
+
η
+
B
)
Φ
q
[
∫
Y
1
‍
a
(
r
)
f
(
u
(
r
)
,
φ
(
μ
(
r
)
)
)
∇
r
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
i
i
i
+
∫
Y
2
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
]
<
T
(
T
+
η
+
B
)
Φ
q
(
∫
0
T
‍
a
(
r
)
∇
r
)
c
μ
=
c
.
Therefore(17)
F
u
∈
P
(
γ
,
c
)
¯
.
We now turn to property (i) of Theorem1. Choosing u
≡
(
T
/
(
T
-
η
)
)
b, k
=
(
T
/
(
T
-
η
)
)
b, it follows that
(18)
α
(
u
)
=
u
(
l
)
=
T
T
-
η
b
>
b
,
θ
(
u
)
=
u
(
η
)
=
T
T
-
η
b
=
k
,
γ
(
u
)
=
u
(
η
)
=
T
T
-
η
b
<
c
,
which shows that {
u
∈
P
(
γ
,
θ
,
α
,
b
,
k
,
c
)
:
α
(
u
)
>
b
}
≠
⌀, and, for u
∈
P
(
γ
,
θ
,
α
,
b
,
(
T
/
(
T
-
η
)
)
b
,
c
), we have
(19)
b
≤
u
(
t
)
≤
(
T
T
-
η
)
2
b
,
for
t
∈
[
0
,
l
]
T
.
From(
H
2
), we have
(20)
α
(
F
u
)
=
(
F
u
)
(
l
)
=
∫
0
T
‍
(
T
-
s
)
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
-
B
0
(
∫
0
η
‍
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
)
+
∫
0
l
‍
(
t
-
s
)
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
≥
-
B
0
(
∫
0
η
‍
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
)
≥
A
∫
0
η
‍
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
≥
A
∫
0
l
‍
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
≥
A
∫
Y
3
‍
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
φ
(
μ
(
r
)
)
)
∇
r
)
∇
s
>
A
∫
Y
3
‍
Φ
q
(
∫
0
s
‍
a
(
r
)
∇
r
)
∇
s
b
δ
=
b
.
We conclude that (i) of Theorem1 is satisfied.
We next address (ii) of Theorem1. If we take u
≡
(
(
T
-
η
)
/
T
)
a, h
=
(
(
T
-
η
)
/
T
)
a, then
(21)
γ
(
u
)
=
u
(
η
)
=
T
-
η
T
a
<
c
,
ψ
(
u
)
=
u
(
η
)
=
T
-
η
T
a
=
h
,
β
(
u
)
=
u
(
l
)
=
T
-
η
T
a
<
a
.
From this we know that{
u
∈
Q
(
γ
,
β
,
ψ
,
h
,
a
,
c
)
:
β
(
u
)
<
a
}
≠
⌀. If u
∈
Q
(
γ
,
β
,
ψ
,
(
(
T
-
η
)
/
T
)
a
,
a
,
c
), then
(22)
0
≤
u
(
t
)
≤
T
T
-
l
a
,
for
t
∈
[
0
,
T
]
T
.
From(
H
3
), we have
(23)
β
(
F
u
)
=
(
F
u
)
(
l
)
=
∫
0
T
‍
(
T
-
s
)
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
-
B
0
(
∫
0
η
‍
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
)
+
∫
0
l
‍
(
t
-
s
)
Φ
q
(
∫
0
s
‍
-
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
≤
∫
0
T
‍
T
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
+
B
∫
0
T
‍
Φ
q
(
∫
0
s
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
+
∫
0
l
‍
T
Φ
q
(
∫
0
T
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
)
∇
s
=
T
(
T
+
l
+
B
)
Φ
q
[
∫
Y
1
‍
a
(
r
)
f
(
u
(
r
)
,
φ
(
μ
(
r
)
)
)
∇
r
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
i
i
i
+
∫
Y
2
‍
a
(
r
)
f
(
u
(
r
)
,
u
(
μ
(
r
)
)
)
∇
r
]
<
T
(
T
+
l
+
B
)
Φ
q
(
∫
0
T
‍
a
(
r
)
∇
r
)
a
λ
=
a
.
Now we show that (iii) of Theorem1 is satisfied. If u
∈
P
(
γ
,
α
,
b
,
c
) and θ
(
F
u
)
=
F
u
(
η
)
>
(
T
/
(
T
-
η
)
)
b, then
(24)
α
(
F
u
)
≥
(
F
u
)
(
l
)
=
T
-
l
T
F
u
(
l
)
≥
T
-
l
T
F
u
(
η
)
>
T
-
l
T
-
η
b
>
b
.
Finally, ifu
∈
Q
(
γ
,
β
,
a
,
c
) and ψ
(
F
u
)
=
F
u
(
η
)
<
(
(
T
-
η
)
/
T
)
a, then from (4) of Lemma 2 we have
(25)
β
(
F
u
)
=
F
u
(
l
)
≤
T
T
-
l
F
u
(
l
)
≤
T
T
-
η
F
u
(
η
)
<
a
,
which shows that condition (iv) of Theorem 1 is fulfilled.
Thus, all the conditions of Theorem1 are satisfied. Hence, F has at least three fixed points u
1, u
2, u
3 satisfying
(26)
β
(
u
1
)
<
a
,
b
<
α
(
u
2
)
,
a
<
β
(
u
3
)
with
α
(
u
3
)
<
b
.
Let(27)
u
(
t
)
=
{
u
i
(
t
)
,
t
∈
[
0
,
T
]
T
,
i
=
1,2
,
3
,
φ
(
t
)
,
t
∈
[
-
r
,
0
]
T
,
which are three positive solutions of BVP (1).



---

*Source: 676052-2014-08-04.xml*