Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 867615, 17 pages
doi:10.1155/2011/867615
Research Article
Multiple Positive Solutions for Second-Order
p-Laplacian Dynamic Equations with Integral
Boundary Conditions
Yongkun Li and Tianwei Zhang
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 13 July 2010; Revised 21 November 2010; Accepted 25 November 2010
Academic Editor: Gennaro Infante
Copyright q 2011 Y. Li and T. Zhang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We are concerned with the following second-order p-Laplacian dynamic equations on time scales
ϕp xΔ t∇ λft, xt, xΔ t 0, t ∈ 0, T T , with integral boundary conditions xΔ 0 0,
T
αxT − βx0 0 gsxs∇s. By using Legget-Williams fixed point theorem, some criteria for the
existence of at least three positive solutions are established. An example is presented to illustrate
the main result.
1. Introduction
Boundary value problems with p-Laplacian have received a lot of attention in recent years.
They often occur in the study of the n-dimensional p-Laplacian equation, non-Newtonian
fluid theory, and the turbulent flow of gas in porous medium 1–7. Many works have been
carried out to discuss the existence of solutions or positive solutions and multiple solutions
for the local or nonlocal boundary value problems.
On the other hand, the study of dynamic equations on time scales goes back to
its founder Stefan Hilger 8 and is a new area of still fairly theoretical exploration in
mathematics. Motivating the subject is the notion that dynamic equations on time scales can
build bridges between continuous and discrete equations. Further, the study of time scales
has led to several important applications, for example, in the study of insect population
models, neural networks, heat transfer, and epidemic models, we refer to 8–10. In addition,
the study of BVPs on time scales has received a lot of attention in the literature, with the
pioneering existence results to be found in 11–16.
However, existence results are not available for dynamic equations on time scales with
integral boundary conditions. Motivated by above, we aim at studying the second-order
2
Boundary Value Problems
p-Laplacian dynamic equations on time scales in the form of
∇
ϕp xΔ t
λf t, xt, xΔ t 0,
t ∈ 0, T T
1.1
with integral boundary condition
xΔ 0 0,
αxT − βx0 T
gsxs∇s,
1.2
0
where λ is positive parameter, ϕp s |s|p−2 s for p > 1 with ϕ−1
p ϕq and 1/p 1/q 1,
Δ is the delta derivative, ∇ is the nabla derivative, T is a time scale which is a nonempty
closed subset of R with the topology and ordering inherited from R, 0 and T are points in T,
an interval 0, T T : 0, T ∩ T, f ∈ C0, T T × R2 , 0, ∞ with ft, 0, 0 / 0 for all t ∈ 0, T T ,
T
g ∈ Cld 0, T T , 0, ∞, α, β > 0 with α − g0 > β, and where g0 0 gs∇s.
The main purpose of this paper is to establish some sufficient conditions for the
existence of at least three positive solutions for BVPs 1.1-1.2 by using Legget-Williams
fixed point theorem. This paper is organized as follows. In Section 2, some useful lemmas
are established. In Section 3, by using Legget-Williams fixed point theorem, we establish
sufficient conditions for the existence of at least three positive solutions for BVPs 1.1-1.2.
An illustrative example is given in Section 4.
2. Preliminaries
In this section, we will first recall some basic definitions and lemmas which are used in what
follows.
Definition 2.1 see 8. A time scale T is an arbitrary nonempty closed subset of the real set R
with the topology and ordering inherited from R. The forward and backward jump operators
σ, ρ : T → T and the graininess μ, ν : T → R are defined, respectively, by
σt : inf{s ∈ T : s > t},
ρt : sup{s ∈ T : s < t},
μt : σt − t,
νt : t − ρt.
2.1
The point t ∈ T is called left-dense, left-scattered, right-dense, or right-scattered if ρt t,
ρt < t, and σt t or σt > t, respectively. Points that are right-dense and left-dense
at the same time are called dense. If T has a left-scattered maximum m1 , defined Tκ T −
{m1 }; otherwise, set Tκ T. If T has a right-scattered minimum m2 , defined Tκ T − {m2 };
otherwise, set Tκ T.
Definition 2.2 see 8. For f : T → R and t ∈ Tκ , then the delta derivative of f at the point
t is defined to be the number f Δ t provided it exists with the property that for each > 0,
there is a neighborhood U of t such that
fσt − fs − f Δ tσt − s ≤ |σt − s|
∀s ∈ U.
2.2
Boundary Value Problems
3
For f : T → R and t ∈ Tκ , then the nabla derivative of f at the point t is defined to
be the number f ∇ t provided it exists with the property that for each > 0, there is a
neighborhood U of t such that
f ρt − fs − f ∇ t ρt − s ≤ ρt − s
∀s ∈ U.
2.3
Definition 2.3 see 8. A function f is rd-continuous provided it is continuous at each rightdense point in T and has a left-sided limit at each left-dense point in T. The set of rdcontinuous functions f will be denoted by Crd T. A function g is left-dense continuous i.e.,
ld-continuous if g is continuous at each left-dense point in T and its right-sided limit exists
finite at each right-dense point in T. The set of left-dense continuous functions g will be
denoted by Cld T.
Definition 2.4 see 8. If F Δ t ft, then we define the delta integral by
b
fsΔs Fb − Fa.
2.4
a
If G∇ t gt, then we define the nabla integral by
b
gs∇s Gb − Ga.
2.5
a
Lemma 2.5 see 8. If f ∈ Crd T and t ∈ Tκ , then
σt
fsΔs μtft.
2.6
gs∇s νtgt.
2.7
t
If g ∈ Cld T and t ∈ Tκ , then
t
ρt
Let the Banach space
1
B Cld
0, T T x : 0, T T −→ R | x is Δ-differentiable on 0, T T , and xΔ is ld-continuous on0, T T
2.8
be endowed with the norm x max{x0 , xΔ 0 }, where
x0 sup |xt|,
t∈0,T T
Δ
x sup xΔ t
0
t∈0,T T
2.9
4
Boundary Value Problems
and choose a cone P ⊂ B defined by
⎧
⎫
Δ
Δ∇
⎪
⎨x ∈ B : xt ≥ 0, x t ≤ 0, x t ≤ 0 ∀t ∈ 0, T T ,⎪
⎬
T
.
P
⎪
⎪
⎩
⎭
αxT − βx0 gsxs∇s
2.10
0
Lemma 2.6. If x ∈ P, then xt ≥ β/α − g0 x0 for all t ∈ 0, T T .
Proof. If x ∈ P, then xΔ ≤ 0. It follows that
x0 x0 max xt.
xT min xt,
t∈0,T T
With αxT − βx0 T
αxT βx0 0
t∈0,T T
2.11
gsxs∇s and xΔ ≤ 0, one obtains
T
T
gsxs∇s ≥ βx0 0
gs∇sxT βx0 g0 xT .
2.12
0
Therefore,
xT ≥
β
β
x0 x0 .
α − g0
α − g0
2.13
From 2.11–2.13, we can get that
β
β
x0 x0 .
α − g0
α − g0
xt ≥ min xt xT ≥
t∈0,T T
2.14
So Lemma 2.6 is proved.
Lemma 2.7. x ∈ B is a solution of BVPs 1.1-1.2 if and only if x ∈ B is a solution of the following
integral equation:
xt T
Θ β V s ϕq
0
T
s
ϕq
t
s
Δ
λf r, xr, x r ∇r Δs
0
Δ
2.15
λf r, xr, x r ∇r Δs,
0
where
1
T
Θ
1
,
α − β − g0
α − β − 0 gs∇s
t
V t gs∇s ∀t ∈ 0, TT .
0
2.16
Boundary Value Problems
5
Proof. First assume x ∈ B is a solution of BVPs 1.1-1.2; then we have
t t ϕp xΔ t ϕp xΔ 0 − λf s, xs, xΔ s ∇s − λf s, xs, xΔ s ∇s.
0
2.17
0
That is,
Δ
x t −ϕq
t
Δ
λf s, xs, x s ∇s
−Ht.
2.18
0
Integrating 2.18 from t to T , it follows that
xt xT T
2.19
HsΔs.
t
Together with 2.19 and αxT − βx0 αxT − β xT T
0
gsxs∇s, we obtain
T
HsΔs
0
T
gs xT 0
T
HrΔr ∇s.
2.20
s
Thus,
α−β−
T
gs∇s xT β
0
T
HsΔs 0
β
β
T
HsΔs T
T
HsΔs −
HrΔr ∇s
s
T T
0
0
β
gs
0
0
T
T
T
∇
V s − V rHrΔr
s
∇s
2.21
V 0 − V sHsΔs
0
HsΔs 0
T
V sHsΔs,
0
namely,
xT βΘ
T
0
HsΔs Θ
T
0
V sHsΔs.
2.22
6
Boundary Value Problems
Substituting 2.22 into 2.19, we obtain
T
xt βΘ
HsΔs Θ
0
T
T
V sHsΔs HsΔs
0
Θ β V s ϕq
0
T
T
λf r, xr, xΔ r ∇r Δs
2.23
0
s
Δ
λf r, xr, x r ∇r Δs.
ϕq
t
s
t
0
The proof of sufficiency is complete.
Conversely, assume x ∈ B is a solution of the following integral equation:
xt T
Θ β V s ϕq
0
T
s
ϕq
t
T
s
λf r, xr, xΔ r ∇r Δs
0
λf r, xr, xΔ r ∇r Δs
0
Θ β V s HsΔs 2.24
T
HsΔs.
0
t
It follows that
Δ
x t −ϕq
t
Δ
λf s, xs, x s ∇s
−Ht,
0
ϕp
2.25
∇
xΔ t
λf t, xt, xΔ t 0.
So xΔ 0 0. Furthermore, we have
αxT − βx0 α
T
Θ β V s HsΔs − β
0
α−β
T
T
Θ β V s HsΔs − β
0
Θ β V s HsΔs − β
gsxs∇s 0
T
T
HsΔs,
gs
0
0
Θ β V r HrΔr 0
HrΔr ∇s
s
T
T
gs∇s
0
T
0
Θ β V s HsΔs HsΔs
0
T
0
T
T
T T
gsHrΔr∇s
0
s
Boundary Value Problems
7
T
T
gs∇s
0
0
T T
0
Θ β V s HsΔs
∇
∇s
V s − V rHrΔr
s
T
T
gs∇s
0
Θ β V s HsΔs T
0
V sHsΔs,
0
2.26
which imply that
αxT − βx0 −
T
gsxs∇s α − β
0
−β
T
T
−
T
Θ β V s HsΔs
0
HsΔs −
T
0
T
gs∇s
0
Θ β V s HsΔs
0
V sHsΔs
0
0.
2.27
The proof of Lemma 2.7 is complete.
Define the operator Ψ : P → B by
Ψxt T
Θ β V s ϕq
0
T
s
ϕq
t
s
Δ
λf r, xr, x r ∇r Δs
0
Δ
2.28
λf r, xr, x r ∇r Δs
0
for all t ∈ 0, T T . Obviously, Ψxt ≥ 0 for all t ∈ 0, T T .
Lemma 2.8. If x ∈ P, then Ψx ∈ P.
Proof. It is easily obtained from the second part of the proof in Lemma 2.7. The proof is
complete.
Lemma 2.9. Ψ : P → P is complete continuous.
Proof. First, we show that Ψ maps bounded set into itself. Assume c is a positive constant and
x ∈ Pc {x ∈ P : x ≤ c}. Note that the continuity of ft, x, xΔ guarantees that there is a
8
Boundary Value Problems
C > 0 such that ft, x, xΔ ≤ ϕp C for all t ∈ 0, T T . So we get from ΨΔ x ≤ 0 and ΨΔ∇ x ≤ 0
that
Ψx0 Ψx0
T
s Θ β V s ϕq
λf r, xr, xΔ r ∇r Δs
0
T
s
ϕq
0
Δ
λf r, xr, x r ∇r Δs
T
2.29
T
Θ β V s Δs Cλq−1 T q ,
0
Δ Ψ x ΨΔ xT ϕq
0
≤ Cλq−1 T q−1
0
0
Δ
λf r, xr, x r ∇r
2.30
0
≤ Cλq−1 T q−1 .
That is, ΨPc is uniformly bounded. In addition, notice that
t1 s Δ
λf r, xr, x r ∇r Δs
|Ψxt1 − Ψxt2 | ϕq
t2
0
≤ Cλ
q−1 q−1
T
2.31
|t1 − t2 |,
which implies that
|Ψxt1 − Ψxt2 | −→ 0 as t1 − t2 −→ 0,
p−1 p−1 Δ
ϕp ΨxΔ t1 − ϕp ΨxΔ t2 ΨxΔ t1 − Ψx t2 t1 Δ
λf r, xr, x r ∇r t2
2.32
≤ λϕp C|t1 − t2 |,
which implies that
p−1 p−1 Δ
ΨxΔ t1 −→ 0 as t1 − t2 −→ 0.
−
t
Ψx
2
2.33
ΨxΔ t1 − ΨxΔ t2 −→ 0 as t1 − t2 −→ 0.
2.34
That is,
Boundary Value Problems
9
So Ψx is equicontinuous for any x ∈ Pc . Using Arzela-Ascoli theorem on time scales 17, we
obtain that Ψ Pc is relatively compact. In view of Lebesgue’s dominated convergence theorem
on time scales 18, it is easy to prove that Ψ is continuous. Hence, Ψ is complete continuous.
The proof of this lemma is complete.
Let υ and ω be nonnegative continuous convex functionals on a pone P, ψ a
nonnegative continuous concave functional on P, and r, a, L positive numbers with r > a
we defined the following convex sets:
Pυ, r; ω, l {x ∈ P : υx < r, ωx < l},
Pυ, r; ω, l {x ∈ P : υx ≤ r, ωx ≤ l},
P υ, r; ω, l; ψ, a x ∈ P : υx < r, ωx < l, ψx > a ,
P υ, r; ω, l; ψ, a x ∈ P : υx ≤ r, ωx ≤ l, ψx ≥ a
2.35
and introduce two assumptions with regard to the functionals υ, ω as follows:
H1 there exists M > 0 such that x ≤ M max{υx, ωx} for all x ∈ P;
H2 Pυ, r; ω, l / ∅ for any r > 0 and l > 0.
The following fixed point theorem duo to Bai and Ge is crucial in the arguments of our
main result.
Lemma 2.10 see 19. Let B be Banach space, P ⊂ B a cone, and r2 ≥ d > b > r1 > 0, l2 ≥ l1 > 0.
Assume that υ and ω are nonnegative continuous convex functionals satisfying (H1) and (H2), ψ is
a nonnegative continuous concave functional on P such that ψx ≤ υx for all x ∈ Pυ, r2 ; ω, l2 ,
and Ψ : Pυ, r2 ; ω, l2 → Pυ, r2 ; ω, l2 is a complete continuous operator. Suppose
C1 {x ∈ Pυ, d; ω, l2 ; ψ, b} / ∅, ψΨx > b for x ∈ Pυ, d; ω, l2 ; ψ, b;
C2 υΨx < r1 , ωΨx < l1 for x ∈ Pυ, r1 ; ω, l1 ;
C3 ψΨx > b for x ∈ Pυ, r2 ; ω, l2 ; ψ, b with υΨx > d.
Then Ψ has at least three fixed points x1 , x2 , x3 ∈ Pυ, r2 ; ω, l2 with
x1 ∈ Pυ, r1 ; ω, l1 ,
x2 ∈ x ∈ P υ, r2 ; ω, l2 ; ψ, b : ψx > b ,
2.36
x3 ∈ Pυ, r2 ; ω, l2 \ P υ, r2 ; ω, l2 ; ψ, b ∪ Pυ, r1 ; ω, l1 .
3. Main Result
In this section, we will give sufficient conditions for the existence of at least three positive
solutions to BVPs 1.1-1.2.
10
Boundary Value Problems
Theorem 3.1. Suppose that there are positive numbers 0 < 0 < < T , l2 ≥ l1 > 0, and r2 > b >
r1 > 0 with 0 , ∈ 0, T T , b/N ≤ min{r2 /K, l2 /L} and αb − g0 b ≤ r2 β such that the following
conditions are satisfied.
H3 ft, u, v ≤ min{ϕp r2 /K, ϕp l2 /L} for all t, u, v ∈ 0, T T × 0, r2 × −l2 , l2 , where
Kλ
q−1
T
Θ β V s sq−1 Δs 0
T
q−1
s
Δs ,
L λq−1 T q−1 .
3.1
0
H4 ft, u, v < min{ϕp r1 /K, ϕp l1 /L} for all t, u, v ∈ 0, T T × 0, r1 × −l1 , l1 .
H5 ft, u, v > ϕp b/N for all t, u, v ∈ 0 , T × b, αb − g0 b/β × −l2 , l2 , where
Nλ
q−1
− 0 q−1
T
Θ β V s Δs.
3.2
Then BVPs 1.1-1.2 have at least three positive solutions.
Proof. By the definition of the operator Ψ and its properties, it suffices to show that the
conditions of Lemma 2.10 hold with respect to the operator Ψ.
Let the nonnegative continuous convex functionals υ, ω and the nonnegative
continuous concave functional ψ be defined on the cone P by
ωx max xΔ t xΔ T ,
υx max |xt| x0,
t∈0,T T
t∈0,T T
ψx min xt xT .
t∈
,T T
3.3
Then it is easy to see that x max{υx, ωx} and H1-H2 hold.
First of all, we show that Ψ : Pυ, r2 ; ω, l2 → Pυ, r2 ; ω, l2 . In fact, if x ∈ Pυ, r2 ; ω, l2 ,
then
υx max |xt| ≤ r2 ,
t∈0,T T
ωx max xΔ t ≤ l2
t∈0,T T
3.4
and assumption H3 implies that
r2
l2
f t, xt, x t ≤ min ϕp
, ϕp
∀t ∈ 0, T T .
K
L
Δ
3.5
Boundary Value Problems
11
On the other hand, for x ∈ P, there is Ψx ∈ P; thus
υΨx max |Ψxt|
t∈0,T T
s T Δ
λf r, xr, x r ∇r Δs
max Θ β V s ϕq
t∈0,T T 0
0
T s Δ
ϕq
λf r, xr, x r ∇r Δs t
0
s T Θ β V s ϕq
λf r, xr, xΔ r ∇r Δs
0
0
T s Δ
ϕq
λf r, xr, x r ∇r Δs
0
0
T
s
r 2
≤
∇r Δs
Θ β V s ϕq
λϕp
K
0
0
T s
r 2
∇r Δs
ϕq
λϕp
K
0
0
s
s
r2 T
r2 T Θ β V s ϕq
λ∇r Δs ϕq
λ∇r Δs
K 0
K 0
0
0
T
T
q−1
r2 q−1
q−1
λ
Θ β V s s Δs s Δs
K
0
0
r2
·K
K
r2 ,
ωΨx max ΨxΔ t
t∈0,T T
t
Δ
max −ϕq
λf r, xr, x r ∇r t∈0,T T 0
ϕq
T
Δ
λf r, xr, x r ∇r
0
≤ ϕq
T
l2
ϕq
L
l2
∇r
λϕp
L
0
T
λ∇r
0
l2 .
Therefore, Ψ : Pυ, r2 ; ω, l2 → Pυ, r2 ; ω, l2 .
3.6
12
Boundary Value Problems
In the same way, if x ∈ Pυ, r1 ; ω, l1 , then assumption H4 implies
r1
l1
f t, xt, xΔ t < min ϕp
, ϕp
∀t ∈ 0, TT .
K
L
3.7
As in the argument above, we can get that Ψ : Pυ, r1 ; ω, l1 → Pυ, r1 ; ω, l1 . Thus, condition
C2 of Lemma 2.10 holds.
To check condition C1 in Lemma 2.10. Let d αb − g0 b/β. We choose xt ≡ d > b
for t ∈ 0, T T . It is easy to see that
xt ≡ d ∈ P υ, d; ω, l2 ; ψ, b ,
3.8
ψx d > b.
Consequently,
x ∈ P υ, d; ω, l2 ; ψ, b : ψx > b / ∅.
3.9
Hence, for x ∈ Pυ, d; ω, l2 ; ψ, b, there are
b ≤ xt ≤ d,
Δ x t ≤ l2
∀t ∈ , T T .
3.10
∀t ∈ 0 , T .
3.11
In view of assumption H5, we have
b
f t, xt, xΔ t > ϕp
N
It follows that
ψΨx min Ψxt
t∈
,T T
ΨxT T
s Δ
Θ β V s ϕq
λf r, xr, x r ∇r Δs
0
≥
T
0
Θ β V s ϕq
≥
T
≥
Θ β V s ϕq
>
λf r, xr, xΔ r ∇r Δs
0
Θ β V s ϕq
T
λf r, xr, xΔ r ∇r Δs
0
T
s
Δ
λf r, xr, x r ∇r Δs
0
Θ β V s ϕq
λϕp
0
b
∇r Δs
N
Boundary Value Problems
13
λq−1 − 0 q−1
T
N·
b
Θ β V s Δs
N
b
N
b.
3.12
Therefore, ψΨx > b for x ∈ Pυ, d; ω, l2 ; ψ, b. So condition C1 in Lemma 2.10 is satisfied.
Finally, we show that C3 in Lemma 2.10 holds. In fact, for x ∈ Pυ, r2 ; ω, l2 ; ψ, b and
υΨx > d αb − g0 b/β, we have
ψΨx min Ψxt ΨxT ≥
t∈
,T T
β
β
max Ψxt υΨx > b.
α − g0 t∈0,T T
α − g0
3.13
Thus by Lemma 2.10 and the assumption that ft, 0, 0 /
0 on 0, T T , BVPs 1.1-1.2 have at
least three positive solutions. The proof is complete.
Theorem 3.2. Suppose that there are positive numbers 0 < ξ < T , l2 ≥ l1 > 0, and r2 > b > r1 > 0
with ξ ∈ 0, T T , b/F ≤ min{r 2 /K, l2 /L}, and αb−g0 b ≤ r2 β such that (H3)-(H4) and the following
condition are satisfied.
H6 ft, u, v > ϕp b/F for all t, u, v ∈ 0, ξT × b, αb − g0 b/β × −l2 , l2 , where
F λq−1 ξq−1 T − ξ.
3.14
Then BVPs 1.1-1.2 have at least three positive solutions.
Proof. Let the nonnegative continuous convex functionals υ, ω be defined on the cone P as
Theorem 3.1 and the nonnegative continuous concave functional ψ be defined on the cone P
by
ψx min xt xξ.
t∈0,ξT
3.15
We will show that condition C1 in Lemma 2.10 holds. Let d αb−g0 b/β. We choose
xt ≡ d > b for t ∈ 0, T T . It is easy to see that
xt ≡ d ∈ P υ, d; ω, l2 ; ψ, b ,
ψx d > b.
3.16
Consequently,
x ∈ P υ, d; ω, l2 ; ψ, b : ψx > b / ∅.
3.17
14
Boundary Value Problems
Hence, for x ∈ Pυ, d; ω, l2 ; ψ, b, there are
Δ x t ≤ l2
∀t ∈ 0, ξT .
3.18
b
f t, xt, xΔ t > ϕp
F
∀t ∈ 0, ξT .
3.19
b ≤ xt ≤ d,
In view of assumption H6, we have
It follows that
ψΨx min Ψxt
t∈0,ξT
Ψxξ
T
s Θ β V s ϕq
λf r, xr, xΔ r ∇r Δs
0
T
s
ϕq
ξ
≥
T
≥
ϕq
λf r, xr, xΔ r ∇r Δs
0
ξ
T
ϕq
ξ
0
T
ξ
>
λf r, xr, xΔ r ∇r Δs
0
s
ξ
0
ϕq
ξ
3.20
λf r, xr, xΔ r ∇r Δs
b
∇r Δs
λϕp
F
0
λq−1 ξq−1 T − ξ
F·
b
F
b
F
b.
Therefore, ψΨx > b for x ∈ Pυ, d; ω, l2 ; ψ, b. So condition C1 in Lemma 2.10 is satisfied.
Using a similar proof to Theorem 3.1, the other conditions in Lemma 2.10 are satisfied. By
Lemma 2.10, BVPs 1.1-1.2 have at least three positive solutions. The proof is complete.
4. An Example
Example 4.1. Consider the following second-order Laplacian dynamic equations on time
scales
∇
ϕ1.5 xΔ t
f t, xt, xΔ t 0,
t ∈ 0, 1T
4.1
Boundary Value Problems
15
with integral boundary condition
xΔ 0 0,
3x1 − x0 1
4.2
es−1 xs∇s,
0
where
ft, u, v ⎧
⎨10−5 t 5|v| 6|u| ∀t, u, v ∈ 0, 1T × 0, 12 × −∞, ∞,
⎩10−5 t 5|v| 72
∀t, u, v ∈ 0, 1T × 12, ∞ × −∞, ∞.
4.3
Then BVPs 4.1-4.2 have at least three positive solutions.
Proof. Take 0 0.25, 0.5, r1 l1 0.009, r2 30000, l2 10000, and b 4. It follows that
Θ
1
1
1
1,
≤
α − β − g0 3 − 1 − 1 es−1 ∇s 3 − 1 − 1
0
1
1
1
≥
0.5.
Θ
α − β − g0 3 − 1 − 1 es−1 ∇s 3 − 1 − e−1
0
4.4
From 4.1-4.2, it is easy to obtain
V t V t K
1
t
t
gs∇s 0
gs∇s t
0
t
0
es−1 ∇s ≤ 1 ∀t ∈ 0, 1 T ,
es−1 ∇s ≥ e−1 ≥ 0.25
0
Θ1 V ss
3−1
0
N 0.5 − 0.253−1
Δs 1
∀t ∈ 0, 1T ,
s3−1 Δs ≤ 3 K,
L 1,
4.5
0
1
Θ1 V sΔs ≤ 0.07 N,
0.5
N 0.5 − 0.25
3−1
1
Θ1 V sΔs > 0.01.
0.5
Hence, we have
r2 l2
b
≤ 400 < 10000 min
,
,
N
K L
αb − g0 b − r2 β ≤ 12 − 30000 < 0.
Moreover, we have
4.6
16
Boundary Value Problems
H3 for all t, u, v ∈ 0, 1T × 0, 30000 × −10000, 10000,
r2
r2
l2
l2
≤ min ϕp
, ϕp
;
ft, u, v < 80 < 100 min ϕ1.5
, ϕ1.5
L
K
L
K
4.7
H4 for all t, u, v ∈ 0, 1T × 0, 0.009 × −0.009, 0.009,
ft, u, v ≤ 0.05401045 < min ϕ1.5
r1
l1
l1
, ϕ1.5
≤ min ϕp
, ϕp
;
L
K
L
K
r1
4.8
H5 for all t, u, v ∈ 0.25, 0.5T × 4, 12 × −10000, 10000,
ft, u, v ≥ 6|u| ≥ 24 > ϕ1.5
b
.
N
4.9
Therefore, conditions H3–H5 in Theorem 3.1 are satisfied. Further, it is easy to verify that
the other conditions in Theorem 3.1 hold. By Theorem 3.1, BVPs 4.1-4.2 have at least three
positive solutions. The proof is complete.
Acknowledgment
This work is supported the by the National Natural Sciences Foundation of China under
Grant no. 10971183.
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