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Boundary Value Problems
Volume 2010, Article ID 878131, 13 pages
doi:10.1155/2010/878131
Research Article
Existence and Multiplicity of Positive
Solutions of a Boundary-Value Problem for
Sixth-Order ODE with Three Parameters
Liyuan Zhang and Yukun An
Nanjing University of Aeronautics and Astronautics, 29 Yudao st., Nanjing 210016, China
Correspondence should be addressed to Liyuan Zhang, [email protected]
Received 13 May 2010; Accepted 14 August 2010
Academic Editor: Kanishka Perera
Copyright q 2010 L. Zhang and Y. An. 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 study the existence and multiplicity of positive solutions of the following boundary-value
problem: −u
6
− γu
4
βu
− αu f t, u,0<t<1, u0u1u
0u
1u
4
0u
4
0
u
L
u
4
0
u
4
L
0,
1.1
where A, B, and C are some given real constants and fx, u is a continuous function on
R
2
, is motivated by the study for stationary solutions of the sixth-order parabolic differential
equations
∂u
∂t
∂
6
u
solution u of problems 1.1 to the interval −L, L yields 2L-spatial periodic solutions of1.2
Gyulov et al. 3 have studied the existence and multiplicity of nontrivial solutions of
BVP 1.1. They gained the following results.
Theorem 1.1. Let fx, u : R
2
→ R be a continuous function and Fx, u
u
0
fx, sds. Suppose
the following assumptions are held:
H
1
Fx, u/u
2
→ ∞ as |u|→∞, uniformly with respect to x in bounded intervals,
H
2
0 ≤ Fx, uou
2
as u → 0, uniformly with respect to x in bounded intervals,
then problem 1.1 has at least two nontrivial solutions provided that there exists a natural number
n such that Pnπ/L < 0,wherePξξ
6
− Aξ
4
Bξ
2
− C is the symbol of the linear differential
operator Lu u
ρu −
1
∂
2
u
∂x
2
2
u − u
3
,ρ>0, 1.4
proposed in 1977.
In much the same way, the existence of spatial periodic solutions of both the EFK
equation and the SH equation was studied by Peletier and Troy 4, Peletier and Rottsch
¨
afer
5, Tersian and Chaparova 6, and other authors. More precisely, in those papers, the
authors studied the following fourth-order boundary-value problem:
u
4
Au
Bu f
x, u
0, 0 <x<L,
u
−u
6
− γu
4
βu
− αu f
t, u
, 0 <t<1,
1.6
u
0
u
1
u
0
u
1
γ
π
2
< 1,
18αβγ − β
2
γ
2
4αγ
3
27α
2
− 4β
3
≤ 0.
1.8
Note. The set of α, β, and γ which satisfies H2 is nonempty. For instance, if γ π
2
,β 0,
then H2 holds for α : −4π
2
/27 <α<0.
To be convenient, we introduce the following notations:
L π
6
− γπ
4
− βπ
2
− α,
u
,
f
∞
lim inf
u →∞
min
t∈
0,1
f
t, u
u
,
f
0
lim sup
u → 0
max
t∈
0,1
2
−
4AC > 0,
4 Boundary Value Problems
3 Equation 2.1 has three real roots, two of which are reroots if ΔB
2
− 4AC 0,
4 Equation 2.1 has three unequal real roots if ΔB
2
− 4AC < 0.
Lemma 2.2. Let λ
1
,λ
2
, and λ
3
be the roots of the polynomial Pλλ
3
γλ
2
− βλ α. Suppose
that condition H2 holds, then λ
1
,λ
2
, and λ
3
are real and greater than −π
2
.
λ
2
λ
3
−γ,
λ
1
λ
2
λ
1
λ
3
λ
2
λ
3
−β.
2.4
Therefore α/π
6
β/π
4
γ/π
2
< 1, γ<3π
2
and 3π
4
− 2γπ
2
λ
3
π
2
> 0,
λ
1
π
2
λ
2
π
2
λ
1
π
2
λ
3
,λ
3
are less than −π
2
. By the first inequality of 2.5, there exist two roots which are less than −π
2
and one which is greater than −π
2
. Without loss of generality, we assume that λ
2
< −π
2
,λ
3
<
−π
2
, then we have λ
1
> −π
2
. Multiplying the second inequality of 2.5 by λ
2
π
2
,onegets
λ
1
π
λ
2
π
2
2
<
λ
1
π
2
λ
3
π
2
< 0, 2.7
which is a contradiction. Hence, the assumption is false. The proof is completed.
Boundary Value Problems 5
Let G
i
t, si 1, 2, 3 be Green’s f unction of the linear boundary-value problem
−u
t
λ
i
t, s ≥ δ
i
G
i
t, tG
i
s, s, for all t, s ∈ 0, 1,whereδ
i
> 0 is a constant.
One denotes the following:
M
i
max
0≤s≤1
G
i
s, s
,m
i
min
1/4≤s≤3/4
G
i
s, s
i 1, 2, 3
3
s, s
ds,
2.9
then M
i
,m
i
,C
12
,C
23
> 0.Let·be the maximum norm of C0, 1, and let C
0, 1 be the cone of
all nonnegative functions in C0, 1.
Let h ∈ C0, 1, then one considers linear boundary-value problem (LBVP) as follows:
−u
6
− γu
4
βu
− αu h
t
,t∈
d
2
dt
2
λ
3
u, 2.11
the solution of LBVP 2.10–1.7 can be expressed by
u
t
1
0
G
1
t, δ
G
2
δ, τ
G
3
3
M
1
M
2
G
1
t, t
u
. 2.13
Proof. From 2.12 and ii of Lemma 2.3, it is easy to see that
u
t
≤ C
1
C
2
C
3
M
1
M
2
G
3
s, s
h
s
ds, 2.15
that is,
1
0
G
3
s, s
h
s
ds ≥
u
C
1
C
τ,s
h
s
ds dτ dδ
≥ δ
1
δ
2
δ
3
C
12
C
23
G
1
t, t
1
0
G
3
s, s
2
u
.
2.17
The proof is completed.
We now define a mapping A : C0, 1
→ C0, 1
by
Au
t
1
0
G
1
t, δ
G
2
δ, τ
t
≥ σ
u
, ∀t ∈
1
4
,
3
4
, 2.19
where σ δ
1
δ
2
δ
3
C
12
C
23
m
1
/C
1
2
C
3
M
1
M
2
G
1
t, t
A
u
≥ σ
A
u
, ∀t ∈
1
4
,
,K1.
Lemma 2.7 see 9. Let A : K → K be a completely continuous mapping. Suppose that the
following two conditions are satisfied:
i inf
u∈∂K
r
Au > 0,
ii μAu
/
u for every u ∈ ∂K
r
and μ ≥ 1,
then iA, K
r
,K0.
Lemma 2.8 see 9. Let X be a Banach space, and let K ⊆ X be a cone in X. For p>0, define
K
p
{u ∈ K |u <p}. Assume that A : K
p
→ K is a completely continuous mapping such that
Au
/
u for every u ∈ ∂K
p
{u ∈ K |u p}.
i If u≤Au for every u ∈ ∂K
p
,theniA, K
p
t, u
≤
L − ε
u, 0 ≤ t ≤ 1, 0 ≤ u ≤ ω. 3.1
Let r ∈ 0,ω, we now prove that μAu
/
u for every u ∈ ∂K
r
and 0 <μ≤ 1. In fact, if there
exist u
0
∈ ∂K
r
and 0 <μ
0
≤ 1 such that μ
0
Au
0
u
0
, then, by definition of A, u
0
t satisfies
differential equation the following:
−u
sin πtdt μ
0
1
0
f
t, u
0
t
sin πtdt ≤
L − ε
1
0
u
0
t
sin πtdt. 3.3
By Lemma 2.4, ut ≥ δ
1
δ
2
δ
A, K
r
,K
1. 3.4
On the other hand, since f
∞
>L, there exist ε ∈ 0,L and H>0 such that
f
t, u
≥
L ε
u, 0 ≤ t ≤ 1,u≥ H. 3.5
Let C max
0≤t≤1, 0≤u≤H
|ft, u − L εu| 1, then it is clear that
f
t, u
≥
L ε
u − C, 0 ≤ t ≤ 1,u≥ 0. 3.6
Choose R>R
3
4
. 3.7
By Lemma 2.5, we have
Au
1
2
1
0
G
1
1
2
,δ
G
2
δ, τ
G
3
τ,s
s, u
s
ds
≥
1
2
δ
1
δ
2
δ
3
C
12
C
23
m
1
m
3
L ε
σ
u
.
σ
u
, 3.9
Boundary Value Problems 9
from which we see that inf
u∈∂K
R
Au > 0, namely the hypotheses i of Lemma 2.7 holds.
Next, we show that if R is large enough, then μAu
/
u for any u ∈ ∂K
R
and μ ≥ 1. In fact, if
there exist u
0
∈ ∂K
R
and μ
0
≥ 1 such that μ
0
Au
0
u
0
, then u
0
t satisfies 3.2 and boundary
0
t
sin πtdt −
2C
π
. 3.10
Consequently, we obtain that
1
0
u
0
t
sin πtdt ≤
2C
πε
. 3.11
By Lemma 2.4,
1
0
u
0
t
t, t
sin πtdt, 3.12
from which and from 3.11 we get that
u
0
≤
2CC
1
C
2
C
3
M
1
M
2
δ
1
δ
2
δ
3
C
12
C
23
Now, by the additivity of fixed point index, combine 3.4 and 3.14 to conclude that
i
A, K
R
\ K
r
,K
i
A, K
R
,K
− i
A, K
r
,K
−1. 3.15
Therefore, A has a fixed point in K
R
\ K
r
, which is the positive solution of BVP1.6-1.7.
Case ii:sincef
0
>L, there exist ε>0andr
1
δ
2
δ
3
C
12
C
23
m
1
m
3
L ε
σ
u
. 3.17
10 Boundary Value Problems
Hence, inf
u∈∂K
r
Au > 0. Next, we show that μAu
/
u for any u ∈ ∂K
r
and μ ≥ 1. In fact, if
f
t, u
0
t
sin πtdt ≥
L ε
1
0
u
0
t
sin πtdt. 3.18
Since
1
0
u
0
t sin πtdt > 0, we see that L ≥ L ε, which is a contradiction. Hence, by
Lemma 2.7, we have
i
If there exist u
0
∈ K and 0 <μ
0
≤ 1 such that μ
0
Au
0
u
0
, then 3.2 is valid. From 3.2 and
3.21, it follows that
L
1
0
u
0
t
sin πtdt μ
0
1
0
f
t, u
0
A, K
R
,K
1. 3.23
From 3.19 and 3.23, it follows that
i
A, K
R
\ K
r
,K
i
A, K
R
,K
− i
A, K
r
,K
1. 3.24
Therefore, A has a fixed point in K
R
M
1
M
2
1
0
G
3
s, sds
−1
.
H4 there is a p>0 such that σp ≤ u ≤ p and 0 ≤ t ≤ 1implyft, u ≥ νp, where
ν
−1
δ
1
δ
2
δ
3
C
12
C
23
m
1
3/4
1/4
,K0.
We now prove that iA, K
p
,K1ifH3 is satisfied. In fact, for every u ∈ ∂K
p
,from
the definition of A we have
Au
max
1
0
G
1
t, δ
G
2
δ, τ
G
1
0
G
3
s, s
f
s, u
s
ds
≤ C
1
C
2
C
3
M
1
M
p
,K
i
A, K
R
,K
− i
A, K
p
,K
−1,
i
A, K
p
\ K
r
,K
i
A, K
p
,K
. The proof is
completed.
Theorem 4.2. If f
0
<Land f
∞
<Land H4 is satisfied, then BVP1.6-1.7 has at least two
positive solutions: u
1
and u
2
, such that 0 ≤u
1
≤p ≤u
2
.
Proof. According to the proof of Theorem 3.1, there exists 0 <ω<p<R
2
< ∞, such that
0 <r<ωimplies iA, K
r
,K1andR ≥ R
2
implies iA, K
R
,K1.
We now prove that iA, K
p
,K0ifH4 is satisfied. In fact, for every u ∈ ∂K
p
G
3
τ,s
f
s, u
s
ds dτ dδ
≥ δ
1
δ
2
δ
3
C
12
C
23
m
1
3/4
1/4
G
3
A, K
p
,K
1,
i
A, K
p
\ K
r
,K
i
A, K
p
,K
− i
A, K
r
,K
−1.
4.5
Therefore, A has the fixed points u
1
and u
1
if 0 ≤ t ≤ 1 and 0 ≤ u ≤ p
1
,
ii ft, u ≥ νp
2
if 0 ≤ t ≤ 1 and σp
2
≤ u ≤ p
2
,
then BVP1.6-1.7 has at least three positive solutions: u
1
, u
2
, and u
3
, such that 0 ≤u
1
≤p
1
≤
u
2
≤p
2
≤u
3
.
Proof. According to the proof of Theorem 3.1, there exists 0 <r
,K
0. 4.6
Boundary Value Problems 13
Combining the four afore-mentioned equations, we have
i
A, K
R
\ K
p
2
,K
i
A, K
R
,K
− i
A, K
p
2
,K
1,
i
,K
i
A, K
p
1
,K
− i
A, K
r
,K
1.
4.7
Therefore, A has the fixed points u
1
, u
2
, and u
3
in K
p
1
\ K
r
, K
p
phase field model,” Indiana University Mathematics Journal, vol. 39, no. 4, pp. 1197–1222, 1990.
2 G. Caginalp and P. C. Fife, “Higher-order phase field models and detailed anisotropy,” Physical Review.
B, vol. 34, no. 7, pp. 4940–4943, 1986.
3 T. Gyulov, G. Morosanu, and S. Tersian, “Existence for a semilinear sixth-order ODE,” Journal of
Mathematical Analysis and Applications, vol. 321, no. 1, pp. 86–98, 2006.
4 L. A. Peletier and W. C. Troy, Spatial Patterns, vol. 45 of Progress in Nonlinear Differential Equations and
their Applications,Birkh
¨
auser Boston, Boston, Mass, USA, 2001.
5 L. A. Peletier and V. Rottsch
¨
afer, “Large time behaviour of solutions of the Swift-Hohenberg equation,”
Comptes Rendus Math
´
ematique. Acad
´
emie des Sciences. Paris, vol. 336, no. 3, pp. 225–230, 2003.
6 S. Tersian and J. Chaparova, “Periodic and homoclinic solutions of extended Fisher-Kolmogorov
equations,” Journal of Mathematical Analysis and Applications, vol. 260, no. 2, pp. 490–506, 2001.
7 Y. Li, “Positive solutions of fourth-order boundary value problems with two parameters,” Journal of
Mathematical Analysis and Applications, vol. 281, no. 2, pp. 477–484, 2003.
8 S. Fan, “The new root formula and criterion of cubic equation,” Journal of Hainan Normal University,
vol. 2, pp. 91–98, 1989 Chinese.
9 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in
Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.
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