Volume 2010, Article ID 878131,13pages 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
Copyrightq2010 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:−u6−γu4βu−αuft, u, 0< t <1,u0 u1 u0 u1 u40 u41
0, wheref:0,1×R → Ris continuous,α,β, andγ∈Rsatisfy some suitable assumptions.
1. Introduction
The following boundary-value problem:
u6Au4BuCu−fx, u 0, 0< x < L,
u0 uL u0 uL u40 u4L 0,
1.1
where A, B, andCare some given real constants and fx, u is a continuous function on
R2, is motivated by the study for stationary solutions of the sixth-order parabolic differential
equations
∂u ∂t
∂6u
∂x6 A
∂4u
∂x4 B
∂2u
∂x2 fx, u. 1.2
transition between the liquid and solid state. Whenfx, u u−u3,it was studied by Gardner
and Jones1as well as by Caginalp and Fife2.
Iff is an even 2L-periodic function with respect to xand odd with respect tou, in order to get the 2L-stationary spatial periodic solutions of1.2, one turns to study the two points boundary-value problem1.1. The 2L-periodic extensionuof the odd extension of the solutionuof problems1.1to the interval−L, Lyields 2L-spatial periodic solutions of1.2 Gyulov et al. 3have studied the existence and multiplicity of nontrivial solutions of BVP1.1. They gained the following results.
Theorem 1.1. Letfx, u:R2 → Rbe a continuous function andFx, u u
0 fx, sds. Suppose
the following assumptions are held:
H1Fx, u/u2 → ∞as|u| → ∞, uniformly with respect to x in bounded intervals,
H20≤Fx, u ou2asu → 0, uniformly with respect toxin bounded intervals,
then problem1.1has at least two nontrivial solutions provided that there exists a natural number nsuch thatPnπ/L<0, wherePξ ξ6−Aξ4Bξ2−Cis the symbol of the linear differential
operatorLuu6Au4BuCu.
At the same time, in investigating such spatial patterns, some other high-order parabolic differential equations appear, such as the extended Fisher-Kolmogorov EFK equation
∂u ∂t −ζ
∂4u
∂x4
∂2u
∂x2 u−u
3, ζ >0, 1.3
proposed by Coullet, Elphick, and Repaux in 1987 as well as by Dee and Van Saarlos in 1988 and Swift-HohenbergSHequation
∂u ∂t ρu−
1 ∂
2u
∂x2
2
u−u3, 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 Troy4, 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:
u4AuBufx, u 0, 0< x < L,
u0 uL u0 uL 0. 1.5
The methods used in those papers are variational method and linking theorems.
to obtain Green’s function and then used the fixed point theorem of cone extension or compression to study the problem.
The purpose of this paper is using the idea of 7 to investigate BVP for sixth-order equations. We will discuss the existence and multiplicity of positive solutions of the boundary-value problem
−u6−γu4βu−αuft, u, 0< t <1, 1.6
u0 u1 u0 u1 u40 u41 0, 1.7
and then we assume the following conditions throughout: H1f:0,1×0,∞→0,∞is continuous, H2α, β, andγ ∈Rsatisfy
γ <3π2, 3π4−2γπ2−β >0,
α π6
β π4
γ π2 <1,
18αβγ−β2γ24αγ327α2−4β3≤0.
1.8
Note. The set ofα, β,andγwhich satisfiesH2is nonempty. For instance, ifγ π2, β0,
thenH2holds forα: −4π2/27< α <0.
To be convenient, we introduce the following notations:
Lπ6−γπ4−βπ2−α,
f
0lim infu→0 tmin∈0,1
ft, u
u
, f∞lim sup
u→ ∞ tmax∈0,1
ft, u
u
,
f
∞lim infu→ ∞ tmin∈0,1
ft, u
u
, f0lim sup u→0
max t∈0,1
ft, u
u
.
1.9
2. Preliminaries
Lemma 2.1see8. Set the cubic equation with one variable as follows:
ax3bx2cxd0, a, b, c, d∈R, a /0. 2.1
Let
Ab2−3ac, Bbc−9ad, Cc2−3bd, Δ B2−4AC, 2.2
one has the following:
1Equation2.1has a triple root ifAB0,
2Equation2.1has a real root and two mutually conjugate imaginary roots ifΔ B2 −
3Equation2.1has three real roots, two of which are reroots ifΔ B2−4AC0,
4Equation2.1has three unequal real roots ifΔ B2−4AC <0.
Lemma 2.2. Letλ1, λ2, andλ3be the roots of the polynomialPλ λ3γλ2−βλα. Suppose
that conditionH2holds, thenλ1, λ2, and λ3are real and greater than−π2.
Proof. There areAγ23β, B−βγ−9α, and Cβ2−3αγ in the equationPλ 0. Since
conditionH2holds, we have
Δ B2−4AC18αβγ−β2γ24αγ327α2−4β3≤0. 2.3
Therefore, the equation has three real roots in reply toLemma 2.1. By Vieta theorem, we have
λ1λ2λ3−α,
λ1λ2λ3 −γ,
λ1λ2λ1λ3λ2λ3−β.
2.4
Thereforeα/π6β/π4γ/π2 <1,γ <3π2and 3π4−2γπ2−β >0 hold if and only if
λ1π2
λ2π2
λ3π2
>0,
λ1π2
λ2π2
λ3π2
>0,
λ1π2
λ2π2
λ1π2
λ3π2
λ2π2
λ3π2
>0.
2.5
Then, we only prove that the system of inequalities2.5holds if and only ifλ1, λ2, andλ3
are all greater than−π2.
In fact, the sufficiency is obvious, we just prove the necessity. Assume thatλ1, λ2, λ3
are less than−π2. By the first inequality of2.5, there exist two roots which are less than−π2
and one which is greater than−π2. Without loss of generality, we assume thatλ2<−π2, λ3< −π2, then we haveλ
1>−π2. Multiplying the second inequality of2.5byλ2π2, one gets
λ1π2
λ2π2
λ2π2
2
λ2π2
λ3π2
<0. 2.6
Compare with the third inequality of2.5, we have
λ2π2
2
<λ1π2
λ3π2
<0, 2.7
LetGit, s i1,2,3be Green’s function of the linear boundary-value problem
−ut λiut 0, u0 u1 0. 2.8
Lemma 2.3see7. Git, si1,2,3has the following properties:
iGit, s>0, for allt, s∈0,1,
iiGit, s≤CiGis, s, for allt, s∈0,1, whereCi>0is a constant, iiiGit, s≥δiGit, tGis, s, for allt, s∈0,1, whereδi>0is a constant.
One denotes the following:
Mimax
0≤s≤1Gis, s, mi1/4min≤s≤3/4Gis, s i1,2,3,
C12 1
0
G1δ, δG2δ, δdδ, C23 1
0
G2s, sG3s, sds,
2.9
thenMi, mi, C12, C23 >0. Let · be the maximum norm ofC0,1, and letC0,1be the cone of
all nonnegative functions inC0,1.
Leth∈C0,1, then one considers linear boundary-value problem (LBVP) as follows:
−u6−γu4βu−αuht, t∈0,1, 2.10
with the boundary condition1.7. Since
−u6−γu4βu−αu
−d2
dt2 λ1
−d2
dt2 λ2
−d2
dt2 λ3
u, 2.11
the solution of LBVP2.10–1.7can be expressed by
ut 1
0
G1t, δG2δ, τG3τ, shsds dτ dδ. 2.12
Lemma 2.4. Leth∈C0,1, then the solution of LBVP2.10–1.7satisfies
ut≥ δ1δ2δ3C12C23
C1C2C3M1M2G1t, t u . 2.13
Proof. From2.12andiiofLemma 2.3, it is easy to see that
ut≤C1C2C3M1M2 1
0
and, therefore,
u ≤C1C2C3M1M2 1
0
G3s, shsds, 2.15
that is,
1
0
G3s, shsds≥ u
C1C2C3M1M2. 2.16
UsingiiiofLemma 2.3, we have
ut 1
0
G1t, δG2δ, τG3τ, shsds dτ dδ
≥δ1δ2δ3C12C23G1t, t 1
0
G3s, shsds
≥ δ1δ2δ3C12C23G1t, t
C1C2C3M1M2 u .
2.17
The proof is completed.
We now define a mappingA:C0,1 → C0,1by
Aut 1
0
G1t, δG2δ, τG3τ, sfs, usds dτ dδ. 2.18
It is clear thatA:C0,1 → C0,1is completely continuous. ByLemma 2.4, the positive solution of BVP1.6-1.7is equivalent to nontrivial fixed point ofA. We will find the nonzero fixed point of Aby using the fixed point index theory in cones. For this, one chooses the subconeKofC0,1by
K
u∈C0,1|ut≥σ u , ∀t∈
1 4,
3 4
, 2.19
whereσδ1δ2δ3C12C23m1/C1C2C3M1M2, we have the following.
Lemma 2.5. HavingAK⊆K,A:K → Kis completely continuous.
Proof. For u ∈ K, let ht ft, ut, then Autis the solution of LBVP2.10–1.7. By Lemma 2.4, one has
Aut≥ δ1δ2δ3C12C23
C1C2C3M1M2G1t, t Au ≥σ Au , ∀t∈
1 4,
3 4
, 2.20
The main results of this paper are based on the theory of fixed point index in cones9. LetEbe a Banach space andK⊂Ebe a closed convex cone inE. Assume thatΩis a bounded open subset ofEwith boundary ∂Ω, andK∩Ω/∅. LetA : K∩Ω → K be a completely continuous mapping. IfAu /ufor everyu∈K∩∂Ω, then the fixed point indexiA, K∩Ω, K is well defined. We have that ifiA, K∩Ω, K/0, thenAhas a fixed point inK∩∂Ω.
LetKr {u∈K| u < r}and∂Kr {u∈K| u r}for everyr >0.
Lemma 2.6see9. LetA:K → Kbe a completely continuous mapping. IfμAu /ufor every
u∈∂Kr and0< μ≤1, theniA, Kr, K 1.
Lemma 2.7 see9. LetA : K → K be a completely continuous mapping. Suppose that the
following two conditions are satisfied:
iinfu∈∂Kr Au >0,
iiμAu /ufor everyu∈∂Kr andμ≥1,
theniA, Kr, K 0.
Lemma 2.8see9. LetX be a Banach space, and letK ⊆ X be a cone inX. For p >0, define
Kp {u∈K| u < p}. Assume thatA:Kp → Kis a completely continuous mapping such that Au /ufor everyu∈∂Kp{u∈K| u p}.
iIf u ≤ Au for everyu∈∂Kp, theniA, Kp, K 0. iiIf u ≥ Au for everyu∈∂Kp, theniA, Kp, K 1.
3. Existence
We are now going to state our existence results.
Theorem 3.1. Assume that (H1) and (H2) hold, then in each of the following case:
if0 < L,f ∞> L, iif
0 > L,f∞< L,
the BVP1.6-1.7has at least one positive solution.
Proof. To prove Theorem 3.1, we just show that the mapping A defined by 2.18 has a nonzero fixed point in the cases, respectively.
Casei: sincef0< L, by the definition off0, we may chooseε >0 andω >0,so that
ft, u≤L−εu, 0≤t≤1, 0≤u≤ω. 3.1
Letr ∈ 0, ω, we now prove thatμAu /ufor everyu∈∂Kr and 0 < μ ≤1. In fact, if there existu0 ∈ ∂Kr and 0 < μ0 ≤1 such thatμ0Au0 u0, then, by definition ofA,u0tsatisfies
differential equation the following:
and boundary condition1.7. Multiplying3.2by sinπtand integrating on0,1, then using integration by parts in the left side, we have
L
1
0
u0tsinπt dtμ0 1
0
ft, u0tsinπt dt≤L−ε 1
0
u0tsinπt dt. 3.3
ByLemma 2.4,ut ≥δ1δ2δ3C12C23/C1C2C3M1M2G1t, t u , and then
1
0u0tsinπt dt >
0. We see that L ≤ L−ε, which is a contradiction. Hence, Asatisfies the hypotheses of Lemma 2.6, inKr. ByLemma 2.6we have
iA, Kr, K 1. 3.4
On the other hand, sincef
∞> L, there existε∈0, LandH >0 such that
ft, u≥Lεu, 0≤t≤1, u≥H. 3.5
LetCmax0≤t≤1,0≤u≤H|ft, u−Lεu|1, then it is clear that
ft, u≥Lεu−C, 0≤t≤1, u≥0. 3.6
ChooseR > R0 max{H/σ, ω}. Letu∈∂KR. Sinceus≥σ u > H, for alls∈1/4,3/4, from3.5we see that
ft, u≥Lεus≥Lεσ u , ∀s∈
1 4,
3 4
. 3.7
ByLemma 2.5, we have
Au
1 2
1
0
G1
1 2, δ
G2δ, τG3τ, sfs, usds dτ dδ
≥δ1δ2δ3C12C23m1 3/4
1/4
G3s, sfs, usds
≥ 1
2δ1δ2δ3C12C23m1m3Lεσ u .
3.8
Therefore,
Au ≥Au
1 2
≥ 1
from which we see that infu∈∂KR Au >0, namely the hypothesesiofLemma 2.7holds.
Next, we show that ifRis large enough, thenμAu /ufor anyu∈∂KR andμ ≥1. In fact, if there existu0∈∂KRandμ0≥1 such thatμ0Au0u0, thenu0tsatisfies3.2and boundary
condition1.7. Multiplying3.2by sinπtand integrating, from3.6we have
L
1
0
u0tsinπt dtμ0 1
0
ft, u0tsinπt dt≥Lε 1
0
u0tsinπt dt−2C
π . 3.10
Consequently, we obtain that
1
0
u0tsinπt dt≤ 2C
πε. 3.11
ByLemma 2.4,
1
0
u0tsinπt dt≥ δ1δ2δ3C12C23
C1C2C3M1M2 u0 1
0
G1t, tsinπt dt, 3.12
from which and from3.11we get that
u0 ≤ 2CC1C2C3M1M2
δ1δ2δ3C12C23πε 1
0
G1t, tsinπt dt
−1
:R. 3.13
Let R > max{R, R0}, then for any u ∈ ∂KR andμ ≥ 1,μAu /u. Hence, hypothesis ii of Lemma 2.7also holds. ByLemma 2.7, we have
iA, KR, K 0. 3.14
Now, by the additivity of fixed point index, combine3.4and3.14to conclude that
iA, KR\Kr, K
iA, KR, K−iA, Kr, K −1. 3.15
Therefore,Ahas a fixed point inKR\Kr, which is the positive solution of BVP1.6-1.7. Caseii: sincef
0 > L, there existε >0 andr0>0 such that
ft, u≥Lεu, 0≤t≤1, 0≤u≤r0. 3.16
Letr∈0, r0, then for everyu∈∂Kr, through the argument used in3.9, we have
Au ≥Au
1 2
≥ 1
Hence, infu∈∂Kr Au >0. Next, we show thatμAu /ufor anyu∈∂Kr andμ≥1. In fact, if
there existu0∈∂Kr andμ0 ≥1 such thatμ0Au0 u0, thenu0tsatisfies3.2and boundary
1.7. From3.2and3.16, it follows that
L
1
0
u0tsinπt dtμ0 1
0
ft, u0tsinπt dt≥Lε 1
0
u0tsinπt dt. 3.18
Since 01u0tsinπt dt > 0, we see that L ≥ Lε, which is a contradiction. Hence, by
Lemma 2.7, we have
iA, Kr, K 0. 3.19
On the other hand, sincef∞< L, there existε∈0, LandH >0 such that
ft, u≤L−εu, 0≤t≤1, u≥H. 3.20
SetCmax0≤t≤1,0≤u≤H|ft, u−L−εu|1, we obviously have
ft, u≤L−εuC, 0≤t≤1, u≥0. 3.21
If there existu0 ∈Kand 0< μ0 ≤1 such thatμ0Au0 u0, then3.2is valid. From3.2and
3.21, it follows that
L
1
0
u0tsinπt dtμ0 1
0
ft, u0tsinπt dt≤L−ε 1
0
u0tsinπt dt2C
π . 3.22
By the proof of3.13, we see that u0 ≤ R. LetR >max{R, r0}, then for anyu∈∂KRand 0< μ≤1,μAu /u. Therefore, byLemma 2.6, we have
iA, KR, K 1. 3.23
From3.19and3.23, it follows that
iA, KR\Kr, K
iA, KR, K−iA, Kr, K 1. 3.24
FromTheorem 3.1, we immediately obtain the following.
Corollary 3.2. Assume thatH1andH2hold, then in each of the following cases:
if0 0,f ∞∞, iif∞0,f
0∞,
the BVP1.6-1.7has at least one positive solution.
4. Multiplicity
Next, we study the multiplicity of positive solutions of BVP1.6-1.7and assume in this section that
H3there is a p > 0 such that 0 ≤ u ≤ p and 0 ≤ t ≤ 1 imply ft, u < ηp, where η C1C2C3M1M2
1
0G3s, sds
−1
.
H4there is a p > 0 such thatσp ≤ u ≤ p and 0 ≤ t ≤ 1 implyft, u ≥ νp, where ν−1δ1δ2δ3C12C23m1
3/4
1/4G3s, sds.
Theorem 4.1. If f
0 > Landf∞ > LandH3is satisfied, then BVP1.6-1.7has at least two positive solutions:u1andu2,such that0≤ u1 ≤p≤ u2 .
Proof. According to the proof ofTheorem 3.1, there exists 0 < r0 < p < R1 < ∞, such that
0< r < r0impliesiA, Kr, K 0 andR≥R1impliesiA, KR, K 0.
We now prove thatiA, Kp, K 1 ifH3is satisfied. In fact, for everyu∈∂Kp, from the definition ofAwe have
Au max
1
0
G1t, δG2δ, τG3τ, sfs, usds dτ dδ
≤C1C2C3M1M2
1
0
G3s, sfs, usds
≤C1C2C3M1M2 1
0
G3s, sηp ds
u .
4.1
FromiiofLemma 2.8, we have
iA, Kp, K
1. 4.2
Combining3.14and3.19, we have
iA, KR\Kp, K
iA, KR, K−i
A, Kp, K
−1,
iA, Kp\Kr, K
iA, Kp, K
−iA, Kr, K 1.
Therefore,Ahas fixed pointsu1andu2inKp\KrandKR\Kp, respectively, which means that u1tandu2tare positive solutions of BVP1.6-1.7and 0≤ u1 ≤p≤ u2 . The proof is
completed.
Theorem 4.2. If f0 < Landf∞ < LandH4is satisfied, then BVP1.6-1.7has at least two
positive solutions:u1andu2,such that0≤ u1 ≤p≤ u2 .
Proof. According to the proof ofTheorem 3.1, there exists 0 < ω < p < R2 < ∞, such that
0< r < ωimpliesiA, Kr, K 1 andR≥R2impliesiA, KR, K 1.
We now prove thatiA, Kp, K 0 ifH4is satisfied. In fact, for everyu∈∂Kp, from the proof ofiofTheorem 3.1, we have
Au
1 2
1
0
G1
1 2, δ
G2δ, τG3τ, sfs, usds dτ dδ
≥δ1δ2δ3C12C23m1 3/4
1/4
G3s, sνp ds
u .
4.4
Therefore, Au ≥ Au1/2 ≥ u , according toiofLemma 2.8,iA, Kp, K 0. Combining3.4and3.23, we have
iA, KR\Kp, K
iA, KR, K−i
A, Kp, K
1,
iA, Kp\Kr, K
iA, Kp, K
−iA, Kr, K −1.
4.5
Therefore,Ahas the fixed pointsu1andu2inKp\KrandKR\Kp, respectively, which means thatu1tand u2tare positive solutions of BVP1.6-1.7and 0 ≤ u1 ≤ p ≤ u2 . The
proof is completed.
Theorem 4.3. Iff
0 > Landf∞< L, and there existsp2> p1>0that satisfies
ift, u< ηp1if0≤t≤1and0≤u≤p1,
iift, u≥νp2if0≤t≤1andσp2≤u≤p2,
then BVP1.6-1.7has at least three positive solutions:u1,u2,andu3,such that0≤ u1 ≤p1 ≤
u2 ≤p2≤ u3 .
Proof. According to the proof ofTheorem 3.1, there exists 0 < r0 < p1 < p2 < R3 <∞, such
that 0< r < r0impliesiA, Kr, K 1 andR≥R3impliesiA, KR, K 1. From the proof of Theorems4.1and4.2, we have
iA, Kp1, K
1, iA, Kp2, K
Combining the four afore-mentioned equations, we have
iA, KR\Kp2, K
iA, KR, K−i
A, Kp2, K
1,
iA, Kp2\Kp1, K
iA, Kp2, K
−iA, Kp1, K
−1,
iA, Kp1\Kr, K
iA, Kp1, K
−iA, Kr, K 1.
4.7
Therefore,Ahas the fixed pointsu1,u2,andu3inKp1\Kr,Kp2\Kp1,andKR\Kp2, which
means thatu1t,u2t,andu3tare positive solutions of BVP1.6-1.7and 0≤ u1 ≤p1 ≤
u2 ≤p2≤ u3 . The proof is completed.
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