Volume 2010, Article ID 106962,9pages doi:10.1155/2010/106962
Research Article
Uniqueness and Parameter Dependence of Positive
Solution of Fourth-Order Nonhomogeneous BVPs
Jian-Ping Sun and Xiao-Yun Wang
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China
Correspondence should be addressed to Jian-Ping Sun,[email protected]
Received 23 February 2010; Accepted 11 July 2010
Academic Editor: Irena Rach ˚unkov´a
Copyrightq2010 J.-P. Sun and X.-Y. Wang. 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 investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville boundary value problem:u4ft, u, t∈0,1,αu0−βu0 λ1, γu1 δu1 λ2,auξ1−buξ1 −λ3, cuξ2 duξ2 −λ4, where 0≤ξ1< ξ2≤1 andλii1,2,3,4are nonnegative parameters.
Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parametersλii1,2,3,4is also studied.
1. Introduction
Boundary value problemsBVPs for short consisting of fourth-order differential equation and four-point homogeneous boundary conditions have received much attention due to their striking applications. For example, Chen et al.1studied the fourth-order nonlinear differential equation
u4ft, u, t∈0,1, 1.1
with the four-point homogeneous boundary conditions
u0 u1 0, 1.2
auξ1−buξ1 0, cuξ2 duξ2 0, 1.3
established. Bai et al.2obtained the existence of solutions for the BVP1.1–1.3by using a nonlinear alternative of Leray-Schauder type. For other related results, one can refer to3–5 and the references therein.
Recently, nonhomogeneous BVPs have attracted many authors’ attention. For instance, Ma 6, 7 and L. Kong and Q. Kong 8–10 studied some second-order multipoint nonhomogeneous BVPs. In particular, L. Kong and Q. Kong 10 considered the following second-order BVP with multipoint nonhomogeneous boundary conditions
uatfu 0, t∈0,1,
u0 m
i1
aiuti λ, u1 m
i1
biuti μ,
1.4
whereλandμare nonnegative parameters. They derived some conditions for the above BVP to have a unique solution and then studied the dependence of this solution on the parameters
λandμ. Sun11discussed the existence and nonexistence of positive solutions to a class of third-order three-point nonhomogeneous BVP. The authors in12studied the multiplicity of positive solutions for some fourth-order two-point nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. For more recent results on higher-order BVPs with nonhomogeneous boundary conditions, one can see13–16.
Inspired greatly by the above-mentioned excellent works, in this paper we are concerned with the following Sturm-Liouville BVP consisting of the fourth-order differential equation:
u4ft, u, t∈0,1 1.5
and the four-point nonhomogeneous boundary conditions
αu0−βu0 λ1, γu1 δu1 λ2, 1.6
auξ1−buξ1 −λ3, cuξ2 duξ2 −λ4, 1.7
where 0≤ξ1 < ξ2≤1 andλi i1,2,3,4are nonnegative parameters. Under the following assumptions:
A1α, β, γ, δ, a, b, c,anddare nonnegative constants withβ >0,δ >0,ρ1 :αγαδγβ > 0,ρ2 :adbcacξ2−ξ1>0,−aξ1b >0,andcξ2−1 d >0;
A2ft, u:0,1×0,∞ → 0,∞is continuous and monotone increasing inufor everyt∈0,1;
A3there exists 0≤θ <1 such that
we prove the uniqueness of positive solution for the BVP 1.5–1.7 and study the dependence of this solution on the parametersλi i1,2,3,4.
2. Preliminary Lemmas
First, we recall some fundamental definitions.
Definition 2.1. LetXbe a Banach space with norm · . Then
1a nonempty closed convex setP ⊆Xis said to be a cone ifmP ⊆Pfor allm≥0 and
P∩−P {0}, where0is the zero element ofX;
2every coneP inXdefines a partial ordering inXbyu≤v⇔v−u∈P;
3a conePis said to be normal if there existsM >0 such that0≤u≤vimplies that
u ≤Mv;
4a conePis said to be solid if the interiorP◦ ofPis nonempty.
Definition 2.2. LetP be a solid cone in a real Banach spaceX, T : P◦ → P◦ an operator, and 0≤θ <1.ThenTis called aθ-concave operator if
Tku≥kθTu for anyk∈0,1, u∈P .◦ 2.1
Next, we state a fixed point theorem, which is our main tool.
Lemma 2.3see17. Assume thatPis a normal solid cone in a real Banach spaceX, 0 ≤θ <1, andT :P◦ → P◦ is aθ-concave increasing operator. ThenThas a unique fixed point inP .◦
The following two lemmas are crucial to our main results.
Lemma 2.4. Assume thatρ1andρ2are defined as in (A1) andρ1ρ2/0. Then for anyh∈C0,1,the BVP consisting of the equation
u4t ht, t∈0,1 2.2
and the boundary conditions1.6and1.7has a unique solution
ut
1
0
G1t, s
ξ2
ξ1
G2s, τhτdτ ds 4
i1
where
G1t, s 1
ρ1
⎧ ⎨ ⎩
αsβγδ−γt, 0≤s≤t≤1,
αtβγδ−γs, 0≤t≤s≤1,
G2t, s 1
ρ2
⎧ ⎨ ⎩
as−ξ1 bcξ2−t d, s≤t, ξ1≤s≤ξ2,
at−ξ1 bcξ2−s d, t≤s, ξ1≤s≤ξ2,
φ1t 1
ρ1
γδ−γt, t∈0,1,
φ2t 1
ρ1
αtβ, t∈0,1,
φ3t 1
ρ2
1
0
cξ2−s dG1t, sds, t∈0,1,
φ4t 1
ρ2
1
0
as−ξ1 bG1t, sds, t∈0,1.
2.4
Proof. Let
ut vt, t∈0,1. 2.5
Then
vt ht, t∈0,1. 2.6
By2.5and1.6, we know that
ut −
1
0
G1t, svsds 1
ρ1
αλ2−γλ1
t 1
ρ1
γδλ1βλ2
, t∈0,1. 2.7
On the other hand, in view of2.5and1.7, we have
avξ1−bvξ1 −λ3, cvξ2 dvξ2 −λ4. 2.8
So, it follows from2.6and2.8that
vt −
ξ2
ξ1
G2t, shsds 1
ρ2
cλ3−aλ4t 1
ρ2
which together with2.7implies that
ut
1
0
G1t, s
ξ2
ξ1
G2s, τhτdτ ds 4
i1
λiφit, t∈0,1. 2.10
Lemma 2.5. Assume that (A1) holds. Then
1G1t, s>0fort, s∈0,1×0,1;
2G2t, s>0fort, s∈0,1×ξ1, ξ2;
3φit>0fort∈0,1, i1,2,3,4.
3. Main Result
For convenience, we denoteλ λ1, λ2, λ3, λ4andμ μ1, μ2, μ3, μ4. In the remainder of this paper, the following notations will be used:
1λ → ∞if at least one ofλii1,2,3,4approaches∞;
2λ → μifλi → μifori1,2,3,4;
3λ>μifλi≥μifori1,2,3,4 and at least one of them is strict.
LetXC0,1. ThenX, · is a Banach space, where · is defined as usual by the sup norm.
Our main result is the following theorem.
Theorem 3.1. Assume that (A1)–(A3) hold. Then the BVP1.5–1.7has a unique positive solution uλtfor anyλ >0, where0 0,0,0,0. Furthermore, such a solutionuλtsatisfies the following properties:
P1limλ→ ∞uλ∞;
P2uλtis strictly increasing inλ, that is,
λ>μ>0⇒uλt> uμt, t∈0,1; 3.1
P3uλtis continuous inλ, that is, for any givenμ>0,
λ−→μ⇒ uλ−uμ −→0. 3.2
Proof. Let P {u ∈ X | ut ≥ 0, t ∈ 0,1}. Then P is a normal solid cone in X with ◦
P {u ∈ X | ut > 0, t ∈ 0,1}.For any λ> 0, if we define an operatorTλ :
◦
P → X as follows:
Tλut
1
0
G1t, s
ξ2
ξ1
G2s, τfτ, uτdτ ds 4
i1
then it is not difficult to verify thatuis a positive solution of the BVP1.5–1.7if and only ifuis a fixed point ofTλ.
Now, we will prove thatTλhas a unique fixed point by usingLemma 2.3.
First, in view ofLemma 2.5, we know thatTλ :
◦
P → P .◦
Next, we claim thatTλ :
◦
P → P◦ is aθ-concave operator.
In fact, for anyk∈0,1andu∈P ,◦ it follows from3.3andA3that
Tλkut
1
0
G1t, s
ξ2
ξ1
G2s, τfτ, kuτdτ ds 4
i1
λiφit
≥kθ
1
0
G1t, s
ξ2
ξ1
G2s, τfτ, uτdτ ds 4
i1
λiφit
≥kθ
1
0
G1t, s
ξ2
ξ1
G2s, τfτ, uτdτ ds 4
i1
λiφit
kθTλut, t∈0,1,
3.4
which shows thatTλisθ-concave.
Finally, we assert thatTλ :
◦
P → P◦ is an increasing operator. Suppose thatu, v∈P◦ andu≤v.By3.3andA2, we have
Tλut
1
0
G1t, s
ξ2
ξ1
G2s, τfτ, uτdτ ds 4
i1
λiφit
≤
1
0
G1t, s
ξ2
ξ1
G2s, τfτ, vτdτ ds 4
i1
λiφit
Tλvt, t∈0,1,
3.5
which indicates thatTλis increasing.
Therefore, it follows fromLemma 2.3thatTλhas a unique fixed pointuλ∈
◦
P ,which is the unique positive solution of the BVP1.5–1.7. The first part of the theorem is proved.
In the rest of the proof, we will prove that such a positive solution uλt satisfies
propertiesP1,P2, andP3. First,
uλt Tλuλt
1
0
G1t, s
ξ2
ξ1
G2s, τfτ, uλτdτ ds
4
i1
λiφit, t∈0,1,
3.6
Next, we showP2. Assume thatλ>μ>0.Let
χsupχ >0 :uλt≥χuμt, t∈0,1. 3.7
Thenuλt≥χuμtfort∈0,1.We assert thatχ≥1.Suppose on the contrary that 0< χ <1.
SinceTλis aθ-concave increasing operator and for givenu∈
◦
P,Tλuis strictly increasing inλ,
we have
uλt Tλuλt≥Tλχuμt> Tμχuμt
≥χθTμuμt χθuμt> χuμt, t∈0,1,
3.8
which contradicts the definition ofχ.Thus, we getuλt≥uμtfort∈0,1.And so,
uλt Tλuλt≥Tλuμt> Tμuμt uμt, t∈0,1, 3.9
which indicates thatuλtis strictly increasing inλ.
Finally, we proveP3. For any givenμ>0,we first suppose thatλ → μwithμ/2 < λ<μ.FromP2, we know that
uλt< uμt, t∈0,1. 3.10
Let
σsupσ >0 :uλt≥σuμt, t∈0,1. 3.11
Then 0< σ <1 anduλt≥σuμtfort∈0,1.If we define
ωλ min
λi
μi
:μi>0
then 0< ωλ<1 and
uλt Tλuλt
≥Tλσuμt
1
0
G1t, s
ξ2
ξ1
G2s, τf
τ, σuμτdτ ds
4
i1
λiφit
≥
1
0
G1t, s
ξ2
ξ1
G2s, τf
τ, σuμτdτ dsωλ
4
i1
μiφit
≥ωλ
1
0
G1t, s
ξ2
ξ1
G2s, τf
τ, σuμτdτ ds
4
i1
μiφit
ωλTμσuμt
≥ωλσθTμuμt
ωλσθuμt, t∈0,1,
3.13
which together with the definition ofσimplies that
ωλσθ≤σ. 3.14
So,
σ≥ωλ1/1−θ. 3.15
Therefore,
uλt≥σuμt≥ωλ1/1−θuμt, t∈0,1. 3.16
In view of3.10and3.16, we obtain that
uλ−uμ ≤1−ωλ1/1−θ uμ , 3.17
which together with the fact thatωλ → 1 asλ → μshows that
uλ−uμ −→0 asλ−→μwithλ<μ. 3.18
Similarly, we can also prove that
uλ−uμ −→0 asλ−→μwithλ>μ. 3.19
Acknowledgment
Supported by the National Natural Science Foundation of China10801068.
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