Discrete Dynamics in Nature and Society Volume 2009, Article ID 976406,14pages doi:10.1155/2009/976406
Research Article
Solutions for m-Point BVP with Sign Changing Nonlinearity
Hua Su
School of Statistics and Mathematic, Shandong University of Finance, Jinan, Shandong 250014, China
Correspondence should be addressed to Hua Su,[email protected] Received 24 October 2008; Accepted 31 January 2009
Recommended by Binggen Zhang
We study the existence of positive solutions for the following nonlinearm-point boundary value problem for an increasing homeomorphism and homomorphism with sign changing nonlinearity:
{φutatft, ut 0, 0< t <1,u0 m−2
i1aiuξi,u1 k
i1biuξi−s
ik1biuξi− m−2
is1biuξi, whereφ : R → R is an increasing homeomorphism and homomorphism and φ0 0. The nonlinear termfmay change sign. As an application, an example to demonstrate our results is given. The conclusions in this paper essentially extend and improve the known results.
Copyrightq2009 Hua Su. 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.
1. Introduction
In this paper, we study the existence of positive solutions of the following nonlinearm-point boundary value problem with sign changing nonlinearity:
φ ut
atft, ut 0, 0< t <1, u0 m−2
i1
aiu ξi
, u1 k
i1
biu ξi
− s
ik1
biu ξi
− m−2
is1
biu ξi
, 1.1
where φ : R → R is an increasing homeomorphism and homomorphism and φ0 0;
ξi∈0,1with 0< ξ1< ξ2<· · ·< ξm−2<1 andai, bi, a, fsatisfy
H1ai, bi ∈0,∞, 0<k
i1bi−s
ik1bi<1,0<m−2
i1 ai<1;
H2at:0,1 → 0,∞does not vanish identically on any subinterval of0,1and satisfies
0<
1
0
atdt <∞; 1.2
H3f ∈C0,1×0,∞,−∞,∞, ft,0≥0 andft,0/0.
Definition 1.1. A projection φ : R → R is called an increasing homeomorphism and homomorphism, if the following conditions are satisfied:
iifx≤y, thenφx≤φy, for allx, y∈R;
iiφis a continuous bijection and its inverse mapping is also continuous;
iiiφxy φxφy, for allx, y∈R.
The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by Il’in and Moiseev 1, 2. Motivated by the study of 1, 2, Gupta 3 studied certain three-point boundary value problems for nonlinear ordinary differential equations. Since then, more general nonlinear multipoint boundary value problems have been studied by several authors. We refer the reader to4–12for some references along this line. Multipoint boundary value problems describe many phenomena in the applied mathematical sciences. For example, the vibrations of a guy wire of a uniform cross-section and composed of N parts of different densities can be set up as a multipoint boundary value problems see Moshinsky13; many problems in the theory of elastic stability can be handled by the method of multipoint boundary value problemssee Timoshenko14.
In 2001, Ma6studiedm-point boundary value problemBVP:
ut htfu 0, 0≤t≤1, u0 0, u1 m−2
i1
αiu ξi
, 1.3
where αi > 0 i 1,2, . . . , m − 2, m−2
i1 αi < 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, and f ∈ C0,∞,0,∞,h ∈ C0,1,0,∞. Author established the existence of positive solutions under the condition thatfis either superlinear or sublinear.
In11, we considered the existence of positive solutions for the following nonlinear four-point singular boundary value problem withp-Laplacian:
φp
ut
atfut 0, 0< t <1, αφpu0−βφp
uξ
0, γφpu1 δφp
uη
0, 1.4
whereφps |s|p−2s, p >1, φq φp−1, 1/p1/q1, α >0, β ≥0, γ >0, δ ≥ 0, ξ, η ∈ 0,1, ξ < η, a :0,1 → 0,∞. By using the fixed-point theorem of cone, the existence of positive solution and many positive solutions for nonlinear singular boundary value problem p-Laplacian is obtained.
Recently, Ma et al.5 used the monotone iterative technique in cones to prove the existence of at least one positive solution form-point boundary value problemBVP:
φp
ut
atft, ut 0, 0< t <1, u0 m−2
i1
aiu ξi
, u1 m−2
i1
biu ξi
, 1.5
where 0 < m−2
i1 bi < 1, 0 < m−2
i1 ai < 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, at ∈ L10,1, f∈C0,1×0,∞,0,∞.
In9, Wang and Hou investigated the followingm-point BVP:
φp
ut
ft, ut 0, t∈0,1, φp
u0 n−2
i1
aiφp
u ξi
, u1 n−2
i1
biu ξi
, 1.6
whereφpu |u|p−2u,p >1,ξi ∈ 0,1with 0 < ξ1 < ξ2 < · · · < ξn−2 < 1 andai, bi satisfy ai, bi∈0,∞, 0<n−2
i1ai<1, 0<n−2
i1bi<1.
However, in all the above-mentioned paper, the authors discuss the boundary value problem BVP under the key conditions that the nonlinear term is positive continuous function. Motivated by the results mentioned above, in this paper we study the existence of positive solutions ofm-point boundary value problem1.1 for an increasing homeomorphism and homomorphism with sign changing nonlinearity. We generalize the results in4–12.
By a positive solution of BVP1.1, we understand a functionuwhich is positive on 0,1and satisfies the differential equation as well as the boundary conditions in BVP1.1.
2. The Preliminary Lemmas
In this section, we present some lemmas which are important to our main results.
Lemma 2.1. LetH1andH2hold. Then foru≥0∈C0,1, the problem φ
ut
atft, ut 0, 0< t <1, u0 m−2
i1
aiu ξi
, u1 k
i1
biu ξi
− s
ik1
biu ξi
− m−2
is1
biu ξi
2.1
has a unique solutionutif and only ifutcan be express as the following equation:
ut −
1
t
ωfsdsB, 2.2
whereA, Bsatisfy
φ−1A m−2
i1
aiφ−1
A− ξi
0
asfs, usds
, 2.3
B− 1
1−k
i1bis
ik1bi
k i1
bi
1
ξi
ωfsds− s
ik1
bi
1
ξi
ωfsds
m−2
is1
biφ−1
A− ξi
0
asfs, usds
,
2.4
where
ωfs φ−1
− s
0
arfr, urdrA
. 2.5
Define l φm−2
i1 ai/1−φm−2
i1 ai ∈ 0,1, then there exists a uniqueA ∈ −l1
0asfs, usds,0satisfying2.3.
Proof. The method of the proof is similar to5, Lemma 2.1, we omit the details.
Lemma 2.2. Let H1 and H2hold. If u ∈ C0,1, the unique solution of the problem 2.1 satisfies
ut≥0, t∈0,1. 2.6
Proof. According toLemma 2.1, we first have
−A s
0
arft, urdr≥0. 2.7
So
u1 B
− 1 1−k
i1bis
ik1bi
k i1
bi
1
ξi
ωfsds− s
ik1
bi
1
ξi
ωfsds
m−2
is1
biφ−1
A− ξi
0
asft, usds
1 1−k
i1bis
ik1bi
k i1
bi
1
ξi
−ωfsds− s
ik1
bi
1
ξi
−ωfsds
m−2
is1
biφ−1
−A ξi
0
asft, usds
≥ 1 1−k
i1bis
ik1bi
k i1
bi
1
ξk
−ωfsds− s
ik1
bi
1
ξk
−ωfsds
k
i1bi−s
ik1bi
1
ξk−ωfsds 1−k
i1bis
ik1bi
≥0.
2.8
Ift∈0,1, we have
ut B− 1
t
φ−1
A− s
0
arfr, urdr
ds
u1 1
t
φ−1
−A s
0
arfr, urdr
ds
≥u1
≥0.
2.9
Sout≥0, t∈0,1. The proof ofLemma 2.2is completed.
Lemma 2.3. Let H1 and H2hold. If u ∈ C0,1, the unique solution of the problem 2.1 satisfies
t∈0,1inf ut≥γu, 2.10
whereγ k
i1bi−s
ik1bi1−ξk/1−k
i1biξks
ik1biξk∈0,1, umaxt∈0,1|ut|.
Proof. Clearly
ut φ−1
A− t
0
asfs, usds
−φ−1
−A t
0
asfs, usds
≤0.
2.11
This implies that
uu0, min
t∈0,1ut u1. 2.12
It is easy to see thatut2≤ut1, for anyt1, t2∈0,1witht1≤t2. Henceutis a decreasing function on0,1. This means that the graph ofutis concave down on0,1. So we have
u ξk
−u1ξk≥ 1−ξk
u0. 2.13
Together withu1 k
i1biuξi−s
ik1biuξi−m−2
is1biuξiandut≤ 0 on0,1, we get
u0≤ k
i1biu ξk
−u1k
i1biξk−s
ik1biu ξk
u1s
ik1biξk
k
i1bi−s
ik1bi 1−ξk
≤ k
i1biu ξi
−u1k
i1biξk−s
ik1biu ξi
u1s
ik1biξk
k
i1bi−s
ik1bi
1−ξk
≤ u1 1−k
i1biξks
ik1biξk
k
i1bi−s
ik1bi 1−ξk
u1 γ .
2.14
The proof ofLemma 2.3is completed.
Lemma 2.4see8. LetKbe a cone in a Banach spaceX. LetDbe an open bounded subset ofX withDKD∩K /φandDK/K. Assume thatA:DK → Kis a compact map such thatx /AK forx∈∂DK. Then the following results hold.
1IfAx ≤ x,x∈∂DK, theniA, DK, K 1.
2If there existsx0 ∈K\ {θ}such thatx /Axλx0, for allx∈∂DKand allx > 0, then iA, DK, K 0.
3LetUbe open inXsuch thatU⊂DK. IfiA, DK, K 1 andiA, DK, K 0, thenAhas a fixed point inDK\UK. The same results hold, ifiA, DK, K 0 andiA, DK, K 1.
LetEC0,1, thenEis Banach space, with respect to the normusupt∈0,1|ut|.
Denote
K
u|u∈C0,1, ut≥0, inf
t∈0,1ut≥γu
, 2.15
whereγis the same as inLemma 2.3. It is obvious thatKis a cone inC0,1.
We defineϕt min{t,1−t}, t∈0,1and
Kρ{ut∈K:u< ρ}, K∗ρ{ut∈K:ρϕt< ut< ρ}, Ωρ
ut∈K: min
ξm−2≤t≤1ut< γρ
ut∈E:u≥0, γu ≤ min
ξm−2≤t≤1ut< γρ .
2.16
Lemma 2.5see13. Ωρdefined above has the following properties:
aKγρ⊂Ωρ⊂Kρ; b Ωρis open relative toK;
cX∈∂Ωρif and only if minξm−2≤t≤1cxt γρ;
dIfx∈∂Ωρ, thenγρ≤xt≤ρfort∈ξm−2,1.
Now, for the convenience, one introduces the following notations:
fγρρ min
ξm−2min≤t≤1
ft, u
φρ :u∈γρ, ρ
, f0ρmax
max0≤t≤1
ft, u
φρ :u∈0, ρ
,
fϕtρρ max
max0≤t≤1
ft, u
φρ :u∈ϕtρ, ρ
,
fα lim
u→αsup max
0≤t≤1
ft, u φu , fα lim
u→αinf min
ξm−2≤t≤1
ft, u
φu ,
α:∞or 0 ,
m
⎧⎨
⎩
1s
ik1bim−2
is1bi
φ−1
l11
0asds
1−k
i1bis
ik1bi
⎫⎬
⎭
−1
,
M ki1bi−s
ik1bi
1−k
i1bis
ik1bi
1
ξk
φ−1 s
0
ardr
ds −1
.
2.17
3. The Main Result
In the rest of the section, we also assume the following conditions.
A1There existρ1, ρ2∈0,∞withρ1< γρ2such that 1 ft, u>0, t∈0,1, u∈
ρ1ϕt,∞ ,
2 fϕtρρ1 1≤φm, fγρρ22≥φMγ. 3.1
A2There existρ1, ρ2∈0,∞withρ1< ρ2such that 3 ft, u>0, t∈0,1, u∈
min
γρ1, ρ2ϕt ,∞
, 4 fγρρ11 ≥φMγ, fϕtρρ2
2 ≤φm. 3.2
A3There existρ1, ρ2, ρ3∈0,∞withρ1< γρ2andρ2< ρ3such that 1 ft, u>0, t∈0,1, u∈
ρ1ϕt,∞ , 2 fϕtρρ1
1 ≤φm, fγρρ22 ≥φMγ, fϕtρρ3
3≤φm. 3.3
A4There existρ1, ρ2, ρ3∈0,∞withρ1< ρ2< γρ3such that 3 ft, u>0, t∈0,1, u∈
min
γρ1, ρ2ϕt ,∞
,
4 fγρρ11 ≥φMγ, fϕtρρ2 2 ≤φm, fγρρ33 ≥φMγ. 3.4
A5There existρ, ρ∈0,∞withρ< γρsuch that 1 ft, u>0, t∈0,1, u∈
ρϕt,∞ ,
2 fϕtρρ ≤φm, fγρρ ≥φMγ, 0≤f∞< φm. 3.5
A6There existρ, ρ∈0,∞withρ< ρsuch that 3 ft, u>0, t∈0,1, u∈
min
γρ, ρϕt ,∞
,
4 fγρρ≥φMγ, fϕtρρ ≤φm, φM< f∞≤ ∞. 3.6
Our main results are the following theorems.
Theorem 3.1. Assume that H1, H2, H3, A3 hold. Then BVP 1.1 has at least three positive solutions.
Theorem 3.2. Assume thatH1, H2, H3, A4hold. Then BVP1.1has at least two positive solutions.
Theorem 3.3. Assume thatH1, H2, H3hold and also assume thatA1orA2hold. Then BVP1.1has at least a positive solution.
Theorem 3.4. Assume thatH1, H2, H3hold and also assume thatA5orA6hold. Then BVP1.1has at least two positive solutions.
Proof ofTheorem 3.1. Without loss of generality, we suppose thatA3hold. Denote
f∗t, u
⎧⎨
⎩
ft, u, u≥ρ1ϕt, f
t, ρ1ϕt
, 0≤u < ρ1ϕt, 3.7
it is easy to check thatf∗t, u∈C0,1×0,∞,0,∞.
Now define an operatorT :K → C0,1by setting
Tut − 1 1−k
i1bis
ik1bi
k i1
bi
1
ξi
ωsds− s
ik1
bi
1
ξi
ωsds
m−2
is1
biφ−1
A− ξi
0
asf∗s, usds
− 1
t
ωsds, 3.8
where
ωs φ−1
− s
0
arf∗τ, urdrA
. 3.9
ByLemma 2.3, we haveTK⊂ K. So by applying Arzela-Ascoli’s theorem, we can obtain thatTKis relatively compact. In view of Lebesgue’s dominated convergence theorem, it is easy to prove thatTis continuous. Hence,T :K → Kis completely continuous.
Now, we consider the following modified BVP1.1:
φ u
atf∗t, ut 0, 0< t <1, u0 m−2
i1
aiu ξi
, u1 k
i1
biu ξi
− s
ik1
biu ξi
− m−2
is1
biu ξi
. 3.10
Obviously, BVP3.10has a solutionutif and only ifuis a fixed point of the operatorT. From the conditionA32, we have
fϕtρ∗ρ1 1 ≤φm, fγρ∗ρ22 ≥φMγ, fϕtρ∗ρ3 3 ≤φm. 3.11
Next, we will show thatiT, K∗ρ1, K 1.
In fact, byfϕtρ∗ρ1 1≤φm, for∀u∈∂K∗ρ1, we have
Tut − 1 1−k
i1bis
ik1bi
× k
i1
bi
1
ξi
ωsds− s
ik1
bi
1
ξi
ωsds m−2
is1
biφ−1
A− ξi
0
asf∗s, usds
− 1
t
ωsds
≤ 1
1−k
i1bis
ik1bi
k
i1
bi
1
0
φ−1
l1 1
0
arf∗r, ur
dr
ds
m−2
is1
biφ−1
l1 1
0
asf∗s, usds
1
0
φ−1
l1 1
0
arf∗r, urdr
ds
≤
1s
ik1bim−2
is1bi
φ−1
l11
0asds
1−k
i1bis
ik1bi
φ−1 φ
ρ1
φm
ρ1u.
3.12
This implies thatTu ≤ uforu∈∂K∗ρ. ByLemma 2.41, we have i
T, Kρ∗1, K
1. 3.13
Furthermore, we will show thatiT, Kρ2, K 1.
Letet≡1, fort∈0,1, thene∈∂K1. We claim that
u /Tuλe, u∈∂Ωρ2, λ >0. 3.14
In fact, if not, there existu0∈∂Ω2andλ0>0 such thatu0Tu0λ0e.
ByA3andLemma 2.1, we have fort∈0,1,
− s
0
aτf∗τ, uτdτA≤ −φ ρ2
φMγ s
0
aτdτ
, 3.15
so that
−ωs φ−1
− s
0
aτf∗τ, uτdτA
≥ρ2Mγφ−1 s
0
aτdτ .
3.16
Then, we have that
u0t Tu0t λ0et
≥ 1
1−k
i1bis
ik1bi
k i1
bi
1
ξk
−ωsds− s
ik1
bi
1
ξk
−ωsds
λ0
≥ k
i1bi−s
ik1bi
1−k
i1bis
ik1bi
ρ2Mγ 1
ξk
φ−1 s
0
ardr
dsλ0
γρ2λ0.
3.17
This implies thatγρ2 ≥ γρ2λ0, this is a contradiction. Hence, byLemma 2.42, it follows that
iT,Ωρ2, K 0. 3.18
Finally, similar to the proof ofiT, K∗ρ1, K 1, we can show thatiT, Kρ∗3, K 1.
By Lemma 2.5a and ρ1 < γρ2 and ρ2 < ρ3, we have Kρ1 ⊂ Kγρ2 ⊂ Ωρ2 ⊂ Kρ2 ⊂Kρ3. It follows fromLemma 2.43thatT has three positive fixed pointsu1, u2, u3 in Kρ∗1,Ωρ2\K∗ρ1, K∗ρ3, respectively. Therefore, BVP3.10has three positive solutionsu1, u2, u3
inKρ∗1,Ωρ2\K∗ρ1, Kρ∗3, respectively.
Then, BVP3.10has three positive solutionsu1, u2, u3∈ρ1ϕt,∞, which means that u1, u2, u3are also the positive solutions of BVP1.1.
Proof ofTheorem 3.2. The proof ofTheorem 3.2is similar to that ofTheorem 3.1, and so we omit it here. The proof ofTheorem 3.2is completed.
Proof ofTheorem 3.3. Theorem 3.3 is corollary of Theorem 3.1. The proof of Theorem 3.3 is completed.
Proof ofTheorem 3.4. We show that conditionA5implies conditionA3. Letk∈f∞, φm, then there existsr > ρsuch that maxt∈0,1ft, u ≤kφu, u ∈ r,∞since 0 ≤ f∞ < φm.
Denote
βmax
t∈0,1maxft, u:ρϕt≤u≤r
, ρ3>max
φ−1 β
φm−k
, ρ
. 3.19
Then we have
t∈0,1maxft, u≤kφu β≤kφ ρ3
β≤φmφ ρ3
, u∈
ρϕt,∞
. 3.20
This implies thatfϕtρρ3
3 ≤ φmandA3holds. Similarly conditionA6implies condition A4.
By an argument similar to thatTheorem 3.1, we can obtain the result ofTheorem 3.4.
The proof ofTheorem 3.4is completed.
4. Examples
Example 4.1. Consider the following five-point boundary value problem withp-Laplacian:
φ u
ft, u 0, 0< t <1, u0 1
128u 1
4
1 256u
1 2
1
64u 3
4
,
u1 1
8u 1
4
− 1 64u
1 2
,
4.1
wherea1 1/128, a2 1/256, a3 1/64, b1 1/8, b2 1/64, b3 0, ξ1 1/4, ξ2 1, /2, ξ33/4:
φu
⎧⎨
⎩
−u2, u≤0, u2, u >0,
ft, u
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩ 1 51t
ut− ϕt 2
30
, t, u∈0,1×0,2, 1
51t
2−ϕt 2
30
, t, u∈0,1×2,∞.
4.2
It is easy to check thatf :0,1×0,∞ → 0,∞is continuous. It follows from a direct calculation that
m
⎧⎨
⎩
1s
ik1bim−2
is1bi
φ−1
l11
0asds
1−k
i1bis
ik1bi
⎫⎬
⎭
−1
0.96,
γ k
i1bi−s
ik1bi 1−ξk
1−k
i1biξks
ik1biξk
21 250,
M ki1bi−s
ik1bi
1−k
i1bis
ik1bi
1
ξk
φ−1 s
0
ardr
ds −1
0.76.
4.3
Chooseρ11, ρ2250, it is easy to check thatρ1< γρ2and ft, u>0, t∈0,1, u∈ϕt,∞,
fρρ1
1ϕtmax
max0≤t≤1
1/51tut−ϕt/230 12
1/511130
12 2
5
< φm m20.92, t∈0,1, u∈
ϕtρ1, ρ1
"
,
fγρρ22min
3/4≤t≤1min
1/51t2−ϕt/230 2502
1/513/42−1/230
2502 1.0742
> φMγ Mγ20.004, t∈ 3
4,1 , u∈ γρ2, ρ2
"
.
4.4 It follows thatfsatisfies the conditionA1ofTheorem 3.3, then problems1.1have at least two positive solutions.
Remark 4.2. Letϕu u, the problem is second-orderm-point boundary value problem.
Remark 4.3. Let φps |s|p−2s, p > 1, the problem is boundary value problem with p- Laplacian operators.
Hence our results generalize boundary value problem withp-Laplacian operators.
Acknowledgment
This research is supported by the Doctor of Scientific Startup Foundation of Shandong University of Finance of China08BSJJ32.
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