Volume 2011, Article ID 604046,11pages doi:10.1155/2011/604046
Research Article
Existence of Positive Solutions for
Nonlocal Fourth-Order Boundary Value Problem
with Variable Parameter
Xiaoling Han, Hongliang Gao, and Jia Xu
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China
Correspondence should be addressed to Xiaoling Han,[email protected]
Received 26 November 2010; Accepted 14 January 2011
Academic Editor: M. Furi
Copyrightq2011 Xiaoling Han et al. 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.
By using the Krasnoselskii’s fixed point theorem and operator spectral theorem, the existence of positive solutions for the nonlocal fourth-order boundary value problem with variable parameter
u4t Btut λft, ut, ut, 0 < t < 1,u0 u1 1
0psusds,u0 u1
1
0qsusdsis considered, wherep, q ∈ L
10,1, λ > 0 is a parameter, andB ∈ C0,1,f ∈
C0,1×0,∞×−∞,0,0,∞.
1. Introduction
The existence of positive solutions for nonlinear fourth-order multipoint boundary value problems has been studied by many authors using nonlinear alternatives of Leray-Schauder, the fixed point theory, and the method of upper and lower solutionssee, e.g.,1–15and references therein. The multipoint boundary value problem is in fact a special case of the boundary value problem with integral boundary conditions.
Recently, Bai16studied the existence of positive solutions of nonlocal fourth-order boundary value problem
u4t βut λft, ut, ut, 0< t <1,
u0 u1
1
0
psusds,
u0 u1
1
0
qsusds.
under the assumption:
A1λ >0 and 0< β < π2,
A2f ∈C0,1×0,∞×−∞,0,0,∞,p, q∈L10,1,ps≥0,qs≥0,1
0psds <1, 1
0qssin
βsds01qssinβ1−sds <sinβ.
In this paper, we study the above generalizing form with variable parameters BVP
u4t Btut λft, ut, ut, 0< t <1,
u0 u1
1
0
psusds,
u0 u1
1
0
qsusds,
1.2
whereB∈C0,1,λ >0 is a parameter.
Obviously, BVP1.1can be regarded as the special case of BVP1.2 withBt β. Since the parametersBtis variable, we cannot expect to transform directly BVP1.2into an integral equation as in16. We will apply the cone fixed point theory, combining with the operator spectra theorem to establish the existence of positive solutions of BVP1.2. Our results generalize the main result in16.
Letβinft∈0,1Bt, and we assume that the following conditions hold throughout the paper:
H1B∈C0,1and 0< β < π2,
H2f ∈ C0,1×0,∞×−∞,0,0,∞, p, q ∈ L10,1, ps ≥ 0, qs ≥ 0 and 1
0psds <1, 1
0qssin
βsds01qssinβ1−sds <sinβ.
2. The Preliminary Lemmas
Setλ10,−π2 < λ2−β <0 andδ11− 1
0
psds, δ2sin
β−
1
0
qssin
βsds−
1
0
qssin
β1−sds. 2.1
ByH1,H2, we getδi/0,i1,2. Denote byK1t, sthe Green’s function of the problem
−ut λ1ut 0, 0< t <1,
u0 u1
1
0
psusds
andK2t, sthe Green’s function of the problem
−ut λ2ut 0, 0< t <1,
u0 u1
1
0
qsusds.
2.3
Then, carefully calculation yield
K1t, s G1t, s ρ1 1
0
G1s, xpxdx,
K2t, s G2t, s ρ2t 1
0
G2s, xqxdx,
G1t, s ⎧ ⎨ ⎩
t1−s, 0≤t≤s≤1,
s1−t, 0≤s≤t≤1,
G2t, s ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sinβtsinβ1−s
βsinβ , 0≤t≤s≤1,
sinβssinβ1−t
βsinβ , 0≤s≤t≤1,
ρ1 1
δ1, ρ2t
sinβtsinβ1−t
δ2 .
2.4
Lemma 2.1see16. Suppose that (A1), (A2) hold. Then, for anyh∈C0,1,usolves the problem
u4t βut ht, 0< t <1,
u0 u1
1
0
psusds,
u0 u1
1
0
qsusds,
2.5
if and only ifut 0101K1t, sK2s, τhτdτds.
LetY C0,1, Y {u∈Y :ut≥0, t∈0,1}, andu0 max0≤t≤1|ut|, foru∈Y.
X{u∈C20,1:u0 u1 1
0psusds,u0 u1 1
0qsusds},u1u0, u2u0u1, foru∈X.
Lemma 2.2see16. · 1≤ · 2≤1δ1 · 1and (X, · 2) is a Banach space.
Lemma 2.3see5. Assume that (A1), (A2) hold. Then,
iKit, s≥0, fort, s∈0,1,i1,2;Kit, s>0, fort, s∈0,1,i1,2,
iiGit, s≥biGit, tGis, s,Git, s≤CiGis, sfort, s∈0,1,i1,2,
whereC11,b11;C21/sin
β,b2
βsinβ.
Denote
di min
1/4≤t≤3/4biGit, t i1,2,
ξ min1/4≤t≤3/4ρ2t
max1/4≤t≤3/4ρ2t,
Di max
t∈0,1 1
0
Kit, sds i1,2.
2.6
Computations yield the following results.
Lemma 2.4see3. D1
i maxt∈0,1 1
0Git, sds >0 i1,2 iwhenλi>0,Di1 1/λi1−1/cosωi/2,
iiwhenλi0,Di11/8,
iiiwhen−π2< λ
i<0,D1i 1/λi1−1/cosωi/2.
Lemma 2.5see16. Suppose that (A1), (A2) hold andρ2t,di,ξare given as above. Then,
imaxt∈0,1ρ2t ρ21/2, ii0< di<1,0< ξ <1.
By Lemmas2.4and2.5,D2maxt∈0,1 1
0K21/2, sds. Takeθmin{d1, d2ξ/C2}, byLemma 2.5, 0< θ <1. Define
Tht
1
0 1
0
K1t, sK2s, τhτdτ ds, t∈0,1,
Aht Tht −
1
0
K2t, τhτdτ, t∈0,1.
2.7
Lemma 2.6. T :Y → X, · 2is completely continuous, andT ≤D2.
Lemma 2.7see17. Let E be a Banach space,P ⊆Ea cone, andΩ1,Ω2be two bounded open sets
ofEwith0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2. Suppose that A : P∩Ω2\Ω1 → P is a completely continuous
operator such that either
iAx ≤ x, x∈P∩∂Ω1andAx ≥ x,x∈P∩∂Ω2, or
iiAx ≥ x, x∈P∩∂Ω1andAx ≤ x,x∈P∩∂Ω2
holds. Then,Ahas a fixed point inP∩Ω2\Ω1.
3. The Main Results
Suppose that K1,K2,G2,ρ2,C2,θ, andD2, are defined as in Section 2, we introduce some notations as follows:
A
1
0 1
0
K1s, sK2s, τdτ ds, B
1
0
G2s, s ρ2
1 2
1
0
G2s, xqxdx
ds,
K sup
t∈0,1
Bt−β, LD2K, η0 1−L
AC2B, η1
1
θ13//44K21/2, τdτ
,
f0lim sup
|u||v| →0 max
t∈0,1
ft, u, v
|u||v| , f0|lim infu||v| →0t∈1min/4,3/4
ft, u, v
|u||v| ,
f∞ lim sup
|u||v| →∞
max
t∈0,1
ft, u, v
|u||v|, f∞|ulim inf||v| →∞t∈1min/4,3/4
ft, u, v
|u||v| .
3.1
Theorem 3.1. Assume that (H1), (H2) hold andL D2K < 1. Then BVP1.2has at least one
positive solution if one of the following cases holds:
if0<1/λη0,f∞>1/λη1,
iif
0>1/λη1,f∞<1/λη0.
Proof. For anyh∈Y, consider the following BVP:
u4t Btut ht, 0< t <1,
u0 u1
1
0
psusds,
u0 u1
1
0
qsusds.
It is easy to see that the above question is equivalent to the following question:
u4t βut −Bt−βut ht, 0< t <1,
u0 u1
1
0
psusds,
u0 u1
1
0
qsusds.
3.3
For anyv∈ X, letGv −Bt−βv. Obviously, the operatorG :X → Y is linear. ByLemma 2.2, for allv ∈X,t ∈0,1,|Gvt| ≤ Bt−βv1 ≤ Kv1 ≤ Kv2. Hence
Gv0 ≤ Kv2, and soG ≤ K. On the other hand,u ∈C20,1∩C40,1is a solution of 3.3if and only ifu∈XsatisfiesuTGuh, that is,
u∈X, I−TGuTh. 3.4
Owing toG:X → Y andT:Y → X, the operatorI−TGmapsXintoX. FromT ≤D2by
Lemma 2.6together withG ≤Kand conditionL <1, applying operator spectral theorem, we have that theI−TG−1exists and is bounded. LetH I−TG−1T, then3.4is equivalent touHh. By the Neumann expansion formula,Hcan be expressed by
HITG· · · TGn· · ·TT TGT· · · TGnT· · ·. 3.5
The complete continuity ofT with the continuity ofI−TG−1yields that the operatorH :
Y → X is completely continuous. For allh∈ Y, letu Th, thenu ∈ X∩Y, andu < 0. So, we haveGut −Bt−βut≥0,t∈0,1. Hence,
∀h∈Y, GTht≥0, t∈0,1, 3.6
and soTGTht TGTht≥0,t∈0,1.
Assume that for allh∈Y,TGkTht≥0,t∈0,1, leth1GTh, by3.6we have
h1 ∈Y, and soTGk1Tht TGkTGTht TGkTh1t ≥ 0,t ∈ 0,1. Thus by induction, it follows thatTGnTht ≥ 0, for alln ≥ 1,h ∈ Y,t ∈ 0,1. By3.5, for all
h∈Y, we have
Hht Tht TGTht · · · TGnTht · · · ≥Tht, t∈0,1,
Hht Aht AGTht · · ·AGTGn−1Tht · · ·
≤Aht Tht≤0, t∈0,1,
3.7
On the other hand, for allh∈Y, we have
Hht≤Tht |TG|Tht · · ·|TG|nTht · · ·
≤1L· · ·Ln· · ·Tht
1
1−LTht t∈0,1,
3.8
Hht≤ |Aht||AGTht|· · ·AGTGn−1
Tht· · ·
≤ |Aht|L|Aht|· · ·Ln|Aht|· · ·
1L· · ·Ln· · ·|Aht|
1
1−LTh
t t∈0,1,
3.9
Hh0≥ Th0, Hh0≤ 1
1−LTh0,
Hh1≥ Th1, Hh1≤ 1
1−LTh1.
3.10
For anyu ∈ Y, defineFu λft, u, u. ByH1andH2, we have thatF : Y → Y is continuous. It is easy to see thatu∈C20,1∩C40,1being a positive solution of BVP1.2 is equivalent tou∈Ybeing a nonzero solution equation as follows:
uHFu. 3.11
Let Q HF. Obviously, Q : Y → Y is completely continuous. We next show that the operatorQhas a nonzero fixed point inY. Let
P
u∈X:u≥0, u≤0, min
1/4≤t≤3/4ut≥1−Ld1u0, 1/max4≤t≤3/4u
t≤ −1−Ld2ξ
C2 u
0
.
3.12
It is easy to know thatPis a cone inX,P ⊂Y. Now, we showQP⊂P.
Forh∈Y, by2.7, there isTh≥0,Th ≤0. Hence, by3.7,Qu≥0,Qu≤0,u∈
P. By proof of Lemma 2.5 in16,
min
1/4≤t≤3/4Tht≥d1Th0, 1/max4≤t≤3/4Th
t≤ −d2ξ
C2
Th
By3.7and3.10,
min
1/4≤t≤3/4Qut≥1/min4≤t≤3/4TFut≥d1TFu0≥1−Ld1Qu0,
max 1/4≤t≤3/4Qu
t≤ max
1/4≤t≤3/4TFu
t≤ −d2ξ
C2
TFu
0≤ −1−L
d2ξ
C2
Qu
0.
3.14
ThusQP ⊂P.
iSincef0<1/λη0, by the definition off0, there existsr1>0 such that
max 0≤t≤1,|ut||ut|≤r1f
t, ut, ut≤ r1
λη0. 3.15
LetΩr1 {u∈P:u2< r1}, one has
ft, ut, ut≤ r1
λη0, u∈∂Ωr1, t∈0,1. 3.16
So, by3.10, we get
Qu0HFu0≤ 1
1−LTFu0
λ 1−L
1
0 1
0
K1t, sK2s, τf
τ, uτ, uτdτ ds
0
≤ r1η0 1−L
1
0 1
0
K1s, sK2s, τdτ ds≤ Aη0r1 1−L,
Qu1HFu1≤ 1
1−LTFu1
≤λC2 1 1−L
1
0
G2τ, τ ρ2
1 2
1
0
G2τ, xqxdx
fτ, uτ, uτdτ
≤ C2Bη0r1 1−L .
3.17
Hence, foru∈∂Ωr1,
Qu2HFu2≤ 1
1−LTFu2≤
ABC2η0r1
On the other hand, sincef
∞>1/λη1, there existsr
2> r1>0 such that
min
1/4≤t≤3/4,θ|ut||ut|≥r 2
ft, ut, ut
|ut||ut| ≥
1
λη1. 3.19
Chooser2 > 1/θr2, let Ωr2 {u ∈ P : u2 < r2}. For u ∈ ∂Ωr2,t ∈ 1/4,3/4, there is r2 ≤θr2≤ |ut||ut| ≤r2. Thus,
ft, ut, ut≥ θr2
λ η1, u∈∂Ωr2, t∈
1 4, 3 4 . TFu 1 2 λ 1 0 K2 1 2, τ
fτ, uτ, uτdτ
≥λ
3/4
1/4
K2
1 2, τ
fτ, uτ, uτdτ ≥η1θr2 3/4
1/4
K2
1 2, τ
dτ r2.
3.20
Hence, foru∈Ωr2,
Qu2≥ TFu2≥ TFu 1 2
≥r2u2. 3.21
By the use of the Krasnoselskii’s fixed point theorem, we know there existsu0∈Ω2\Ω1such thatQu0u0, namely,u0is a solution of1.2and satisfiedu0≥0,u0≤0,r1≤ u02≤r2.
iiThe proof is similar toi, so we omit it.
Corollary 3.2. Assume that (H1), (H2) hold, andL < 1. Then that1.2has at least two positive solution, iffsatisfy
if0<1/λη0,f∞<1/λη0,
iiThere existsR0>0such thatft, u, v≥θR0/λη1, fort∈1/4,3/4,|u||v| ≥θR0.
Proof. By the proof ofTheorem 3.1, we know that1from the conditionf0 <1/λη0, there existsΩr1 {u∈ P :u2 < r1}, such thatQu2 ≤ u2,u ∈ ∂Ωr1,2from the condition f∞<1/λη0, there existsΩr2 {u∈P:u2< r2},r2> r1, such thatQu2≤ u2,u∈∂Ωr2,
3 from the conditionii, there exists Ωr3 {u ∈ P : u2 < r3},r2 > r3 > r1, such that
Qu2 ≥ u2,u∈∂Ωr3. By the use of Krasnoselskii’s fixed point theorem, it is easy to know
that1.2has at least two positive solutions.
Corollary 3.3. Assume (H1), (H2) hold, andL < 1. Then problem1.2has at least two positive solution, iffsatisfy
if
0>1/λη1,f∞>1/λη1,
iiThere existsR0>0such thatft, u, v≤θR0/λη0, fort∈0,1,|u||v| ≤R0.
Example 3.4. Consider the following boundary value problem
u4t
π2 4 t
ut π218ut−ut−17.9 sinut−ut, 0< t <1,
u0 u1
1
0
susds,
u0 u1 0.
3.22
In this problem, we know thatBt π2/4t,pt t,qt 0, λ π2, then we can get
C11,C21,ρ11,ρ2 √
2,βπ2/4,K1,D 2 4
√
2−1/π2. Further more, we obtain
A 48−13π2/π3,B2/π2, thenη
0 1−Lπ3/48−11π,η14π2/ √
2 cosπ/8−1, so
f0 0.1< 1
π2η0≈0.19, f∞18> 1
π2η1≈13.3. 3.23
Thus,Bt,pt,qt, andf satisfy the conditions ofTheorem 3.1, and there exists at least a positive solution of the above problem.
Acknowledgments
This work is sponsored by the NSFCno. 11061030, NSFCno. 11026060, and nwnu-kjcxgc-03-69, 03-61.
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