Volume 2009, Article ID 937064,19pages doi:10.1155/2009/937064
Research Article
Positive Solutions to Singular and
Delay Higher-Order Differential Equations on Time Scales
Liang-Gen Hu,
1Ti-Jun Xiao,
2and Jin Liang
31Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
2School of Mathematical Sciences, Fudan University, Shanghai 200433, China
3Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
Correspondence should be addressed to Jin Liang,[email protected] Received 21 March 2009; Accepted 1 July 2009
Recommended by Juan Jos´e Nieto
We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.
Copyrightq2009 Liang-Gen Hu 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.
1. Introduction
In this paper, we are concerned with the following singular three-point boundary value problemBVP for shortfor delay higher-order dynamic equations on time scales:
−1nuΔ2nt wtft, ut−c, t∈a, b, ut ψt, t∈a−c, a,
uΔ2ia−βi1uΔ2i1a αi1uΔ2i, γi1uΔ2i uΔ2ib, 0≤i≤n−1,
1.1
wherec ∈ 0,b−a/2, ∈ a, b,βi ≥ 0, 1 < γi < b−aβi/ −aβi, 0 ≤ αi <
b−γi γi −1a−βi/b−,i 1,2, . . . , n and ψ ∈ Ca−c, a. The functional w : a, b → 0,∞is continuous andf : a, b×0,∞ → 0,∞is continuous. Our
nonlinearityw may have singularity att aand/ort b,and f may have singularity at u0.
To understand the notations used in1.1, we recall the following definitions which can be found in1,2.
aA time scaleTis a nonempty closed subset of the real numbersR.Thas the topology that it inherits from the real numbers with the standard topology. It follows that the jump operatorsσ, ρ:T → T,
σt inf{τ∈T:τ > t}, ρt sup{τ∈T:τ < t} 1.2 supplemented by inf∅ : supT and sup∅ : infTare well defined. The point t ∈ Tis left-dense, left-scattered, right-dense, right-scattered ifρt t,ρt < t, σt t,σt< t, respectively. IfThas a left-scattered maximumt1right-scattered minimumt2, defineTkT− {t1}Tk T− {t2}; otherwise, setTkTTk T.
By an intervala, bwe always mean the intersection of the real intervala, bwith the given time scale, that is,a, b∩T. Other types of intervals are defined similarly.
bFor a functionf :T → Randt∈Tk, theΔ-derivative off att, denoted byfΔt, is the numberprovided it existswith the property that, given anyε >0, there is a neighborhoodU⊂Toftsuch that
fσt−fs−fΔtσt−s≤ε|σt−s|, ∀s∈U. 1.3
cFor a functionf :T → Randt∈Tk, the∇-derivative off att, denoted byf∇t, is the numberprovided it existswith the property that, given anyε >0, there is a neighborhoodU⊂Toftsuch that
f
ρt
−fs−f∇t
ρt−s≤ερt−s, ∀s∈U. 1.4
dIfFΔt ftΦ∇t gt,then we define the integral t
a
fΔ Ft−Fa
t
a
g∇ Φt−Φa . 1.5
Theoretically, dynamic equations on time scales can build bridges between continuous and discrete mathematics. Practically, dynamic equations have been proposed as models in the study of insect population models, neural networks, and many physical phenomena which include gas diffusion through porous media, nonlinear diffusion generated by nonlinear sources, chemically reacting systems as well as concentration in chemical of biological problems 2. Hence, two-point and multipoint boundary value problems for dynamic equations on time scales have attracted many researchers’ attentionsee, e.g.,1–19 and references therein. Moreover, singular boundary value problems have also been treated in many paperssee, e.g.,4,5,12–14,18and references therein.
In 2004, J. J. DaCunha et al. 13 considered singular second-order three-point boundary value problems on time scales
uΔΔt ft, ut 0, 0,1∩T, u0 0, u
p u
σ21 1.6
and obtained the existence of positive solutions by using a fixed point theorem due to Gatica et al.14, wheref :0,1×0,∞ → 0,∞is decreasing inufor everyt∈ 0,1and may have singularity atu0.
In 2006, Boey and Wong11were concerned with higher-order differential equation on time scales of the form
−1n−1yΔnt −1p1F t, y
σn−1t
, t∈a, b, yΔia 0, 0≤i≤p−1,
yΔiσb 0, p≤i≤n−1,
1.7
wherep, nare fixed integers satisfyingn ≥ 2, 1 ≤ p ≤ n−1. They obtained some existence theorems of positive solutions by using Krasnosel’skii fixed point theorem.
Recently, Anderson and Karaca8studied higher-order three-point boundary value problems on time scales and obtained criteria for the existence of positive solutions.
The purpose of this paper is to investigate further the singular BVP for delay higher- order dynamic equation 1.1. By the use of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to the BVP 1.1 are established.
The paper is organized as follows. InSection 2, we give some lemmas, which will be required in the proof of our main theorem. In Section 3, we prove some theorems on the existence of positive solutions for BVP1.1. Moreover, we give an example to illustrate our main result.
2. Lemmas
For 1 ≤ i ≤ n, let Git, s be Green’s function of the following three-point boundary value problem:
−uΔΔt 0, t∈a, b,
ua−βiuΔa αiu, γiu ub, 2.1
where ∈a, bandαi, βi, γisatisfy the following condition:
C
βi≥0, 1< γi< b−aβi
−aβi, 0≤αi< b−γi γi−1
a−βi
b− . 2.2
Throughout the paper, we assume thatσb b.
From8, we know that for anyt, s∈a, b×a, band 1≤i≤n,
Git, s
⎧⎨
⎩
Gi1t, s, s∈a, ,
Gi2t, s, s∈, b, 2.3 where
Gi1t, s 1 di
⎧⎨
⎩
γit− b−t
σs βi−a
, σs≤t,
γiσs−ω b−σs
tβi−a
αi−bt−σs, t≤s,
Gi2t, s 1 di
⎧⎨
⎩
σs1−αi αiβi−a
b−t γi
−aβi
t−σs, σs≤t, t1−αi αiβi−a
b−σs, t≤s, di
γi−1 a−βi
1−αib αi−γi
.
2.4 The following four lemmas can be found in8.
Lemma 2.1. Suppose that the condition (C) holds. Then the Green function ofGit, sin2.3satisfies Git, s>0, t, s∈a, b×a, b. 2.5 Lemma 2.2. Assume that the condition (C) holds. Then Green’s functionGit, sin2.3satisfies
Git, s≤max{Gib, s, Giσs, s}, t, s∈a, b×a, b. 2.6 Remark 2.3. 1Ifs∈γi−aβi−αi−βia/1−αi, b, s≤t, we know thatGit, sis nonincreasing intand
Gib, s
Giσs, s γi
−aβi
b−σs σs1−αi αiβi−a
b−σs
≥ γi
−aβi
b1−αi αiβi−a >0.
2.7
Therefore, we have
Gib, s≤Git, s≤Giσs, s≤δiGib, s, 2.8 where
δi b1−αi αiβi−a γi
−aβi
>1. 2.9
2Iftandssatisfy the other cases, then we get thatGit, sis nondecreasing intand
Git, s≤Gib, s. 2.10
Lemma 2.4. Assume that (C) holds. Then Green’s functionGit, sin 2.3 verifies the following inequality:
Git, s≥min t−a
b−a , b−t γib−a
Gib, s
≥min
t−a
δib−a , b−t γib−a
max{Gib, s, Giσs, s}.
2.11
Remark 2.5. Ifs∈,γi−aβi−αi−βia/1−αi, s≤t, then we find γi−1
a−βi
1−αiσs αi−γi
<0. 2.12
So there exists a misprint on8, Page 2431, line 23. From2.3, it follows that
Git, s Gib, s
σs1−αi αiβi−a
b−t γi
−aβi
t−σs γi
−aβi
b−σs
≥
βi−a
b−t γi
−aβi
t−σs γi
−aβi
b−a ≥ b−t γib−a.
2.13
Consequently, we get
Git, s≥ b−t
γib−aGib, s. 2.14
Ifs∈γi−aβi−αi−βia/1−αi, b,s≤t, then, from2.8, we obtain
Git, s≥ t−a
b−aGib, s≥ t−a
δib−aGiσs, s. 2.15
Remark 2.6. If we sethit:min{t−a/δib−a,b−t/γib−a}, then we have
Git, s≥hitmax{Gib, s, Giσs, s}, t, s∈a, b×a, b. 2.16
Denote
Gi·, s max
t∈a,b|Git, s|, s∈a, b. 2.17
Thus we have
Git, s≥hitGi·, s, t, s∈a, b×a, b. 2.18 Lemma 2.7. Assume that condition (C) is satisfied. ForGit, sas in2.3, putH1t, s:G1t, s and recursively define
Hjt, s b
a
Hj−1t, rGjr, sΔr 2.19
for 2≤j≤n. ThenHnt, sis Green’s function for the homogeneous problem
−1nuΔ2nt 0, t∈a, b, uΔ2ia−βi1uΔ2i1a αi1uΔ2i, γi1uΔ2i uΔ2ib, 0≤i≤n−1.
2.20
Lemma 2.8. Assume that (C) holds. Denote
K:n−1
j1
kj, L:n−1
j1
lj, 2.21
then Green’s functionHnt, sinLemma 2.7satisfies
h1tLGn·, s ≤Hnt, s≤KGn·, s, t, s∈a, b×a, b, 2.22
where
kj b
a
Gj·, sΔs >0, lj b
a
Gj·, shj1sΔs, 1≤j ≤n−1. 2.23
Proof. We proceed by induction onn≥2. We denote the statement byPn. FromLemma 2.7, it follows that
H2t, s
b
a
H1t, rG2r, sΔr
≤ b
a
G1·, rG2·, sΔrk1G2·, s,
2.24
and from2.18, we have
H2t, s b
a
H1t, rG2r, sΔr
≥ b
a
h1tG1·, r ×h2rG2·, sΔr h1tl1G2·, s.
2.25
SoP2is true.
We now assume thatPmis true for some positive integerm≥2. FromLemma 2.7, it follows that
Hm1t, s
b
a
Hmt, rGm1r, sΔr
≤ b
a
Hmt, rGm1r, sΔr
≤
⎛
⎝b
a m−1
j1
kj× Gm·, rΔr
⎞
⎠Gm1·, s
m
j1
kjGm1·, s,
Hm1t, s b
a
Hmt, rGm1r, sΔr
≥ b
a
h1t×m−1
j1
ljGm·, rhm1rGm1·, sΔr
h1tm
j1
ljGm1·, s.
2.26
SoPm1holds. ThusPnis true by induction.
Lemma 2.9see20. LetE, · be a real Banach space and P ⊂ Ea cone. Assume thatT : Pζ,η → Pis completely continuous operator such that
iTu uforu∈∂PζandTu uforu∈∂Pη, iiTu uforu∈∂PζandTu uforu∈∂Pη. ThenT has a fixed pointu∗∈Pwithζ≤ u∗ ≤η.
3. Main Results
We assume that{am}m≥1and{bm}m≥1are strictly decreasing and strictly increasing sequences, respectively, with limm→ ∞ama, limm→ ∞bmbanda1 < b1. A Banach spaceECa, bis the set of real-valued continuousin the topology ofTfunctionsutdefined ona, bwith the norm
u max
t∈a,b|ut|. 3.1
Define a cone by
P
u∈E:ut≥ h1tL
K u, t∈a, b
. 3.2
Set
Pξ{u∈P :u< ξ}, ∂Pξ {u∈P :uξ}, ξ >0, Pζ,η
u∈P :ζ <u< η
, 0< ζ < η,
Y1{t∈a, b:t−c < a}, Y2{t∈a, b:t−c≥a}, Ym{t∈Y2 :t−c∈a, am∪bm, b}.
3.3
Assume that
C1ψ :a−c, a → 0,∞is continuous;
C2we have
0< K q
p
Gn·, swsΔs, K b
a
Gn·, swsΔs <∞, 3.4
for constantspandqwithac < p < q < b;
C3the function f : a, b× 0,∞ → R is continuous and w : a, b → R is continuous satisfying
mlim→ ∞sup
u∈Pζ,η
K
Ym
Gn·, swsfs, us−cΔs0, ∀0< ζ < η. 3.5
We seek positive solutionsu:a, b → R, satisfying1.1. For this end, we transform 1.1into an integral equation involving the appropriate Green function and seek fixed points of the following integral operator.
Define an operatorT :Ca, b → Ca, bby
Tut b
a
Hnt, swsfs, us−cΔs, ∀u∈Ca, b, 3.6
whereCa, b {u∈Ca, b| ut≥0, t∈a, b}.
Proposition 3.1. Let (C1), (C2), and (C3) hold, and letζ,ηbe fixed constants with 0< ζ < η. Then T :Pζ,η → Pis completely continuous.
Proof. We separate the proof into four steps.
Step 1. For eachu∈Pζ,η,Tuis bounded.
By conditionC3, there exists some positive integerm0satisfying
sup
u∈Pζ,η
K
Ym0
Gn·, swsfs, us−cΔs≤1, 3.7
where
Ym0{t∈Y2:t−c∈a, am0∪bm0, b}; 3.8
here, we used the fact that for eachu∈Pζ,ηandt∈am0c, bm0c∩a, b,
η≥ut−c≥ h1t−cL
K u ≥ζmin
h1am0L
K ,h1bm0L
K ,h1b−cL K
ζh >0, 3.9
where
hmin
h1am0L
K ,h1bm0L
K ,h1b−cL K
. 3.10
Set
D:max f
t, ψt−c :t∈Y1
, Q:max
ft, ut−c:t∈Y2, ζh≤ut−c≤η
. 3.11
Then we obtain
Tut≤ sup
t∈a,b
sup
u∈Pζ,η
b
a
Hnt, swsfs, us−cΔs
≤Ksup
u∈Pζ,η
Y1
Gn·, swsfs, us−cΔs
sup
u∈Pζ,η
K
Ym0
Gn·, swsfs, us−cΔs
sup
u∈Pζ,η
K
Y2\Ym0
Gn·, swsfs, us−cΔs
≤1max{D, Q}K b
a
Gn·, swsΔs <∞.
3.12
Consequently,Tuis bounded and well defined.
Step 2. T :Pζ,η → P. For everyu∈Pζ,η, we get from2.22
Tu sup
t∈a,b
b
a
Hnt, swsfs, us−cΔs
≤K b
a
Gn·, swsfs, us−cΔs.
3.13
Then by the above inequality
Tut b
a
Hnt, swsfs, us−cΔs
≥ b
a
h1tLGn·, swsfs, us−cΔs
≥ h1tL
K Tu.
3.14
This leads toTu∈P.
Step 3. We will show thatT : Pζ,η → P is continuous. Let{um}m≥1 be any sequence inPζ,η
such that limm→ ∞umu∈Pζ,η. Notice also that asm → ∞,
φms fs, ums−c−fs, us−cws−→0, fors∈ac, b, fs, ums−c−fs, us−cws
f
s, ψs−c
−f
s, ψs−cws 0, fors∈a, ac,
Y2
Hnt, sφmsΔs≤ sup
x∈Pζ,η
2K
Y2
Gn·, swsfs, xsΔs <∞.
3.15
Now these together withC2and the Lebesgue dominated convergence theorem10yield that asm → ∞,
Tum−Tu sup
t∈a,b
b
a
Hnt, swsfs, ums−c−fs, us−cΔs−→0. 3.16 Step 4. T :Pζ,η → Pis compact.
Define
wmt
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
min{wt, wam}, a≤t≤am,
wt, am≤t≤bm,
min{wt, wbm}, bm≤t≤b,
fmt, ut−c
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎩ f
t, ψt−c
, a≤t < ac,
min
ft, ut−c, ft, uam
, ac≤t≤amc,
ft, ut−c, t∈amc, bmc∩a, b, min
ft, ut−c, ft, ubm
, t∈bmc, b∩a, b,
3.17
and an operator sequence{Tm}for a fixedmby
Tmut b
a
Hnt, swmsfms, us−cΔs, ∀t∈a, b. 3.18
Clearly, the operator sequence{Tm} is compact by using the Arzela-Ascoli theorem 3, for eachm∈N. We will prove thatTmconverges uniformly toTonPζ,η. For anyu∈Pζ,η,
we obtain
Tmu−Tu sup
t∈a,b
b
a
Hnt, s
wmsfms, us−c−wsfs, us−c Δs
≤K b
a
Gn·, swmsfms, us−c−wsfs, us−cΔs
≤K
Y1
Gn·, s|wms−ws|f
s, ψs−c Δs
K
Y2
Gn·, swmsfms, us−c−wsfs, us−cΔs.
3.19
From C1,C2, and the Lebesgue dominated convergence theorem10, we see that the right-hand side3.19can be sufficiently small formbeing big enough. Hence the sequence {Tm}of compact operators converges uniformly toT onPζ,η so that operatorT is compact.
Consequently,T : Pζ,η → P is completely continuous by using the Arzela-Ascoli theorem 3.
Proposition 3.2. It holds thatv∈Pζ,ηis a solution of1.1if and only ifTvv.
Proof. Ifv∈Pζ,ηandTvv, then we have
−1nvΔ2nt −1nTvΔ2nt wtft, vt−c, 3.20
and for any 0≤i≤n−1,
vΔ2ia−βi1vΔ2i1a αi1vΔ2i, γi1vΔ2i vΔ2ib. 3.21
From8, Lemma 3.1, we know thatvt≥0 ona, b. So we conclude thatvis the solution of BVP1.1.
For convenience, we list the following notations and assumptions:
R
μK q
p
Gn·, swsΔs
−1
, μmin
h1 p
L K ,h1
q L K
;
κ
K b
a
Gn·, swsΔs −1
;
3.22
fμξξ : ft, ut−c
ut , t∈
p, q , u∈
μξ, ξ
; 3.23
fρζ: ft, ut−c
ut , t∈Y2, u∈ ρ, ζ
; 3.24
S ρ
sup
u∈∂Pρ
K
Y2
Gn·, swsfs, us−cΔs, ρ >0. 3.25
From conditionC2and3.12, we haveSρ<∞.
Theorem 3.3. Assume that there exist positive constants ρ, ζ, ξ, r with ζ < μξ,r < κ and ζ ≥ κSρ/κ−rsuch that
ifμξξ > Randfρζ< r;
iift, ψt−c/ut< r, for allt∈Y1andu∈ρ, ζ.
If (C1), (C2), and (C3) hold, then the boundary value problem1.1has at least one positive solution
usuch that
ut
⎧⎨
⎩
ψt, ift∈a−c, a, u∗t, ift∈a, b,
ζ≤ u∗ ≤ξ.
3.26
Proof. Define the operatorT : Pζ,ξ → P by3.6. From iand 3.23, it follows that there existsε1>0 such that
ft, ut−c≥Rε1ut, fort∈ p, q
, u∈ μξ, ξ
. 3.27
We claim that
Tu u, ∀u∈∂Pξ. 3.28
If it is false, then there exists someu1 ∈∂PξwithTu1≤u1, that is,u1−Tu1 ∈Pwhich implies thatu1t≥Tu1tfort∈a, b.
Set
λmin
u1t:t∈ p, q
≥min h1
p L K ,h1
q L K
u1μξ. 3.29
We know from2.22and3.27that fort∈p, q, u1t≥Tu1t
b
a
Hnt, swsfs, u1s−cΔs
Y1
Hnt, swsfs, u1s−cΔs
Y2
Hnt, swsfs, u1s−cΔs
≥ q
p
Hnt, swsfs, u1s−cΔs
≥min h1
p , h1
q L
q
p
Gn·, swsfs, u1s−cΔs
≥Rε1min
t∈p,qu1tμK q
p
Gn·, swsΔs
≥λR
μK q
p
Gn·, swsΔs
λε1μK q
p
Gn·, swsΔs
λλε1μK q
p
Gn·, swsΔs,
3.30
the first inequality ofC2implies that
u1t> λ, ∀t∈ p, q
. 3.31
Clearly,3.31contradicts3.29. This means that3.28holds.
Next we will show that
Tu u, ∀u∈Pζ. 3.32
Suppose on the contrary that there exists someu2∈∂Pζwithu2≤Tu2for allt∈a, b.
Fort, u∈Y2×ρ, ζ, fromiand3.24, there existsε2>0 such that
ft, ut−c≤r−ε2ut. 3.33
and fort, u∈Y1×ρ, ζ, there existsε2>0, fromii, such that f
t, ψt−c
≤r−ε2ut. 3.34
Put
Y3:
t∈Y2:u2t> ρ
, u2t
⎧⎨
⎩ min
u2t, ρ
, t∈Y2,
ρ, t∈Y1. 3.35
IfY3 ∅, then we takeu2t ρ. It is easy to see thath1t−cLζ/K ≤u2t−c≤ u2 ζ fort∈Y2andu2t∈Ca, b,u2ρ, that is,u2∈∂Pρ. From3.33and3.34, we find that
Tu2 sup
t∈a,b
b
a
Hnt, swsfs, u2s−cΔs
≤K b
a
Gn·, swsfs, u2s−cΔs K
Y1
Gn·, swsf
s, ψs−c ΔsK
Y3
Gn·, swsfs, u2s−cΔs
K
Y2\Y3
Gn·, swsfs, u2s−cΔs
≤r−ε2max
t∈Y1
u2t
Y1
Gn·, swsΔs
sup
t,u2∈Y3×ρ,ζft, u2t−cK
Y3
Gn·, swsΔs
sup
u2∈∂Pρ
K
Y2
Gn·, swsfs,u2s−cΔs
≤ζrK b
a
Gn·, swsΔsS ρ
−ζε2K b
a
Gn·, swsΔs ζrκ−1−ζε2κ−1S
ρ
< ζu2
3.36
yielding a contradiction with u2 ≤ Tu2 for all t ∈ a, b. This means that 3.32 holds.
Therefore, from3.28,3.32andLemma 2.9, we conclude that the operatorT has at least one fixed pointu∗ ∈Pζ,ξ. From the definition of the conePand2.18, we see thatu∗t >0 for allt∈a, b. Thus,Proposition 3.2implies thatu∗is a solution of BVP1.1. So we obtain the desired result.
Adopting the same argument as inTheorem 3.3, we obtain the following results.
Corollary 3.4. Let ρ, ζ, r, fρζ be as in Theorem 3.3. Suppose that (ii) of Theorem 3.3 holds and limξ→ ∞fμξξ ∞. If (C1), (C2), and (C3) holds , then boundary value problem1.1has at least one positive solutionu∈Pζ,ηsuch that
ut
⎧⎨
⎩
ψt, ift∈a−c, a, u∗∗t, ift∈a, b, ζ≤ u∗∗ ≤η, ζ < μη.
3.37
Theorem 3.5. Assume that there exist positive constantsρi, ζi, ξi, r withζi < μξi,r < κ andζi ≥ κSρi/κ−r,i1,2, . . . , msuch that
iiifμξξi
i> Randfρζii < r;
ivft, ψt−c/ut< r, for allt∈Y1andu∈ρi, ζi.
If (C1), (C2), and (C3) hold, then boundary value problem 1.1has at least m positive solutions
ui∈Pζi,ξi such that fori1,2, . . . , m
uit
⎧⎨
⎩
ψt, ift∈a−c, a, u∗it, ift∈a, b
ζ≤u∗i≤ξ.
3.38
Example 3.6. LetTR. Consider the following singular three-point boundary value problems for delay four-order dynamic equations:
u4t ft, ut−1 0, t∈0,4,
u0 1
2u1, 2u1 u4,
u0 1
2u1, 2u1 u4, ut et, t∈−1,0,
3.39
where, for anyt∈0,4,ρ1,ζ1480,μ0.112, ξ13500,M1 1 andM21/502,
ft, ut−1
⎧⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎪
⎪⎩
2M1ut, t, u∈1,4×ξ,∞,
M1ut
1sinπ
ut−ϑφ 2
ϑ−ϑφ cosπ
ut−ϑφ 2
ϑ−ϑφ , t, u∈1,4×
μξ, ξ , 1
2M2utcosπ
ut−μ 2
ϑφ−μ 2M1ϑφsinπ
ut−μ 2
ϑφ−μ , t, u∈1,4×
ζ, μξ , 1
2M2ut
2−sinπ
ut− 2
μ− −cosπ
ut− 2
μ−
, t, u∈1,4× ρ, ζ
, ρut−1/2−ut1/21
2M2ρ, t, u∈1,4×
0, ρ , 1
2M2ut, t, u∈0,1×R.
3.40
Clearly, we know that
α 1
2, β0, γ2, η1, δ 5
4, d 1 2, p 3
2, q 7
2, hit min t
5,4−t 8
, i1,2, G4, s 12s s∈0,1, G4, s 44−s s∈1,3,
Gs, s 4−s1s s∈3,4.
3.41
Simple computations yield
K 4
0
G1·, sds 1
0
12sds 3
1
44−sds 4
3
1s4−sds24.17,
L 4
0
G1·, sh2sds
1
0
12ss 5ds
20/13
1
44−ss 5ds
3
20/13
44−s4−s 8 ds
4
3
4−s21s
8 ds
4.695, μmin
h1 p
L K ,h1
q L K
0.112,
R
μK 7/2
3/2
G2·, sds
−1
0.282,
κ
K 4
0
G2·, sds −1
1 24.172.
3.42
Obviously,
mlim→ ∞sup
u∈Pζ,η
K
Ym
G2·, sfs, us−cds0, ∀0< ζ < η. 3.43
Ift, u∈1,4×0,1, then we have
ht htρ≤ut≤ρ1. 3.44
Therefore, we get
ft, ut−1≤ht−1/2−ht1/21
2M2, fort, u∈1,4×0,1. 3.45
From3.25, it follows that S1 sup
u∈∂P1
K 4
1
G2·, sfs, us−1ds
≤K 20/13
1
12s 5
s
!1/2
− s 5
!1/2 1
2M2 ds K
3
20/13
44−s 8
4−s
!1/2
− 4−s 8
!1/2 1
2M2 ds K
4
3
1s4−s 8 4−s
!1/2
− 4−s 8
!1/2 1
2M2 ds
≤1120.
3.46
Thus,
ζ1480≥ κS1
κ−r ≈1461.37, ξ13500, ζ < μξ. 3.47 Therefore, byTheorem 3.3, the BVP3.39has at least one positive solutionusuch that
ut
⎧⎨
⎩
et, ift∈−1,0, u∗t, ift∈0,4, 1480≤ u∗ ≤13500.
3.48
Acknowledgments
The authors would like to thank the referees for helpful comments and suggestions. The work was supported partly by the NSF of China10771202, the Research Fund for Shanghai Key Laboratory of Modern Applied Mathematics08DZ2271900, and the Specialized Research Fund for the Doctoral Program of Higher Education of China2007035805.
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