doi:10.1155/2010/235296
Research Article
Existence of Positive Solutions for Semipositone
Higher-Order BVPS on Time Scales
Yuguo Lin and Minghe Pei
Department of Mathematics, Beihua University, JiLin City 132013, China
Correspondence should be addressed to Minghe Pei,peiminghe@ynu.ac.kr
Received 4 December 2009; Revised 16 March 2010; Accepted 29 March 2010
Academic Editor: Alberto Cabada
Copyrightq2010 Y. Lin and M. Pei. 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 offer conditions on semipositone functionft, u0, u1, . . . , un−2such that the boundary value problem,uΔnt ft, uσn−1t, uΔσn−2t, . . . , uΔn−2
σt 0,t∈0,1∩T,n≥2,uΔi0 0, i0,1, . . . , n−3,αuΔn−20−βuΔn−10 0,γuΔn−2σ1 δuΔn−1σ1 0, has at least one positive solution, whereTis a time scale andft, u0, u1, . . . , un−2 ∈ C0,1 ×R0,∞n−1,R−∞,∞is continuous withft, u0, u1, . . . , un−2≥ −Mfor some positive constantM.
1. Introduction
Throughout this paper, letTbe a time scale, for anya, b∈R −∞,∞b > a, the interval a, b defined asa, b {t ∈ T | a ≤ t ≤ b}. Analogous notations for open and half-open intervals will also be used in the paper. We also use the notationRc, d to denote the real interval{t∈R |c≤ t≤ d}. To understand further knowledge about dynamic equations on time scales, the reader may refer to1–3 for an introduction to the subject.
In this paper, we present results governing the existence of positive solutions to the differential equation on time scales of the form
uΔnt ft, uσn−1t, uΔσn−2t, . . . , uΔn−2σt0, t∈0,1, n≥2 1.1
subject to the two-point boundary conditions
uΔi0 0, i0,1, . . . , n−3,
αuΔn−20−βuΔn−10 0,
γuΔn−2σ1 δuΔn−1σ1 0,
whereα, γ, β, δ≥0, d:αδγβαγσ1>0 and
δγσ1−σ21≥0. 1.3
Throughout, we assume that ft, u0, u1, . . . , un−2 ∈ C0,1 × R0,∞n−1,R−∞,∞ is
continuous withft, u0, u1, . . . , un−2≥ −Mfor some positive constantM.
LetCrdn0,1 denote the space of functions
Cnrd0,1 u:u∈C0, σn1 , . . . , uΔn−1∈C0, σ1 , uΔn ∈Crd0,1
. 1.4
We say thatutis a positive solution of BVP1.1and1.2, ifut∈Cnrd0,1 is a solution of BVP1.1and1.2anduΔit>0, t∈0, σn−i1, i0,1, . . . , n−2.
Various cases of BVP1.1and1.2have attracted a lot of attention in the literature. When n 2, BVP 1.1 and 1.2 has been studied by many specialists. For example, Agarwal et al. 4 have established the existence of positive solutions for continuous case of the semipositone Sturm-Liouville BVPs. Erbe and Peterson 5 andHao et al. 6 dealt with Sturm-Liouville BVPs on time scale of positone nonlinear term. In addition, Agarwal and O’Regan7 obtained positive solution of second-order right focal BVPs on time scale by using nonlinear alternative of Leray-Schauder type. In 2005, Chyan and Wong 8
obtained triple solutions of the same BVPs with7 . Recently,Sun and Li9,10 investigated semipositone Dirichlet BVPs on time scale. For higher-order BVPs, continuous case of BVP 1.1and1.2have been investigated by Agarwal and Wong11 , Wong and Agarwal12
and Wong13 . The discrete positone case of BVP1.1and1.2has been tackled by using a fixed point theorem for mappings that are decreasing with respect to a cone in14 . Especially,
time-scale case of 1.1 with four-point boundary condition has been studied by Liu and Sang15 . Besides, BVP1.1and1.2of nonlinear positone termft, ut, ut, . . . , un−1t which satisfied Nagumo-type conditions have been dealt with in 16 . Motivated by the
works mentioned above, the purpose of this paper is to tackle semipositone BVP1.1and 1.2. In fact, BVPs appeared in7–14 can be looked at as special case of BVP1.1and1.2
in this paper. For other related works, we also refer to17–19 .
The paper is outlined as follows. In Section2, we will present some notations and lemmas which will be used later. In Section3, by using Krasnoselskii’s fixed point theorem in a cone, we offer criteria for the existence of positive solution of BVP1.1and1.2.
2. Preliminary
In this section, we offer some notations and lemmas, which will be used in main results. Throughout this paper, we always use the following notations:
C1Kt, s is the Green’s function of the differential equation −uΔ n
t 0, t ∈ 0,1 subject to the boundary conditions1.2;
C2kt, s is the Green’s function of the differential equation−uΔΔt 0, t ∈ 0,1
subject to the boundary conditions
C3DefineTi:0,1 → R,i0,1, . . . , n−2 as
T0t≡qt, Tit t
0
Ti−1τΔτ, i1,2, . . . , n−2, 2.2
where
qt: min
t∈0,σ21
αtβ
ασ21 β,
γσ1−t δ γσ1 δ
. 2.3
Lemma 2.1. For the Green’s functionkt, sthe following hold:
0≤qtkσs, s≤kt, s≤kσs, s, t, s∈ 0, σ21×0,1 . 2.4
Proof. It is clear that
kt, s KΔnt−2t, s
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
1 d
αtβγσ1−σs δ, t≤s, 1
d
ασs βγσ1−t δ, σs≤t.
2.5
From the expression ofkt, s, we can easily obtain
0≤qtkσs, s≤kt, s≤kσs, s, t, s∈ 0, σ21×0,1 . 2.6
Lemma 2.2. Letwtbe the solution of BVP
−uΔnt M, t∈0,1 ,
uΔi0 0, i0,1, . . . , n−3,
αuΔn−20−βuΔn−10 0,
γuΔn−2σ1 δuΔn−1σ1 0.
2.7
Then
0≤wΔit≤cMTn−2−it, t∈ 0, σn−i1
, i0,1, . . . , n−2, 2.8
where
c:=
γσ1 δασ21 β
d σ1, 2.9
Proof. Fort≤s,
kt, s 1 d
αtβγσ1−σs δ≤ 1 d
αtβγσ1−t δ
αtβ ασ21 β·
γσ1−t δ
γσ1 δ ·
γσ1 δα2σ1 β
d
≤ cqtσ1.
2.10
Forσs≤t,
kt, s 1 d
ασs βγσ1−t δ≤ 1 d
αtβγσ1−t δ
γσ1γσ1 −t δ
δ ·
αtβ ασ21 β·
γσ1 δασ21 β
d
≤ cqtσ1.
2.11
So
0≤kt, s≤ cqt
σ1, t, s∈ 0, σ21
×0,1 . 2.12
By definingwtaswt 0σ1Kt, sM ds, t∈0, σn1 , it is clear that
wΔn−2t
σ1
0
kt, sM ds, t∈ 0, σ21. 2.13
Then
0≤wΔn−2t≤ cqt σ1
σ1
0
MΔscMqt, t∈ 0, σ21. 2.14
Further, sincewΔit 0, i0,1, . . . , n−3, we get
0≤wΔit≤cMTn−2−it, t∈ 0, σn−i1
, i0,1, . . . , n−2. 2.15
Lemma 2.3see20 . LetEbe a Banach space, and letC⊂Ebe a cone inE. Assume thatΩ1,Ω2
are open subsets ofEwith0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2, and letT : C∩Ω2\Ω1 → Cbe a completely
iTu ≤ u, u∈C∩∂Ω1;Tu ≥ u, u∈C∩∂Ω2or
iiTu ≥ u, u∈C∩∂Ω1;Tu ≤ u, u∈C∩∂Ω2.
Then,T has a fixed point inC∩Ω2\Ω1.
3. Main Results
In this section, by using Lemma2.3, we offer criteria for the existence of positive solution of
BVP1.1and1.2.
LetEdenote the space of functions
Eu:u∈C0, σn1 , . . . , uΔn−3 ∈C 0, σ31, uΔn−2∈C 0, σ21. 3.1
Let B {u ∈ E : uΔi0 0, i 0,1, . . . , n−3} be a Banach space with the norm u supt∈0,σ21 |uΔ
n−2
t|, and let
Cu∈ B:uΔn−2t≥qtu, t∈ 0, σ21. 3.2
It is obvious thatCis a cone inB. FromuΔi0 0, i 0,1, . . . , n−3, it follows that for all u∈C,
Tn−2−itu ≤uΔ
i
t≤σ0u, t∈ 0, σn−i1
, i0,1, . . . , n−2, 3.3
where
σ0: σn1 n−2. 3.4
Throughout the rest of the section, we assume that the set0, σ1 is such that
ξmin
τ ∈T:τ≥ σ1 4
, ζmin
τ∈T:τ ≤ 3σ1 4
3.5
exist and satisfy
σ1
4 ≤ξ < ζ≤ 3σ1
4 . 3.6
In addition, we denote that
ηi min
In order to obtain positive solutions of BVP1.1and1.2, we need to consider the
following boundary value problem:
uΔnt f∗t, vσn−1t, vΔσn−2t, . . . , vΔn−2σt0, t∈0,1,
uΔi0 0, i0,1, . . . , n−3,
αuΔn−20−βuΔn−10 0, γuΔn−2σ1 δuΔn−1σ1 0,
3.8
where
vt ut−wt, wtis as in Lemma 2.2,
f∗t, u0, u1, . . . , un−2 ft, ρ0, ρ1, . . . , ρn−2M
3.9
and for alli0,1, . . . , n−2,
ρi ⎧ ⎨ ⎩
ui, ui≥0,
0, ui<0.
3.10
Let the operatorS:C → Bbe defined by
Sut
σ1
0
Kt, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs, t∈0, σn1 ,
SuΔn−2 t
σ1
0
kt, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs, t∈ 0, σ21. 3.11
Lemma 3.1. The operatorSmapsCintoC.
Proof. From Lemma2.1, we know that fort∈0, σ21 ,
SuΔn−2 t
σ1
0
kt, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs
≤
σ1
0
kσs, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs.
3.12
So fort∈0, σ21 ,
Su ≤
σ1
0
kσs, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
From Lemma2.1again, it follows that fort∈0, σ21 ,
Lemma 3.2. The operatorS:C → Cis completely continuous.
Proof. First we shall prove that the operator S is continuous. Let um,u ∈ C be such that
limm→ ∞um−u0. FromuΔ
Next, to show complete continuity, we will apply Arzela-Ascoli theorem. LetΩbe a bounded subset ofC. Then there existsL >0 such that for allu∈Ω,
whereσ0is given in3.4. Let
M sup
s,ρ0,ρ1,...,ρn−2∈0,1×R0,σ0L n−2×R0,L
fs, ρ0, ρ1, . . . , ρn−2
M. 3.19
Clearly, we have fort∈0, σ21 ,
SuΔn−2
t≤M
σ1
0
kt, sΔs≤M
σ1
0
kσs, sΔs 3.20
and fort, t∈0, σ21 ,
SuΔn−2
t−SuΔn−2
t ≤M
σ1
0
kt, s−k
t, sΔs. 3.21
The Arzela-Ascoli theorem guarantees that SΩ is relatively compact, so S : C → C is completely continuous.
Theorem 3.3. Assume that the following conditions hold:
ithere exist r ∈ RcM,∞ such that for any u0, u1, . . . , un−2 ∈ Γr :R0, σ0r n−2 ×
R0, r ,
Au0, u1, . . . , un−2:
σ1
0
kσs, sfs, u0, u1, . . . , un−2 MΔs≤r, 3.22
iithere existR ∈ RcM,∞ with R /r such that for any u0, u1, . . . , un−2 ∈ ΓR :
Rη0R, σ0R ×Rη1R, σ0R × · · · ×Rηn−2R, R ,
Bu0, u1, . . . , un−2:ηn−2
ζ
ξ
kσs, sfs, u0, u1, . . . , un−2 MΔs≥R, 3.23
whereσ0is given in3.4,ξ, ζare given in3.5,ηi, i0,1, . . . , n−2are given in3.7, and
1−cM
R . 3.24
Then BVP1.1and1.2has a positive solution.
Proof. Without loss of generality, we assume thatr < R. Now we seek positive solutions of BVP3.8. Let
Foru∈∂Ω1∩C, it follows from3.3that
0≤uΔit≤σ0r, t∈ 0, σn−i1
, i0,1, . . . , n−3. 3.26
Fromi, we obtain that foru∈∂Ω1∩C,
SuΔn−2 t
σ1
0
kt, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs
≤
σ1
0
kσs, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs
≤r.
3.27
So
Su ≤ u, u∈∂Ω1∩C. 3.28
Let
Ω2{u∈B:u ≤R}. 3.29
Foru∈∂Ω2∩C, it follows from Lemma2.2and3.3that fors∈0, σ1 ,
vΔiσn−i−1suΔiσn−i−1s−wΔiσn−i−1s
≥uΔiσn−i−1s−cMTn−2−i
σn−i−1s
uΔiσn−i−1s−cMTn−2−i
σn−i−1su
R
≥uΔiσn−i−1s−cMu
Δi
σn−i−1s
R
uΔiσn−i−1s≥RTn−2−i
σn−i−1s, i0,1, . . . , n−2.
3.30
So
vΔiσn−i−1s≥ηiR, s∈ξ, ζ , i0,1, . . . , n−2, 3.31
Combining Lemma2.1,3.3, andiiwith3.31, we obtain that foru∈∂Ω2∩C,
SuΔn−2 t
σ1
0
kt, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs
≥
ζ
ξ
kt, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2σsΔs
≥ηn−2
ζ
ξ
kσs, sf∗s, vσn−1s, vΔσn−2s, . . . , vΔn−2
σsΔs
≥R.
3.32
So
Su ≥ u, u∈∂Ω2∩C. 3.33
Therefore, it follows from Lemma2.3that BVP3.8has a solutionu1∈Csuch thatr≤ u1 ≤
R.
Finally, we will prove thatu1t−wtis a positive solution of BVP1.1and1.2. Let
ut u1t−wt, then we have from Lemma2.2and3.3that fori0,1, . . . , n−2,
uΔit uΔ1it−wΔit≥uΔ1it−cMTn−2−it
≥uΔ1it−cMTn−2−itu1 r
≥uΔ1it−cMu
Δi
1 t
r
1−cM r
uΔ1it≥r−cMTn−2−it>0, t∈
0, σn−i1.
3.34
In addition,
uΔnt uΔ1nt M
−f∗t, u1
σn−1t−wσn−1t, . . . , u1Δn−2σt−wΔn−2σtM
−ft, u1
σn−1t−wσn−1t, . . . , u1Δn−2σt−wΔn−2σt
−ft, uσn−1t, . . . , uΔn−2σt.
3.35
Corollary 3.4. Assume that
afor anyt, u0, u1, . . . , un−2∈0,1 ×R0,∞n−1,
ft, u0, u1, . . . , un−2 M≤μtgu0, u1, . . . , un−2, 3.36
whereg : R0,∞n−1 → R0,∞is a continuous function which is nondecreasing inuj for each fixedu0, . . . , uj−1, uj1, . . . , un−2andμtis a continuous nonnegative function
on0,1 ,
bfor anyt, u0, u1, . . . , un−2∈ξ, ζ ×R0,∞n−1,
ft, u0, u1, . . . , un−2 M≥νthu0, u1, . . . , un−2, 3.37
whereh:R0,∞n−1 → R0,∞is a continuous function which is nondecreasing inuj for each fixedu0, . . . , uj−1, uj1, . . . , un−2andνtis a continuous nonnegative function on0,1 ,
cthere existsr ∈RcM,∞such that
gσ0r, . . . , σ0r, r
σ1
0
kσs, sμsΔs≤r, 3.38
dthere existsR∈RcM,∞withR /rsuch that
hη0R, η1R, . . . , ηn−2Rηn−2
ζ
ξ
kσs, sνsΔs≥R. 3.39
Then BVP1.1and1.2has a positive solution.
Proof. Fromaandc, we obtain that foru0, u1, . . . , un−2∈Γr,
Au0, u1, . . . , un−2
σ1
0
kσs, sfs, u0, u1, . . . , un−2 MΔs
≤
σ1
0
kσs, sμsgu0, u1, . . . , un−2Δs
≤gσ0r, . . . , σ0r, r
σ1
0
kσs, sμsΔs≤r.
So, conditioniof Theorem3.3is satisfied. Frombandd, we obtain that foru0, u1, . . . ,
So, conditioniiof Theorem3.3is satisfied.
Therefore, from Theorem3.3, BVP1.1and1.2has a positive solution.
Corollary 3.5. Assume that conditionsaandcof Corollary3.4and the following condition hold:
lim
Proof. We only need to prove that3.42implies conditioniiof Theorem3.3. From3.42,
we know that there existsRRmay be chosen arbitrary largesuch that foru0, u1, . . . , un−2∈
So, conditioniiof Theorem3.3is satisfied.
Example 3.6. Consider the following boundary value problem:
Hence, conditions c and d in Corollary 3.4are satisfied. Therefore from Corollary 3.4, 3.46has at least one positive solution.
Remark 3.7. In Example3.6, because nonlinear termfmay attain negative value, the result in 15 is not applicable.
Acknowledgment
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