Volume 2008, Article ID 123823,11pages doi:10.1155/2008/123823
Research Article
Existence and Uniqueness of Solutions for
Singular Higher Order Continuous and Discrete Boundary Value Problems
Chengjun Yuan,1, 2Daqing Jiang,1and You Zhang1
1School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, Jilin, China
2School of Mathematics and Computer, Harbin University, Harbin 150086, Heilongjiang, China
Correspondence should be addressed to Chengjun Yuan,[email protected] Received 4 July 2007; Accepted 31 December 2007
Recommended by Raul Manasevich
By mixed monotone method, the existence and uniqueness are established for singular higher-order continuous and discrete boundary value problems. The theorems obtained are very general and complement previous known results.
Copyrightq2008 Chengjun Yuan 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 recent years, the study of higher-order continuous and discrete boundary value problems has been studied extensively in the literaturesee 1–17and their references. Most of the results told us that the equations had at least single and multiple positive solutions.
Recently, some authors have dealt with the uniqueness of solutions for singular higher- order continuous boundary value problems by using mixed monotone method, for example, see6,14,15. However, there are few works on the uniqueness of solutions for singular dis- crete boundary value problems.
In this paper, we state a unique fixed point theorem for a class of mixed monotone op- erators, see6, 14,18. In virtue of the theorem, we consider the existence and uniqueness of solutions for the following singular higher-order continuous and discrete boundary value problems1.1and1.2by using mixed monotone method. We first discuss the existence and uniqueness of solutions for the following singular higher-order continuous boundary value problem
yntλqtgyhy0, 0< t <1, λ >0,
yi0 yn−21 0, 0≤i≤n−2, 1.1
wheren≥2,qt∈C0,1,0,∞,g:0,∞→0,∞is continuous and nondecreasing;
h:0,∞→0,∞is continuous and nonincreasing, andhmay be singular aty0.
Next, we consider the existence and uniqueness of solutions for the following singular higher-order discrete boundary value problem
Δnyiλqin−1gyin−1 hyin−1 0, i∈N{0,1,2, . . . , T−1}, λ >0, Δky0 Δn−2yT1 0, 0≤k≤n−2,
1.2 wheren≥2,N{0,1,2, . . . , Tn},qi∈CN,0,∞,g:0,∞→0,∞is continuous and nondecreasing;h : 0,∞ → 0,∞is continuous and nonincreasing, and hmay be singular aty0. Throughout this paper, the topology onNwill be the discrete topology.
2. Preliminaries
LetPbe a normal cone of a Banach spaceE, ande∈Pwithe ≤1, e /θ.Define
Qe{x∈P|x /θ, there exist constantsm, M >0 such thatm e≤x≤Me}. 2.1 Now we give a definitionsee18.
Definition 2.1see18. AssumeA:Qe×Qe→Qe.Ais said to be mixed monotone ifAx, y is nondecreasing inxand nonincreasing iny, that is, ifx1≤x2x1, x2∈QeimpliesAx1, y≤ Ax2, yfor anyy ∈Qe, andy1≤y2y1, y2 ∈QeimpliesAx, y1≥Ax, y2for anyx∈Qe. x∗∈Qeis said to be a fixed point ofAifAx∗, x∗ x∗.
Theorem 2.2 see 6,14. Suppose that A: Qe ×Qe → Qe is a mixed monotone operator and
∃a constantα,0≤α <1, such that A
tx,1
ty
≥tαAx, y, for x, y∈Qe, 0< t <1. 2.2 ThenAhas a unique fixed pointx∗∈Qe. Moreover, for anyx0, y0∈Qe×Qe,
xnA
xn−1, yn−1
, ynA
yn−1, xn−1
, n1,2, . . . , 2.3 satisfy
xn−→x∗, yn−→x∗, 2.4
where
xn−x∗o 1−rαn
, yn−x∗o 1−rαn
, 2.5
0< r <1,ris a constant fromx0, y0.
Theorem 2.3see6,14,18. Suppose that A:Qe×Qe→Qeis a mixed monotone operator and∃a constantα∈0,1such that2.2holds. Ifx∗λis a unique solution of equation
Ax, x λx, λ >0 2.6
inQe, thenx∗λ−xλ∗
0 →0, λ→λ0.If 0< α <1/2,then 0< λ1< λ2impliesx∗λ
1≥x∗λ
2,x∗λ
1/x∗λ
2, and
λ→∞limx∗λ0, lim
λ→0x∗λ ∞. 2.7
3. Uniqueness positive solution of differential equations1.1
This section discusses singular higher-order boundary value problem1.1. Throughout this section, we letGt, sbe the Green’s function to−y0, y0 y1 0,we note that
Gt, s
⎧⎨
⎩
t1−s, 0≤t≤s≤1,
s1−t, 0≤s≤t≤1, 3.1
and one can show that
Gt, tGs, s≤Gt, s≤Gt, t, forGt, s≤Gs, s, t, s∈0,1×0,1. 3.2 Suppose thatyis a positive solution of1.1. Let
xt yn−2t, 3.3
fromyi0 yn−21 0, 0≤i≤n−2, and Taylor Formula, we define operatorT :C20,1→ Cn0,1, by
yt Txt
t
0
t−sn−3
n−3! xsds, for 3≤n, yt Txt xt, forn2..
3.4
Then we have
x2t λft, Txt 0, 0< t <1, λ >0,
x0 x1 0. 3.5
Then from3.4, we have the next lemma.
Lemma 3.1. Ifxtis a solution of 3.5, thenytis a solution of 1.1.
Further, ifytis a solution of1.1, imply thatxtis a solution of3.5.
LetP {x∈C0,1|xt≥0, for allt∈0,1}. Obviously,Pis a normal cone of Banach spaceC0,1.
Theorem 3.2. Suppose that there existsα∈0,1such that
gtx≥tαgx, 3.6
h t−1x
≥tαhx, 3.7
for anyt∈0,1andx >0, andq∈C0,1,0,∞satisfies 1
0
sn−1n−2s−α
qsds <∞. 3.8
Then1.1has a unique positive solutionyλ∗t.And moreover, 0< λ1< λ2impliesyλ∗
1≤y∗λ
2, y∗λ
1/yλ∗
2. Ifα∈0,1/2, then
λ→0limy∗λ0, lim
λ→∞yλ∗ ∞. 3.9
Proof. Since3.7holds, lett−1xy,one has
hy≥tαhty. 3.10
Then
hty≤ 1
tαhy, fort∈0,1, y >0. 3.11
Lety1.The above inequality is ht≤ 1
tαh1, fort∈0,1. 3.12
From3.7,3.11, and3.12, one has h
t−1x
≥tαhx, h 1
t
≥tαh1, htx≤ 1
tαhx, ht≤ 1
tαh1,fort∈0,1, x >0.
3.13 Similarly, from3.6, one has
gtx≥tαgx, gt≥tαg1, fort∈0,1, x >0. 3.14 Lett 1/x, x >1,one has
gx≤xαg1, forx≥1. 3.15
Letet Gt, t t1−t, and we define Qe
x∈C0,1| 1
MGt, t≤xt≤MGt, t, t∈0,1
, 3.16
whereM >1 is chosen such that M >max
λg1
1
0
qsdsλh1
1
0
sn−1n−2s n!
−α
qsds 1/1−α
,
λg1 1
0
Gs, s
sn−1n−2s n!
α
qsdsλh1
1
0
Gs, sqsds
−1/1−α .
3.17
First, from3.4and3.16, for anyx∈Qe, we have the following.
When 3≤n, 1
M
tn−1n−2t
n! ≤
t
0
1
MGs, st−sn−3
n−3! ds≤Txt
≤ t
0
MGs, st−sn−3
n−3! ds≤Mtn−1n−2t
n! ≤M, fort∈0,1,
3.18
whenn2, 1 M
tn−1n−2t
n! ≤Txt xt≤Mtn−2t
n! ≤M, fort∈0,1, 3.19
then
1 M
tn−1n−2t
n! ≤Txt≤Mtn−1n−2t
n! ≤M, fort∈0,1. 3.20
For anyx, y∈Qe,we define Aλx, yt λ
1
0
Gt, sqsgTxs hTysds, fort∈0,1. 3.21
First, we show thatAλ:Qe×Qe→Qe.
Letx, y∈Qe,from3.14,3.15, and3.20, we have
gTxt≤gM≤Mαg1, fort∈0,1, 3.22
and from3.13, we have hTyt≤h
1 M
tn−1n−2t n!
≤
tn−1n−2t n!
−α h
1 M
≤Mα
tn−1n−2t n!
−α
h1, fort∈0,1.
3.23
Then, from3.2,3.21,3.22and3.23, we have Aλx, yt≤λGt, t
1
0
Mαg1qsds 1
0
Mα
sn−1n−2s n!
−α
h1qsds
≤MGt, t, fort∈0,1.
3.24
On the other hand, for anyx, y∈Qe, from3.13and3.14, we have gTxt≥g
1 M
tn−1n−2t n!
≥tn−1n−2t n!
α
g 1
M
≥
tn−1n−2t n!
α 1 Mαg1,
hTyt≥hM h
1 1/M
≥ 1
Mαh1, fort∈0,1.
3.25 Thus, from3.2,3.21and3.25, we have
Aλx, yt
≥λGt, t 1
0
Gs, sqsM−α
sn−1n−2s n!
α
g1ds
1
0
Gs, sqsM−αh1ds
≥ 1
MGt, t, fort∈0,1.
3.26
So,Aλis well defined andAλQe×Qe⊂Qe.
Next, for anyl∈0,1, one has
Aλ
lx, l−1y t λ
1
0
Gt, sqs
glTxs h
l−1Tys ds
≥λ 1
0
Gt, sqs
lαgTxs lαhTys ds lαAλx, yt, fort∈0,1.
3.27
So the conditions of Theorems2.2and2.3hold. Therefore, there exists a uniquex∗λ∈Qe such thatAλx∗, x∗ xλ∗. It is easy to check thatx∗λis a unique positive solution of3.5for givenλ >0. Moreover,Theorem 2.3means that if 0< λ1< λ2,thenx∗λ
1t≤xλ∗
2t,xλ∗
1t/xλ∗
2t
and ifα∈0,1/2, then
λ→0limx∗λ0, lim
λ→∞x∗λ ∞. 3.28
Next, fromLemma 3.1 and3.4, we get that y∗λ Tx∗λ is a unique positive solution of1.1for given λ > 0. Moreover, if 0 < λ1 < λ2,theny∗λ
1t ≤ y∗λ
2t,y∗λ
1t/y∗λ
2tand if α∈0,1/2, then
λ→0limy∗λ0, lim
λ→∞yλ∗ ∞. 3.29
This completes the proof.
Example 3.3. Consider the following singular boundary value problem:
ynt λ
μyat y−bt
0, t∈0,1, yi0 yn−21 0, 0≤i≤n−2,
3.30
whereλ, a, b >0,μ≥0, max{a, b}< 1/n−1.
ApplyingTheorem 3.2, letαmax{a, b}< 1/n−1,qt 1,gy μya,hy y−b, then
gty≥tαgy, h t−1
≥tαhy, 1
0
sn−1n−2s−α
ds <∞. 3.31
Thus all conditions inTheorem 3.2are satisfied. We can find3.30has a unique positive so- lutiony∗λt. In addition, 0 < λ1 < λ2 impliesy∗λ
1 ≤ y∗λ
2, y∗λ
1/yλ∗
2. Ifα max{a, b} ∈ 0,1/2, then
λ→0limy∗λ0, lim
λ→∞yλ∗ ∞. 3.32
4. Uniqueness positive solution of difference equations1.2
This section discusses singular higher-order boundary value problem1.2. Throughout this section, we letKi, jbe Green’s function to−Δ2yi ui1 0,i∈N,y0 yT1 0, we note that
Ki, j
⎧⎪
⎪⎨
⎪⎪
⎩
jT1−i
T1 , 0≤j≤i−1, iT1−j
T1 , i≤j ≤T1,
4.1
and one can show that
Ki, i≥Ki, j, Kj, j≥Ki, j, Ki, j≥ Ki, i
T1, for 0≤i≤T1, 1≤j≤T. 4.2 Suppose thatyis a positive solution of1.2. Let
xi Δn−2yi, for 0≤i≤T1. 4.3
FromΔiy0 Δn−2yT1 0, 0≤i≤n−2, andΔmyi−1 Δm−1yi−Δm−1yi−1, so we define operatorT, by
Txi yin−1 i1
l1
Cn−2i−ln−1xl, for 0≤i≤T. 4.4
Then
Δ2xi λFin−1, Txi 0, 0≤i≤T−1, λ >0,
x0 xT1 0. 4.5
Lemma 4.1. Ifxiis a solution of 4.5, thenyiis a solutionn of 1.2.
Proof. Since we remark thatxiis a solution of4.5, if and only if xi T
j1
Ki, jλFjn−1, Txj, for 0≤i≤T1. 4.6
Let
Txi yin−1, for 0≤i≤T. 4.7
From4.4we findΔiy0 Δn−2yT1 0,0≤i≤n−2, andxi Δn−2yi, so thatyiis a solution of1.2.
Further, ifyiis a solution of1.2, imply thatxiis a solution of4.5.
LetP {x∈CN,0,∞|xi≥0, for alli∈N}. Obviously,Pis a normal cone of Banach spaceCN,0,∞.
Theorem 4.2. Suppose that there existsα∈0,1such that gtx≥tαgx, h
t−1x
≥tαgx, 4.8
for anyt∈0,1andx >0, andq∈CN,0,∞.
Then1.2has a unique positive solutionyλ∗i.And moreover, 0< λ1 < λ2impliesy∗λ
1 ≤yλ∗
2, y∗λ
1/y∗λ
2. Ifα∈0,1/2, then
λ→0limy∗λ0, lim
λ→∞yλ∗ ∞. 4.9
Proof. The proof is the same as that ofTheorem 3.2, from4.12and4.13, one has h
t−1x
≥tαhx, h 1
t
≥tαh1, htx≤ 1
tαhx, ht≤ 1
tαh1, fort∈0,1, x >0;
gtx≥tαgx, gt≥tαg1, fort∈0,1, x >0.
4.10 gtx≥tαgx, gt≥tαg1, fort∈0,1, x >0. 4.11 Lett1/x, x >1,one has
gx≤xαg1, forx≥1. 4.12
Letei Ki, i/T1,and we define Qe
x∈P| 1
Mei≤xi≤Mei, for 0≤i≤T1
, 4.13
whereM >1 is chosen such that M >max
λT1g1T
j1
qjn−1 j1
l1
Cn−2j−ln−1 α
λT11αh1T
j1
K−αj, jqjn−1 1/1−α
;
λg1T
j1
qjn−1
Kj, j T1
α
λh1T
j1
qjn−1 j1
l1
Cn−2j−ln−1
−α−1/1−α . 4.14
From4.4and4.13, for anyx∈Qe, we have 1
Mej≤Txj j1
l1
Cn−2j−ln−1xl≤Mej j1
l1
Cn−2j−ln−1, for 0≤j ≤T. 4.15
For anyx, y∈Qe,we define Aλx, yi λ
T j1
Ki, jqjn−1gTxj hTyj, for 0≤i≤T1. 4.16 First we show thatAλ:Qe×Qe→Qe.
Letx, y∈Qe,from4.11and4.12, we have
gTxj≤g
Mejj1
l1Cn−2j−ln−1
≤g
M
j1
l1Cn−2j−ln−1
≤Mα j1
l1Cn−2j−ln−1 α
g1, for 1≤j≤T , 4.17 and from4.10, we have
hTyj≤h 1
Mej
≤e−αjh 1
M
≤Mαe−αjh1, for 1≤j≤T. 4.18 Then, from4.2and the above, we have
Aλx, yi≤λKi, iT
j1
qjn−1gTxj T1hTyj
≤eiMαλT1
g1T
j1
qjn−1 j1
l1
Cn−2j−ln−1 α
h1T
j1
e−αjqjn−1
≤eiMαλT1
g1T
j1
qjn−1 j1
l1
Cn−2j−ln−1 α
h1T
j1
Kj, j T1
−α
qjn−1
≤Mei, for 0≤i≤T1.
4.19
On the other hand, for anyx, y∈Qe, from4.10and4.12, we have
gxj≥g
1 Mej
≥eαj 1
Mαg1, for 1≤j≤T, 4.20
hyj≥h
Mejj1
l1
Cn−2j−ln−1
≥M−α j1
l1
Cn−2j−ln−1 −α
h1, for 1≤j≤T. 4.21 Thus, from4.2and4.16, we have
Aλx, yi≥λei T
j0
qjn−1gTxj T
j0
qjn−1hTyj
≥λeiM−α
g1T
j0
qj
Ki, i T1
α
h1T
j0
qjn−1 j1
l1
Cj−ln−1n−2 −α
≥ 1
Mei, for 0≤i≤T1.
4.22
So,Aλis well defined andAλQe×Qe⊂Qe. Next, for anyl∈0,1, one has
Aλ
lx, l−1y i λ
T j1
Ki, jqjn−1
gTlxj h
T l−1
yj
λ T j1
Ki, jqjn−1
glTxj h
l−1Tyj
≥λ T j1
Ki, jqjn−1
lαgTxj lαhTyj ds lαAλx, yi, for 0≤i≤T1.
4.23
So the conditions of Theorems2.2and2.3hold. Therefore, there exists a uniquex∗λ∈Qesuch thatAλx∗, x∗ x∗λ.It is easy to check thatx∗λis a unique positive solution of4.5for given λ >0. Moreover,Theorem 2.3means that if 0< λ1 < λ2,thenx∗λ
1t≤x∗λ
2t,x∗λ
1t/xλ∗
2tand
ifα∈0,1/2, then
λ→0limx∗λ0, lim
λ→∞x∗λ ∞. 4.24
Next, on usingLemma 3.1, from4.5, we get thaty∗λTxλ∗is a unique positive solution of1.2for given λ > 0. Moreover, if 0 < λ1 < λ2,theny∗λ
1t ≤ y∗λ
2t,y∗λ
1t/y∗λ
2tand if α∈0,1/2, then
λ→0limy∗λ0, lim
λ→∞yλ∗ ∞. 4.25
This completes the proof.
Example 4.3. Consider the following singular boundary value problem:
Δnyi−1 λ
μyai y−bi
0, i∈N,
Δiy0 Δn−2y1 0, 0≤i≤n−2, 4.26
whereλ, a, b >0,μ≥0, max{a, b}<1.
Letqi 1,gy μya,hy y−b,αmax{a, b}<1, then gty≥tαgy, h
t−1y
≥tαhy, 4.27
thus all conditions inTheorem 4.2are satisfied. We can find4.26has a unique positive so- lutionyλ∗t. In addition, 0 < λ1 < λ2 impliesyλ∗
1 ≤ yλ∗
2, yλ∗
1/y∗λ
2. Ifα max{a, b} ∈0,1/2, then
λ→0limy∗λ0, lim
λ→∞yλ∗ ∞. 4.28
Acknowledgments
The work was supported by the National Natural Science Foundation of ChinaGrants no.
10571021 and 10701020. The work was supported by Subject Foundation of Harbin University Grant no. HXK200714.
References
1 Y. Guo and J. Tian, “Positive solutions of m-point boundary value problems for higher order ordinary differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 66, no. 7, pp. 1573–1586, 2007.
2 D. Jiang, “Multiple positive solutions to singular boundary value problems for superlinear higher- order ODEs,” Computers and Mathematics with Applications, vol. 40, no. 2-3, pp. 249–259, 2000.
3 P. W. Eloe and J. Henderson, “Singular nonlinear boundary value problems for higher order ordi- nary differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 17, no. 1, pp. 1–10, 1991.
4 P. W. Eloe and J. Henderson, “Postive solutions for higher order differential equations,” Journal of Differential Equations, vol. 3, pp. 1–8, 1995.
5 C. J. Chyan and J. Henderson, “Positive solutions for singular higher order nonlinear equations,”
Differential Equations and Dynamical Systems, vol. 2, no. 2, pp. 153–160, 1994.
6 X. Lin, D. Jiang, and X. Li, “Existence and uniqueness of solutions for singulark, n−kconjugate boundary value problems,” Computers & Mathematics with Applications, vol. 52, no. 3-4, pp. 375–382, 2006.
7 R. P. Agarwal and D. O’Regan, “Existence theory for single and multiple solutions to singular positone boundary value problems,” Journal of Differential Equations, vol. 175, no. 2, pp. 393–414, 2001.
8 P. R. Agarwal and D. O’Regan, “Singular discrete boundary value problems,” Applied Mathematics Letters, vol. 12, no. 4, pp. 127–131, 1999.
9 R. P. Agarwal and F.-H. Wong, “Existence of positive solutions for higher order difference equations,”
Applied Mathematics Letters, vol. 10, no. 5, pp. 67–74, 1997.
10 Z. Du, C. Xue, and W. Ge, “Triple solutions for a higher-order difference equation,” Journal of Inequali- ties in Pure and Applied Mathematics, vol. 6, no. 1, Article 10, pp. 1–11, 2005.
11 P. J. Y. Wong and R. P. Agarwal, “On the existence of solutions of singular boundary value problems for higher order difference equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 28, no. 2, pp. 277–287, 1997.
12 R. P. Agarwal and P. J. Y. Wong, “Existence of solutions for singular boundary problems for higher order differential equations,” Rendiconti del Seminario Matem`atico e Fisico di Milano, vol. 65, pp. 249–
264, 1995.
13 R. P. Agarwal and F.-H. Wong, “Existence of positive solutions for higher order difference equations,”
Applied Mathematics Letters, vol. 10, no. 5, pp. 67–74, 1997.
14 X. Lin, D. Jiang, and X. Li, “Existence and uniqueness of solutions for singular fourth-order boundary value problems,” Journal of Computational and Applied Mathematics, vol. 196, no. 1, pp. 155–161, 2006.
15 Z. Zengqin, “Uniqueness of positive solutions for singular nonlinear second-order boundary-value problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 23, no. 6, pp. 755–765, 1994.
16 R. P. Agarwal and I. Kiguradze, “Two-point boundary value problems for higher-order linear differ- ential equations with strong singularities,” Boundary Value Problems, vol. 2006, Article ID 83910, 32 pages, 2006.
17 X. Hao, L. Liu, and Y. Wu, “Positive solutions for nonlinearnth-order singular nonlocal boundary value problems,” Boundary Value Problems, vol. 2007, Article ID 74517, 10 pages, 2007.
18 D. Guo, The Order Methods in Nonlinear Analysis, Shandong Technical and Science Press, Jinan, China, 2000.
