Volume 2010, Article ID 973731,22pages doi:10.1155/2010/973731
Research Article
Solutions and Green’s Functions for
Boundary Value Problems of Second-Order
Four-Point Functional Difference Equations
Yang Shujie and Shi Bao
Institute of Systems Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China
Correspondence should be addressed to Yang Shujie,yangshujie@163.com
Received 23 April 2010; Accepted 11 July 2010
Academic Editor: Irena Rach ˚unkov´a
Copyrightq2010 Y. Shujie and S. Bao. 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 consider the Green’s functions and the existence of positive solutions for a second-order functional difference equation with four-point boundary conditions.
1. Introduction
In recent years, boundary value problems BVPs of differential and difference equations have been studied widely and there are many excellent resultssee Gai et al.1, Guo and Tian2, Henderson and Peterson3, and Yang et al.4. By using the critical point theory, Deng and Shi5studied the existence and multiplicity of the boundary value problems to a class of second-order functional difference equations
Lunfn, un1, un, un−1 1.1
with boundary value conditions
Δu0A, uk1B, 1.2
where the operatorLis the Jacobi operator
Ntouyas et al.6and Wong7investigated the existence of solutions of a BVP for functional differential equations
xt ft, xt, xt
, t∈0, T,
α0x0−α1x0 φ∈Cr,
β0xT β1xT A∈Rn,
1.4
wheref:0, T×Cr×Rn → Rnis a continuous function,φ∈Cr C−r,0,Rn, A∈Rn, and
xtθ xtθ, θ∈−r,0.
Weng and Guo8considered the following two-point BVP for a nonlinear functional difference equation withp-Laplacian operator
ΔΦpΔxt rtfxt 0, t∈ {1, . . . , T},
x0φ∈C, ΔxT1 0,
1.5
whereΦpu |u|p−2u,p >1,φ0 0,T,τ ∈N,C{φ|φk≥0, k∈−τ,0},f:C → R
is continuous,Ttτ1rt>0.
Yang et al. 9 considered two-point BVP of the following functional difference equation withp-Laplacian operator:
ΔΦpΔxt rtfxt, xt 0, t∈ {1, . . . , T},
α0x0−α1Δx0 h,
β0xT1 β1ΔxT1 A,
1.6
whereh ∈ Cτ {φ ∈ Cτ | φθ ≥ 0, θ ∈ {−τ, . . . ,0}},A ∈ R, andα0,α1,β0, andβ1 are
nonnegative real constants. Fora, b∈Nanda < b, let
R{x|x∈R, x≥0},
a, b {a, a1, . . . , b}, a, b {a, a1, . . . , b−1}, a,∞ {a, a1, . . . ,},
Cτ
φ|φ:−τ,0 → R, Cτ φ∈Cτ|φθ≥0, θ∈−τ,0
.
1.7
ThenCτandCτ are both Banach spaces endowed with the max-norm
φ
τ kmax∈−τ,0φk. 1.8
In this paper, we consider the following second-order four-point BVP of a nonlinear functional difference equation:
−Δ2ut−1 rtft, u
t, t∈1, T,
u0αu
ηh, t∈−τ,0,
uT1 βuξ γ,
1.9
whereξ, η ∈ 1, Tand ξ < η, 0 < τ < T,Δut ut1−ut,Δ2ut ΔΔut,f :
R×Cτ → Ris a continuous function,h∈Cτ andht≥h0≥0 fort∈−τ,0,α,β, andγ
are nonnegative real constants, andrt≥0 fort∈1, T.
At this point, it is necessary to make some remarks on the first boundary condition in
1.9. This condition is a generalization of the classical condition
u0 αuηC 1.10
from ordinary difference equations. Here this condition connects the historyu0with the single
uη. This is suggested by the well-posedness of BVP1.9, since the functionfdepends on the termuti.e., past values ofu.
As usual, a sequence{u−τ, . . . , uT1}is said to be a positive solution of BVP1.9 if it satisfies BVP1.9anduk≥0 fork∈−τ, Twithuk>0 fork∈1, T.
2. The Green’s Function of
1.9
First we consider the nonexistence of positive solutions of1.9. We have the following result.
Lemma 2.1. Assume that
βξ > T 1, 2.1
or
αT1−η> T1. 2.2
Then1.9has no positive solution.
Proof. FromΔ2ut−1 −rtft, u
t≤0, we know thatutis convex fort∈0, T1.
Assume thatxtis a positive solution of1.9and2.1holds.
IfxT1>0, thenxξ>0. It follows that
xT1−x0
T1
βxξ−x0
T1
> xξ ξ −
x0
T1
≥ xξ−x0
ξ ,
2.3
which is a contradiction to the convexity ofxt.
IfxT1 0, thenxξ 0. Ifx0>0, then we have
xT1−x0
T1 −
x0
T1,
xξ−x0
ξ − x0
ξ .
2.4
Hence
xT1−x0
T1 >
xξ−x0
ξ , 2.5
which is a contradiction to the convexity ofxt. Ifxt≡0 fort∈1, T, thenxtis a trivial solution. So there exists at0∈1, ξ∪ξ, Tsuch thatxt0>0.
We assume thatt0∈1, ξ. Then
xT1−xt0
T1−t0 −
xt0
T1−t0
,
xξ−xt0
ξ−t0 −
xt0
ξ−t0
.
2.6
Hence
xT1−xt0
T1−t0
> xξ−xt0 ξ−t0
, 2.7
which is a contradiction to the convexity ofxt.
Ift0∈ξ, T, similar to the above proof, we can also get a contradiction.
Now we have
xT1−x0
T1
βxξ−x0 γ T1
≥ xξ
ξ − x0
T1
γ T1
≥ xξ−x0
ξ
γ T1
> xξ−x0 ξ ,
2.8
which is a contradiction to the convexity ofxt.
Assume thatxtis a positive solution of1.9and2.2holds.
1Consider thath0 0. IfxT1>0, then we obtain
xT1−x0
T1
xT1−αxη T1
< xT1 T1−η −
αxη T1
≤ xT1−x
η T1−η ,
2.9
which is a contradiction to the convexity ofxt.
Ifxη>0, similar to the above proof, we can also get a contradiction.
IfxT1 xη 0, and sox0 0, then there exists at0 ∈1, η∪η, Tsuch that
xt0>0. Otherwise,xt≡0 is a trivial solution. Assume thatt0 ∈1, η, then
xT1−xt0
T1−t0 −
xt0
T1−t0
,
xη−xt0
η−t0 −
xt0
η−t0
,
2.10
which implies that
xT1−xt0
T1−t0
> x
η−xt0
η−t0
. 2.11
A contradiction to the convexity ofxtfollows. Ift0∈η, T, we can also get a contradiction.
Now we obtain
xT1−x0
T1
xT1−αxη−h0
T1
≤ xT1
T1−η − xη T1−η −
h0
T1
< xT1−x
η T1−η ,
2.12
which is a contradiction to the convexity ofxt.
Next, we consider the existence of the Green’s function of equation
−Δ2ut−1 ft,
u0 αuη,
uT1 βuξ.
2.13
We always assume that
H10≤α,β≤1 andαβ <1.
Motivated by Zhao10, we have the following conclusions.
Theorem 2.2. The Green’s function for second-order four-point linear BVP2.13is given by
G1t, s Gt, s
αT1−t αη 1−αT1×
αηβ1−αt 1−αT1−βξ
1−βαη 1−αT1−βξ G
η, s
β1−αtαβη
1−βαη 1−αT1−βξGξ, s,
2.14
where
Gt, s ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
sT1−t
T1 , 0≤s≤t−1,
tT1−s
T1 , t≤s≤T1.
2.15
Proof. Consider the second-order two-point BVP
−Δ2ut−1 ft, t∈1, T,
u0 0,
uT1 0.
It is easy to find that the solution of BVP2.16is given by
ut
T
s1
Gt, sfs, 2.17
u0 0, uT1 0, uη
T
s1
Gη, sfs. 2.18
The three-point BVP
−Δ2ut−1 ft, t∈1, T,
u0 αuη, t∈−τ,0,
uT1 0
2.19
can be obtained from replacingu0 0 byu0 αuηin2.16. Thus we suppose that the solution of2.19can be expressed by
vt ut cdtuη, 2.20
wherecanddare constants that will be determined. From2.18and2.20, we have
v0 u0 cuη,
vηuηcdηuη1cdηuη,
vT1 uT1 cdT1uη cdT1uη.
2.21
Putting the above equations into2.19yields
1−αc−αηdα,
c T1d0. 2.22
ByH1, we obtaincanddby solving the above equation:
c αT1 αη 1−αT1,
d −α
αη 1−αT1.
By2.19and2.20, we have
v0 αvη,
vT1 0,
vξ uξ cdξuη.
2.24
The four-point BVP2.13can be obtained from replacinguT1 0 byuT1 βuξin
2.19. Thus we suppose that the solution of2.13can be expressed by
wt vt abtvξ, 2.25
whereaandbare constants that will be determined. From2.24and2.25, we get
w0 v0 avξ αvηavξ,
wηvηabηvξ,
wT1 vT1 abT1vξ abT1vξ,
wξ vξ abξvξ.
2.26
Putting the above equations into2.13yields
1−αa−αηb0,
1−βaT1−βξbβ. 2.27
ByH1, we can easily obtain
a αβη
1−βαη 1−αT1−βξ,
b β1−α
1−βαη 1−αT1−βξ.
2.28
Then by2.17,2.20,2.23,2.25, and2.28, the solution of BVP2.13can be expressed by
wt
T
s1
G1t, sfs, 2.29
Remark 2.3. ByH1, we can see thatG1t, s>0 fort, s∈0, T12. Let
m min
t,s∈1,T2
G1t, s, M max
t,s∈1,T2
G1t, s. 2.30
ThenM≥m >0.
Lemma 2.4. Assume that (H1) holds. Then the second-order four-point BVP 2.13has a unique
solution which is given in2.29.
Proof. We need only to show the uniqueness.
Obviously, wt in 2.29 is a solution of BVP 2.13. Assume that vt is another solution of BVP2.13. Let
zt vt−wt, t∈−τ, T1. 2.31
Then by2.13, we have
−Δ2zt−1 −Δ2vt−1 Δ2wt−1≡0, t∈1, T, 2.32
z0 v0−w0 αzη,
zT1 vT1−wT1 βzξ. 2.33
From2.32we have, fort∈1, T,
zt c1tc2, 2.34
which implies that
z0 c2, z
ηc1ηc2, zξ c1ξc2, zT1 c1T1 c2. 2.35
Combining2.33with2.35, we obtain
αηc1−1−αc20,
T1−βξc1
1−βc2 0.
2.36
ConditionH1implies that2.36has a unique solutionc1c20. Thereforevt≡wtfor
3. Existence of Positive Solutions
In this section, we discuss the BVP1.9. Assume thath0 0,γ0. We rewrite BVP1.9as
−Δ2ut−1 rtft, u
t, t∈1, T,
u0αu
ηh, t∈−τ,0,
uT1 βuξ
3.1
withh0 0.
Suppose thatutis a solution of the BVP3.1. Then it can be expressed as
ut ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
T
s1
G1t, srsfs, us, t∈1, T,
αuηht, t∈−τ,0,
βuξ, tT1.
3.2
Lemma 3.1see Guo et al.11. Assume thatEis a Banach space andK ⊂Eis a cone inE. Let
Kp{u∈K| up}. Furthermore, assume thatΦ:K → Kis a completely continuous operator andΦu /uforu∈∂Kp{u∈K| up}. Thus, one has the following conclusions:
(1) ifu ≤ Φu foru∈∂Kp, theniΦ, Kp, K 0; (2) ifu ≥ Φu foru∈∂Kp, theniΦ, Kp, K 1.
Assume thatf≡0. Then3.1may be rewritten as
−Δ2ut−1 0, t∈1, T,
u0 αu
ηh,
uT1 βuξ.
3.3
Letutbe a solution of3.3. Then by3.2andξ, η∈1, T, it can be expressed as
ut ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
0, t∈1, T,
ht, t∈−τ,0,
0, tT1.
Letutbe a solution of BVP3.1andyt ut−ut. Then fort∈1, Twe have
yt≡utand
yt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
T
s1
G1t, srsf
s, ysus
, t∈1, T,
αyη, t∈−τ,0,
βyξ, tT1.
3.5
Let
u max
t∈−τ,T1|ut|, E{u|u:−τ, T1 → R},
K
u∈E| min
t∈1,Tut≥
m
Mu, ut αu
η, t∈−τ,0, uT1 βuξ
.
3.6
ThenEis a Banach space endowed with norm · andKis a cone inE. Fory∈K, we have byH1and the definition ofK,
y max
t∈−τ,T1yttmax∈1,Tyt. 3.7
For everyy ∈ ∂Kp,s ∈ 1, T,and k ∈ −τ,0, by the definition of K and 3.5, if
sk≤0, we have
ysysk αy
η. 3.8
IfT ≥sk≥1, we have, by3.4,
ususk 0, ysysk≥ min t∈1,Tyt≥
m
My, 3.9
hence by the definition of · τ, we obtain fors∈τ1, T
ysτ ≥ m
My. 3.10
Lemma 3.2. For everyy∈K, there ist0∈τ1, T, such that
yt
Proof. Fors∈τ1, T, k ∈−τ,0, andsk∈1, T, by the definitions of · τand · , we
have
ysτ max
k∈−τ,0ysk,
y max
t∈1,Tyt.
3.12
Obviously, there is at0∈τ1, T, such that3.11holds.
Define an operatorΦ:K → Eby
Φyt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T
s1
G1t, srsf
s, ysus
, t∈1, T,
αΦyη, t∈−τ,0,
βΦyξ, tT1.
3.13
Then we may transform our existence problem of positive solutions of BVP3.1into a fixed point problem of operator3.13.
Lemma 3.3. Consider thatΦK⊂K.
Proof. Ift∈−τ,0andtT1,Φyt αΦηandΦyT1 βΦξ, respectively. Thus,
H1yields
Φy max
t∈−τ,T1Φy
t max
t∈1,TΦy
tΦy1,T. 3.14
It follows from the definition ofKthat
min
t∈1,T
Φyt min
t∈1,T T
s1
G1t, srsf
s, ysus
≥m
T
s1
rsfs, ysus
≥ m
M
T
s1
max
1≤s,t≤TG1t, s
rsfs, ysus
≥ m
Mtmax∈1,T T
s1
G1t, srsf
s, ysus
m
MΦy,
3.15
Lemma 3.4. Suppose that (H1) holds. ThenΦ:K → Kis completely continuous.
We assume that
H2
T
t1rt>0,
H3hhτ max
t∈−τ,0ht>0.
We have the following main results.
Theorem 3.5. Assume that (H1)–(H3) hold. Then BVP3.1has at least one positive solution if the
following conditions are satisfied:
H4there exists ap1> hsuch that, fors∈1, T, ifφτ ≤p1h, thenfs, φ≤R1p1;
H5there exists ap2> p1such that, fors∈1, T, ifφτ≥m/Mp2, thenfs, φ≥R2p2
or
H61> α >0;
H7there exists a0< r1 < p1such that, fors∈1, T, ifφτ ≤r1, thenfs, φ≥R2r1;
H8 there exists an r2 ≥ max{p2h,Mh/mα}, such that, for s ∈ 1, T, ifφτ ≥
mα/Mr2−h, thenfs, φ≤R1r2,
where
R1≤
1
MTs1rs, R2≥
1
mTsτ1rs. 3.16
Proof. Assume thatH4andH5hold. For everyy∈∂Kp1, we haveysusτ ≤p1h, thus
ΦyΦy1,T
≤M
T
s1
rsfs, ysus
≤MR1p1
T
s1
rs
≤p1
y,
3.17
which implies byLemma 3.1that
iΦ, Kp1, K
For everyy∈∂Kp2, by3.8–3.10andLemma 3.2, we have, fors∈τ1, T,ysτ ≥
m/My m/Mp2. Then by3.13andH5, we have
ΦyΦy1,T≥m
T
sτ1
rsfs, ysus
m
T
sτ1
rsfs, ys
≥mR2p2
T
sτ1
rs≥p2y,
3.19
which implies byLemma 3.1that
iΦ, Kp2, K
0. 3.20
So by3.18and3.20, there exists one positive fixed pointy1of operatorΦwithy1 ∈Kp2\
Kp1.
Assume thatH6–H8hold, for everyy∈∂Kr1ands∈τ1, T,ysusτ ysτ ≤
yr1, byH7, we have
Φy≥y. 3.21
Thus we have fromLemma 3.1that
iΦ, Kr1, K 0. 3.22
For everyy∈∂Kr2, by3.8–3.10, we haveysusτ ≥ ysτ−h≥mα/Mr2−h >0,
Φy ≤ y. 3.23
Thus we have fromLemma 3.1that
iΦ, Kr2, K 1. 3.24
So by 3.22 and 3.24, there exists one positive fixed pointy2 of operator Φwith
y2∈Kr2\Kr1.
Consequently,u1y1uoru2y2uis a positive solution of BVP3.1.
Theorem 3.6. Assume that (H1)–(H3) hold. Then BVP3.1has at least one positive solution if (H4)
and (H7) or (H5) and (H8) hold.
Theorem 3.7. Assume that (H1)–( H3) hold. Then BVP3.1has at least two positive solutions if
Theorem 3.8. Assume that (H1)–(H3) hold. Then BVP3.1has at least three positive solutions if
(H4)–(H8) hold.
Assume thath0>0,γ >0, and
H9 1−βh0−1−αγ >0.
DefineHt:−τ, T1 → Ras follows:
Ht ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
ht, t∈−τ,0,
0, t∈1, T,
HT1, tT1,
3.25
which satisfies
H10 1−αHT1−1−βh0>0.
Obviously,Htexists.
Assume thatutis a solution of1.9. Let
wt ut pHt B, 3.26
where
p
1−βh0−1−αγ
1−αHT1−1−βh0, B
h0γ−HT1
1−αHT1−1−βh0. 3.27
By1.9,3.26,3.27,H7,H8, and the definition ofHt, we have
w0 u0 ph0 B
αwηph0 1−αBh0
αwη,
3.28
wT1 uT1 phT1 B
βwξ pHT1 1−βBγ
βwξ,
3.29
and, fort∈1, T,
−Δ2wt−1 −Δ2ut−1−pΔ2Ht−1
rtft, ut−pΔ2Ht−1
rtft, wt−pHt−B
−p{Ht1−Ht−1}.
3.30
Let
Ft, wt rtf
t, wt−pHt−B
Then by3.27,H9,H10, and the definition ofHt, we haveFt, wt > 0 fort ∈
1, T.Thus, the BVP1.9can be changed into the following BVP:
−Δ2wt−1 Ft, w
t, t∈1, T,
w0αw
ηg, t∈−τ,0,
wT1 βwξ,
3.32
withg−BαhpH0B∈Cτ andg0 0.
Similar to the above proof, we can show that1.9has at least one positive solution. Consequently,1.9has at least one positive solution.
Example 3.9. Consider the following BVP:
−Δ2ut−1 t
120ft, ut, t∈1,5,
u0u2 t 2
4, t∈−2,0,
uT1 1 2u4.
3.33
That is,
T5, τ 2, α 1
2, β1, ξ2, η4, ht
t2
4, rt
t
120.
3.34
Then we obtain
h1, 21
24≤G1t, s≤ 163
40 ,
5
s1
rt 1
8,
5
s3
rt 1
10. 3.35
Let
ft, φ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
2R2
p2−r1
π arctan
s− m Mp2
R2p2, s≤ m
Mp2,
2R1r2−R2p2
π arctan
s− m Mp2
R2p2, s > m
Mp2,
R1
3
2, R2 12, r11, r2400, p14, p240,
3.36
wheresφτ.
By calculation, we can see thatH4–H8hold, then byTheorem 3.8, the BVP3.33
4. Eigenvalue Intervals
In this section, we consider the following BVP with parameterλ:
−Δ2ut−1 λrtft, u
t, t∈1, T,
u0αu
ηh, t∈−τ,0,
uT1 βuξ
4.1
withh0 0.
The BVP4.1is equivalent to the equation
ut ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ T
s1
G1t, srsfs, us, t∈1, T,
αuηht, t∈−τ,0,
βuξ, tT1.
4.2
Letutbe the solution of3.3,yt ut−ut. Then we have
yt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ T
s1
G1t, srsf
s, ysus
, t∈1, T,
αyη, t∈−τ,0,
βyξ, tT1.
4.3
LetEandKbe defined as the above. DefineΦ:K → Eby
Φyt ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ T
s1
G1t, srsf
s, ysus
, t∈1, T,
αΦyη, t∈−τ,0,
βΦyξ, tT1.
4.4
Then solving the BVP 4.1 is equivalent to finding fixed points in K. Obviously Φ is completely continuous and keeps theKinvariant forλ≥0.
Define
f0 lim inf
φτ→0tmin∈1,T
ft, φ
φτ
, f∞lim inf
φτ→ ∞tmin∈1,T
ft, φ
φτ
, f∞lim sup
φτ→ ∞
max
t∈1,T
ft, φ
φτ
,
4.5
Theorem 4.1. Assume that (H1), (H2), (H6),
H11 r min
t∈1,Trt>0,
H12 min{1/mrf0, M/m2f0
T
sτ1rs}< λ <1/Mδf∞
T
s1rs
hold, whereδ max{1,1μα}, then BVP4.1has at least one positive solution, whereμis a positive constant.
Proof. Assume that conditionH12holds. If λ >1/mrf0and f0 < ∞, there exists an > 0
sufficiently small, such that
λ≥ 1 mrf0−
. 4.6
By the definition off0, there is anr1>0, such that for 0<φτ ≤r1,
min
t∈1,T
ft, φ
φτ > f0−. 4.7
It follows that, fort∈1, Tand 0<φτ ≤r1,
ft, φ>f0−φτ. 4.8
For everyy∈∂Kr1ands∈τ1, T, by3.9, we have
ysusτysτ ≤yr1. 4.9
Therefore by3.13andLemma 3.2, we have
Φy max
t∈1,Tλ T
s1
G1t, srsf
s, ysus
≥λmax
t∈1,T T
sτ1
G1t, srsf
s, ysus
≥mλrf0−yt0τ
mλrf0−y
≥ y.
If λ > M/m2f
0Tsτ1rs, then for a sufficiently small > 0, we have λ ≥ M/m2f0 −
Tsτ1rs.Similar to the above, for everyy∈∂Kr1, we obtain by3.10
Φy ≥mλ
T
sτ1
rsf0−ysτ
≥mλ
T
sτ1
rsf0−
m My
≥ m2λ
f0−
M
T
sτ1
rsy
≥y.
4.11
Iff0 ∞, chooseK >0 sufficiently large, such that
m2λK
M
T
sτ1
rs≥1. 4.12
By the definition off0, there is anr1>0,such that, fort∈1, Tand 0<φτ ≤r1,
ft, φ> Kφτ. 4.13
For everyy∈∂Kr1, by3.8–3.10and3.13, we have
Φy ≥y, 4.14
which implies that
iΦ, Kr1, K 0. 4.15
Finally, we consider the assumptionλ < 1/Mδf∞Ts1rs. By the definition off∞, there is
r > max{r1, h/μα}, such that, fort∈1, Tandφ ≥r,
ft, φ<f∞1φ. 4.16
We now show that there is r2 ≥ r, such that, for y ∈ ∂Kr2,Φy ≤ y.In fact, for
we have
Φy≤y, 4.17
which implies that
iΦ, Kr2, K 1. 4.18
Theorem 4.2. Assume that (H1),(H2), and (H11) hold. Iff∞ ∞orf0 ∞, then there is aλ0 >0
such that for0< λ≤λ0, BVP4.1has at least one positive solution.
Proof. Letr > hbe given. Define
Lmaxft, φ|t, φ∈1, T×Cr τ
. 4.19
ThenL >0, whereCr
τ{φ∈Cτ | φτ ≤r}.
For everyy ∈ ∂Kr−h, we know thaty r−h. By the definition of operatorΦ, we obtain
ΦyΦy
1,T≤λLM T
s1
rs. 4.20
It follows that we can takeλ0 r−h/ML
T
s1rs>0 such that, for all 0< λ≤λ0and all
y∈∂Kr−h,
Φy≤y. 4.21
Fix 0< λ≤ λ0. Iff∞ ∞, forC 1/λmr, we obtain a sufficiently largeR > r such
that, forφτ ≥R,
min
t∈1,T
ft, φ
Φτ
> C. 4.22
It follows that, forφτ ≥Randt∈1, T,
For everyy∈∂KR, by the definition of·,·τand the definition ofLemma 3.2, there
exists at0∈τ1, Tsuch thatyyt0τ Randut0 0, thusyt0ut0τ ≥R. Hence
Φy max
t∈1,Tλ T
s1
G1t, srsf
s, ysus
≥ max
t∈1,TλG1t, t0rt0f
t0, yt0ut0
≥λmrCyt0τ
≥mCRλr
R
y.
4.24
Iff0 ∞,there iss < r, such that, for 0<φτ ≤sandt∈1, T,
ft, φ> Tφτ, 4.25
whereT >1/λmr.
For everyy∈∂Ks, by3.8–3.10andLemma 3.2,
Φy ≥mλ
T
sτ1
rsfs, ys
≥Tmλ
T
sτ1
rsysτ
≥Tmλryt0τ
Tmλry
≥y,
4.26
which by combining with4.21completes the proof.
Example 4.3. Consider the BVP3.33inExample 3.9with
ft, φ ⎧ ⎪ ⎨ ⎪ ⎩
Aarctans, s≤ m Mp2, AarctansC
1000 , s >
m Mp2,
C1000− m
Mp2
Aarctanm
Mp2
,
4.27
wheresφτ,Ais some positive constant,p240, m 21/24, and M 163/40.
By calculation,f0A,f∞πA/2000, andr1/120; letδ1. Then by Theorem4.1,
Acknowledgments
The authors would like to thank the editor and the reviewers for their valuable comments and suggestions which helped to significantly improve the paper. This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.
References
1 M. J. Gai, B. Shi, and D. C. Zhang, “Boundary value problems for second-order singular functional differential equations,”Chinese Annals of Mathematics, vol. 23A, no. 6, pp. 1–10, 2001Chinese. 2 Y. Guo and J. Tian, “Two positive solutions for second-order quasilinear differential equation
boundary value problems with sign changing nonlinearities,” Journal of Computational and Applied Mathematics, vol. 169, no. 2, pp. 345–357, 2004.
3 J. Henderson and A. Peterson, “Boundary value problems for functional difference equations,” Applied Mathematics Letters, vol. 9, no. 3, pp. 57–61, 1996.
4 S. J. Yang, B. Shi, and M. J. Gai, “Boundary value problems for functional differential systems,”Indian Journal of Pure and Applied Mathematics, vol. 36, no. 12, pp. 685–705, 2005.
5 X. Deng and H. Shi, “On boundary value problems for second order nonlinear functional difference equations,”Acta Applicandae Mathematicae, vol. 110, no. 3, pp. 1277–1287, 2009.
6 S. K. Ntouyas, Y. G. Sficas, and P. Ch. Tsamatos, “Boundary value problems for functional-differential equations,”Journal of Mathematical Analysis and Applications, vol. 199, no. 1, pp. 213–230, 1996. 7 F.-H. Wong, “Existence of positive solutions for m-Laplacian boundary value problems,” Applied
Mathematics Letters, vol. 12, no. 3, pp. 11–17, 1999.
8 P. X. Weng and Z. H. Guo, “Existence of positive solutions to BVPs for a nonlinear functional difference equation withp-Laplacian operator,”Acta Mathematica Sinica, vol. 49, no. 1, pp. 187–194, 2006.
9 S. J. Yang, B. Shi, and D. C. Zhang, “Existence of positive solutions for boundary value problems of nonlinear functional difference equation withp-Laplacian operator,”Boundary Value Problems, vol. 2007, Article ID 38230, 12 pages, 2007.
10 Z. Zhao, “Solutions and Green’s functions for some linear second-order three-point boundary value problems,”Computers and Mathematics with Applications, vol. 56, no. 1, pp. 104–113, 2008Chinese. 11 D. J. Guo, J. X. Sun, and Z. L. Liu,Functional Methods of Nonlinear Ordinary Differential Equations,
