Hindawi Publishing Corporation
Boundary Value Problems
Volume 2009, Article ID 937064, 19 pages
doi:10.1155/2009/937064
Research Article
Positive Solutions to Singular and
Delay Higher-Order Differential Equations on
Time Scales
Liang-Gen Hu,1 Ti-Jun Xiao,2 and Jin Liang3
1
Department of Mathematics, University of Science and Technology of China, Hefei 230026, China
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
3
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
2
Correspondence should be addressed to Jin Liang, jinliang@sjtu.edu.cn
Received 21 March 2009; Accepted 1 July 2009
Recommended by Juan José 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.
Copyright q 2009 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
problem BVP for short for delay higher-order dynamic equations on time scales:
−1n uΔ t wtft, ut − c,
2n
ut ψt,
uΔ a − βi1 uΔ
2i
t ∈ a − c, a,
2i1
a αi1 uΔ ,
γi1 uΔ uΔ b,
2i
t ∈ a, b,
2i
2i
1.1
0 ≤ i ≤ n − 1,
where c ∈ 0, b − a/2, ∈ a, b, βi ≥ 0, 1 < γi < b − a βi / − a βi , 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 and f : a, b × 0, ∞ → 0, ∞ is continuous. Our
2
Boundary Value Problems
nonlinearity w may have singularity at t a and/or t b, and f may have singularity at
u 0.
To understand the notations used in 1.1, we recall the following definitions which
can be found in 1, 2.
a A time scale T is a nonempty closed subset of the real numbers R. T has the topology
that it inherits from the real numbers with the standard topology. It follows that the
jump operators σ, ρ : T → T,
σt inf{τ ∈ T : τ > t},
ρt sup{τ ∈ T : τ < t}
1.2
supplemented by inf ∅ : sup T and sup ∅ : inf T are well defined. The point
t ∈ T is left-dense, left-scattered, right-dense, right-scattered if ρt t, ρt < t,
σt t, σt < t, respectively. If T has a left-scattered maximum t1 right-scattered
minimum t2 , define Tk T − {t1 } Tk T − {t2 }; otherwise, set Tk T Tk T.
By an interval a, b we always mean the intersection of the real interval a, b with
the given time scale, that is, a, b ∩ T. Other types of intervals are defined similarly.
b For a function f : T → R and t ∈ Tk , the Δ-derivative of f at t, denoted by f Δ t,
is the number provided it exists with the property that, given any ε > 0, there is a
neighborhood U ⊂ T of t such that
fσt − fs − f Δ tσt − s ≤ ε|σt − s|,
∀s ∈ U.
1.3
c For a function f : T → R and t ∈ Tk , the ∇-derivative of f at t, denoted by f ∇ t,
is the number provided it exists with the property that, given any ε > 0, there is a
neighborhood U ⊂ T of t such that
f ρt − fs − f ∇ t ρt − s ≤ ερt − s,
∀s ∈ U.
1.4
d If F Δ t ftΦ∇ t gt, then we define the integral
t
a
fΔ Ft − Fa
t
g∇ Φt − Φa .
1.5
a
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’ attention see, e.g., 1–19
and references therein. Moreover, singular boundary value problems have also been treated
in many papers see, e.g., 4, 5, 12–14, 18 and references therein.
Boundary Value Problems
3
In 2004, J. J. DaCunha et al. 13 considered singular second-order three-point
boundary value problems on time scales
uΔΔ t ft, ut 0,
u0 0,
0, 1 ∩ T,
u p u σ 2 1
1.6
and obtained the existence of positive solutions by using a fixed point theorem due to Gatica
et al. 14, where f : 0, 1 × 0, ∞ → 0, ∞ is decreasing in u for every t ∈ 0, 1 and may
have singularity at u 0.
In 2006, Boey and Wong 11 were concerned with higher-order differential equation
on time scales of the form
n
−1n−1 yΔ t −1p1 F t, y σ n−1 t ,
yΔ a 0,
i
t ∈ a, b,
0 ≤ i ≤ p − 1,
yΔ σb 0,
i
1.7
p ≤ i ≤ n − 1,
where p, n are fixed integers satisfying n ≥ 2, 1 ≤ p ≤ n − 1. They obtained some existence
theorems of positive solutions by using Krasnosel’skii fixed point theorem.
Recently, Anderson and Karaca 8 studied 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 higherorder 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. In Section 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 BVP 1.1. Moreover, we give an example to illustrate our
main result.
2. Lemmas
For 1 ≤ i ≤ n, let Gi t, s be Green’s function of the following three-point boundary value
problem:
−uΔΔ t 0,
ua − βi uΔ a αi u,
t ∈ a, b,
γi u ub,
2.1
where ∈ a, b and αi , βi , γi satisfy the following condition:
C
βi ≥ 0,
b − a βi
1 < γi <
,
− a βi
b − γi γi − 1 a − βi
.
0 ≤ αi <
b−
2.2
4
Boundary Value Problems
Throughout the paper, we assume that σb b.
From 8, we know that for any t, s ∈ a, b × a, b and 1 ≤ i ≤ n,
Gi t, s ⎧
⎨Gi1 t, s,
s ∈ a, ,
⎩G t, s,
i2
s ∈ , b,
2.3
where
⎧
σs ≤ t,
1 ⎨ γi t − b − t σs βi − a ,
Gi1 t, s di ⎩ γ σs − ω b − σs t β − a α − bt − σs, t ≤ s,
i
i
i
⎧
1 ⎨ σs1 − αi αi βi − a b − t γi − a βi t − σs, σs ≤ t,
Gi2 t, s di ⎩t1 − α α β − ab − σs,
t ≤ s,
i
i
i
di γi − 1 a − βi 1 − αi b αi − γi .
2.4
The following four lemmas can be found in 8.
Lemma 2.1. Suppose that the condition (C) holds. Then the Green function of Gi t, s in 2.3 satisfies
Gi t, s > 0,
t, s ∈ a, b × a, b.
2.5
Lemma 2.2. Assume that the condition (C) holds. Then Green’s function Gi t, s in 2.3 satisfies
Gi t, s ≤ max{Gi b, s, Gi σs, s},
t, s ∈ a, b × a, b.
2.6
Remark 2.3. 1 If s ∈ γi − a βi − αi − βi a/1 − αi , b, s ≤ t, we know that Gi t, s is
nonincreasing in t and
γi − a βi b − σs
Gi b, s
Gi σs, s
σs1 − αi αi βi − a b − σs
γi − a βi
> 0.
≥
b1 − αi αi βi − a
2.7
Gi b, s ≤ Gi t, s ≤ Gi σs, s ≤ δi Gi b, s,
2.8
Therefore, we have
where
δi b1 − αi αi βi − a
> 1.
γi − a βi
2.9
Boundary Value Problems
5
2 If t and s satisfy the other cases, then we get that Gi t, s is nondecreasing in t and
Gi t, s ≤ Gi b, s.
2.10
Lemma 2.4. Assume that (C) holds. Then Green’s function Gi t, s in 2.3 verifies the following
inequality:
b−t
t−a
,
Gi b, s
Gi t, s ≥ min
b − a γi b − a
t−a
b−t
≥ min
,
max{Gi b, s, Gi σs, s}.
δi b − a γi b − a
2.11
Remark 2.5. If s ∈ , γi − a βi − αi − βi a/1 − αi , s ≤ t, then we find
γi − 1 a − βi 1 − αi σs αi − γi < 0.
2.12
So there exists a misprint on 8, Page 2431, line 23. From 2.3, it follows that
σs1 − αi αi βi − a b − t γi − a βi t − σs
Gi t, s
Gi b, s
γi − a βi b − σs
βi − a b − t γi − a βi t − σs
b−t
.
≥
≥
γi b − a
γi − a βi b − a
2.13
Consequently, we get
Gi t, s ≥
b−t
Gi b, s.
γi b − a
2.14
If s ∈ γ i − a βi − αi − βi a/1 − αi , b, s ≤ t, then, from 2.8, we obtain
Gi t, s ≥
t−a
t−a
Gi b, s ≥
Gi σs, s.
b−a
δi b − a
2.15
6
Boundary Value Problems
Remark 2.6. If we set hi t : min{t − a/δi b − a, b − t/γi b − a}, then we have
Gi t, s ≥ hi t max{Gi b, s, Gi σs, s},
t, s ∈ a, b × a, b.
2.16
Denote
Gi ·, s max |Gi t, s|,
s ∈ a, b.
t∈a,b
2.17
Thus we have
Gi t, s ≥ hi tGi ·, s,
t, s ∈ a, b × a, b.
2.18
Lemma 2.7. Assume that condition (C) is satisfied. For Gi t, s as in 2.3, put H1 t, s : G1 t, s
and recursively define
Hj t, s b
2.19
Hj−1 t, rGj r, sΔr
a
for 2 ≤ j ≤ n. Then Hn t, s is Green’s function for the homogeneous problem
−1n uΔ t 0,
2n
uΔ a − βi1 uΔ
2i
2i1
t ∈ a, b,
a αi1 uΔ ,
2i
γi1 uΔ uΔ b,
2i
2i
2.20
0 ≤ i ≤ n − 1.
Lemma 2.8. Assume that (C) holds. Denote
K :
n−1
kj ,
L :
j1
n−1
lj ,
2.21
j1
then Green’s function Hn t, s in Lemma 2.7 satisfies
h1 tLGn ·, s ≤ Hn t, s ≤ KGn ·, s,
t, s ∈ a, b × a, b,
2.22
where
kj b
a
Gj ·, sΔs > 0,
lj b
a
Gj ·, shj1 sΔs,
1 ≤ j ≤ n − 1.
2.23
Boundary Value Problems
7
Proof. We proceed by induction on n ≥ 2. We denote the statement by P n. From Lemma 2.7,
it follows that
b
H2 t, s H1 t, rG2 r, sΔr a
≤
b
2.24
G1 ·, rG2 ·, sΔr k1 G2 ·, s,
a
and from 2.18, we have
H2 t, s b
H1 t, rG2 r, sΔr
a
≥
b
h1 tG1 ·, r × h2 rG2 ·, sΔr
2.25
a
h1 tl1 G2 ·, s.
So P 2 is true.
We now assume that P m is true for some positive integer m ≥ 2. From Lemma 2.7, it
follows that
b
Hm1 t, s Hm t, rGm1 r, sΔr a
≤
b
Hm t, rGm1 r, sΔr
a
⎛
⎞
b
m−1
≤⎝
kj × Gm ·, rΔr ⎠Gm1 ·, s
a j1
m
kj Gm1 ·, s,
j1
Hm1 t, s b
Hm t, rGm1 r, sΔr
a
≥
b
h1 t ×
a
h1 t
m−1
lj Gm ·, rhm1 rGm1 ·, sΔr
j1
m
lj Gm1 ·, s.
j1
So P m 1 holds. Thus P n is true by induction.
2.26
8
Boundary Value Problems
Lemma 2.9 see 20. Let E, · be a real Banach space and P ⊂ E a cone. Assume that T :
Pζ,η → P is completely continuous operator such that
i T u u for u ∈ ∂Pζ and T u u for u ∈ ∂Pη ,
ii T u u for u ∈ ∂Pζ and T u u for u ∈ ∂Pη .
Then T has a fixed point u∗ ∈ P with ζ ≤ u∗ ≤ η.
3. Main Results
We assume that {am }m≥1 and {bm }m≥1 are strictly decreasing and strictly increasing sequences,
respectively, with limm → ∞ am a, limm → ∞ bm b and a1 < b1 . A Banach space E Ca, b is
the set of real-valued continuous in the topology of T functions ut defined on a, b with
the norm
u max |ut|.
3.1
t∈a,b
Define a cone by
P
h1 tL
u ∈ E : ut ≥
u, t ∈ a, b .
K
3.2
Set
Pξ {u ∈ P : u < ξ}, ∂Pξ {u ∈ P : u ξ},
Pζ,η u ∈ P : ζ < u < η , 0 < ζ < η,
Y1 {t ∈ a, b : t − c < a},
ξ > 0,
Y2 {t ∈ a, b : t − c ≥ a},
3.3
Ym {t ∈ Y2 : t − c ∈ a, am ∪ bm , b}.
Assume that
C1 ψ : a − c, a → 0, ∞ is continuous;
C2 we have
q
0<K
Gn ·, swsΔs,
p
b
K
Gn ·, swsΔs < ∞,
3.4
a
for constants p and q with a c < p < q < b;
C3 the function f : a, b × 0, ∞ → R is continuous and w : a, b → R is
continuous satisfying
Gn ·, swsfs, us − cΔs 0,
lim sup K
m → ∞ u∈P
ζ,η
Ym
∀0 < ζ < η.
3.5
Boundary Value Problems
9
We seek positive solutions u : a, b → R , satisfying 1.1. For this end, we transform
1.1 into an integral equation involving the appropriate Green function and seek fixed points
of the following integral operator.
Define an operator T : C a, b → Ca, b by
T ut b
Hn t, swsfs, us − cΔs,
∀u ∈ C a, b,
3.6
a
where C a, b {u ∈ Ca, b | ut ≥ 0, t ∈ a, b}.
Proposition 3.1. Let (C1), (C2), and (C3) hold, and let ζ, η be fixed constants with 0 < ζ < η. Then
T : Pζ,η → P is completely continuous.
Proof. We separate the proof into four steps.
Step 1. For each u ∈ Pζ,η , T u is bounded.
By condition C3, there exists some positive integer m0 satisfying
Gn ·, swsfs, us − cΔs ≤ 1,
sup K
u∈Pζ,η
Ym0
3.7
where
Ym0 {t ∈ Y2 : t − c ∈ a, am0 ∪ bm0 , b};
3.8
here, we used the fact that for each u ∈ Pζ,η and t ∈ am0 c, bm0 c ∩ a, b,
h1 am0 L h1 bm0 L h1 b − cL
h1 t − cL
,
,
η ≥ ut − c ≥
ζh > 0,
u ≥ ζ min
K
K
K
K
3.9
where
h min
h1 am0 L h1 bm0 L h1 b − cL
,
,
.
K
K
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
10
Boundary Value Problems
Then we obtain
b
T ut ≤ sup sup
Hn t, swsfs, us − cΔs
a
t∈a,b u∈Pζ,η
Gn ·, swsfs, us − cΔs
≤ K sup
Y1
u∈Pζ,η
Gn ·, swsfs, us − cΔs
sup K
3.12
Ym0
u∈Pζ,η
sup K
u∈Pζ,η
Y2 \Ym0
Gn ·, swsfs, us − cΔs
≤ 1 max{D, Q}K
b
Gn ·, swsΔs < ∞.
a
Consequently, T u is bounded and well defined.
Step 2. T : Pζ,η → P . For every u ∈ Pζ,η , we get from 2.22
T u sup
b
Hn t, swsfs, us − cΔs
t∈a,b a
≤K
b
3.13
Gn ·, swsfs, us − cΔs.
a
Then by the above inequality
T ut b
Hn t, swsfs, us − cΔs
a
≥
b
h1 tLGn ·, swsfs, us − cΔs
a
≥
This leads to T u ∈ P .
h1 tL
T u.
K
3.14
Boundary Value Problems
11
Step 3. We will show that T : Pζ,η → P is continuous. Let {um }m≥1 be any sequence in Pζ,η
such that limm → ∞ um u ∈ Pζ,η . Notice also that as m → ∞,
φm s fs, um s − c − fs, us − cws −→ 0,
for s ∈ a c, b,
fs, um s − c − fs, us − cws
f s, ψs − c − f s, ψs − c ws 0,
for s ∈ a, a c,
3.15
Hn t, sφm sΔs ≤ sup 2K
Y2
x∈Pζ,η
Gn ·, swsfs, xsΔs < ∞.
Y2
Now these together with C2 and the Lebesgue dominated convergence theorem 10 yield
that as m → ∞,
T um − T u sup
b
Hn t, swsfs, um s − c − fs, us − cΔs −→ 0.
3.16
t∈a,b a
Step 4. T : Pζ,η → P is compact.
Define
⎧
⎪
min{wt, wam }, a ≤ t ≤ am ,
⎪
⎪
⎪
⎨
wm t wt,
am ≤ t ≤ bm ,
⎪
⎪
⎪
⎪
⎩
min{wt, wbm }, bm ≤ t ≤ b,
fm t, ut − c ⎧ ⎪
⎪
⎪f t, ψt − c ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨min ft, ut − c, ft, uam ,
⎪
⎪
⎪
ft, ut − c,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩minft, ut − c, ft, ub ,
m
a ≤ t < a c,
3.17
a c ≤ t ≤ am c,
t ∈ am c, bm c ∩ a, b,
t ∈ bm c, b ∩ a, b,
and an operator sequence {Tm } for a fixed m by
Tm ut b
Hn t, swm sfm s, us − cΔs,
∀t ∈ a, b.
3.18
a
Clearly, the operator sequence {Tm } is compact by using the Arzela-Ascoli theorem
3, for each m ∈ N. We will prove that Tm converges uniformly to T on Pζ,η . For any u ∈ Pζ,η ,
12
Boundary Value Problems
we obtain
b
Tm u − T u sup Hn t, s wm sfm s, us − c − wsfs, us − c Δs
a
t∈a,b
≤K
b
Gn ·, swm sfm s, us − c − wsfs, us − cΔs
3.19
a
Gn ·, s|wm s − ws|f s, ψs − c Δs
≤K
Y1
Gn ·, swm sfm s, us − c − wsfs, us − cΔs.
K
Y2
From C1, C2, and the Lebesgue dominated convergence theorem 10, we see that the
right-hand side 3.19 can be sufficiently small for mbeing big enough. Hence the sequence
{Tm } of compact operators converges uniformly to T on Pζ,η so that operator T is compact.
Consequently, T : Pζ,η → P is completely continuous by using the Arzela-Ascoli theorem
3.
Proposition 3.2. It holds that v ∈ Pζ,η is a solution of 1.1 if and only if T v v.
Proof. If v ∈ Pζ,η and T v v, then we have
−1n vΔ t −1n T vΔ t wtft, vt − c,
2n
2n
3.20
and for any 0 ≤ i ≤ n − 1,
vΔ a − βi1 vΔ
2i
2i1
a αi1 vΔ ,
2i
γi1 vΔ vΔ b.
2i
2i
3.21
From 8, Lemma 3.1, we know that vt ≥ 0 on a, b. So we conclude that v is the solution
of BVP 1.1.
For convenience, we list the following notations and assumptions:
R
q
μK
−1
Gn ·, swsΔs
p
b
−1
κ K Gn ·, swsΔs ;
a
h1 p L h1 q L
,
;
μ min
K
K
,
3.22
Boundary Value Problems
ξ
fμξ :
13
ft, ut − c
,
ut
t ∈ p, q , u ∈ μξ, ξ ;
ft, ut − c
, t ∈ Y2 , u ∈ ρ, ζ ;
ut
S ρ sup K Gn ·, swsfs, us − cΔs,
3.23
ζ
3.24
fρ :
u∈∂Pρ
ρ > 0.
3.25
Y2
From condition C2 and 3.12, we have Sρ < ∞.
Theorem 3.3. Assume that there exist positive constants ρ, ζ, ξ, r with ζ < μξ, r < κ and ζ ≥
κSρ/κ − r such that
ξ
ζ
i fμξ > R and fρ < r;
ii ft, ψt − c/ut < r, for all t ∈ Y1 and u ∈ ρ, ζ.
If (C1), (C2), and (C3) hold, then the boundary value problem 1.1 has at least one positive solution
u
such that
u
t ⎧
⎨ψt,
if t ∈ a − c, a,
⎩u∗ t,
if t ∈ a, b,
3.26
ζ ≤ u∗ ≤ ξ.
Proof. Define the operator T : Pζ,ξ → P by 3.6. From i and 3.23, it follows that there
exists ε1 > 0 such that
ft, ut − c ≥ R ε1 ut,
for t ∈ p, q , u ∈ μξ, ξ .
3.27
We claim that
T u u,
∀u ∈ ∂Pξ .
3.28
If it is false, then there exists some u1 ∈ ∂Pξ with T u1 ≤ u1 , that is, u1 − T u1 ∈ P which implies
that u1 t ≥ T u1 t for t ∈ a, b.
Set
h1 p L h1 q L
,
λ min u1 t : t ∈ p, q ≥ min
u1 μξ.
K
K
3.29
14
Boundary Value Problems
We know from 2.22 and 3.27 that for t ∈ p, q,
u1 t ≥ T u1 t
b
Hn t, swsfs, u1 s − cΔs
a
Hn t, swsfs, u1 s − cΔs Y1
≥
q
Hn t, swsfs, u1 s − cΔs
Y2
Hn t, swsfs, u1 s − cΔs
p
≥ min h1 p , h1 q L
q
3.30
Gn ·, swsfs, u1 s − cΔs
p
≥ R ε1 min u1 tμK
t∈p,q
≥ λR μK
q
q
Gn ·, swsΔs
p
Gn ·, swsΔs λε1 μK
q
p
λ λε1 μK
Gn ·, swsΔs
p
q
Gn ·, swsΔs,
p
the first inequality of C2 implies that
∀t ∈ p, q .
u1 t > λ,
3.31
Clearly, 3.31 contradicts 3.29. This means that 3.28 holds.
Next we will show that
T u u,
3.32
∀u ∈ Pζ .
Suppose on the contrary that there exists some u2 ∈ ∂Pζ with u2 ≤ T u2 for all t ∈ a, b.
For t, u ∈ Y2 × ρ, ζ, from i and 3.24, there exists ε2 > 0 such that
ft, ut − c ≤ r − ε2 ut.
3.33
and for t, u ∈ Y1 × ρ, ζ, there exists ε2 > 0, from ii, such that
f t, ψt − c ≤ r − ε2 ut.
3.34
Put
Y3 : t ∈ Y2 : u2 t > ρ ,
u
2 t ⎧
⎨min u2 t, ρ ,
t ∈ Y2 ,
⎩ρ,
t ∈ Y1 .
3.35
Boundary Value Problems
15
2 t ρ. It is easy to see that h1 t − cLζ/K ≤ u2 t − c ≤ u2 ζ
If Y3 ∅, then we take u
2 t ∈ C a, b, u2 ρ, that is, u
2 ∈ ∂Pρ . From 3.33 and 3.34, we find that
for t ∈ Y2 and u
T u2 sup
b
Hn t, swsfs, u2 s − cΔs
t∈a,b a
≤K
b
Gn ·, swsfs, u2 s − cΔs
a
Gn ·, swsf s, ψs − c Δs K
K
Gn ·, swsfs, u2 s − cΔs
Y1
Y3
K
Y2 \Y3
Gn ·, swsfs, u2 s − cΔs
≤ r − ε2 max u2 t
t∈Y1
Gn ·, swsΔs
Y1
ft, u2 t − cK
sup
3.36
t,u2 ∈Y3 ×ρ,ζ
Gn ·, swsΔs
Y3
2 s − cΔs
Gn ·, swsfs, u
sup K
u
2 ∈∂Pρ
≤ ζrK
b
Y2
Gn ·, swsΔs S ρ − ζε2 K
a
b
Gn ·, swsΔs
a
ζrκ−1 − ζε2 κ−1 S ρ
< ζ u2 yielding a contradiction with u2 ≤ T u2 for all t ∈ a, b. This means that 3.32 holds.
Therefore, from 3.28, 3.32 and Lemma 2.9, we conclude that the operator T has at least
one fixed point u∗ ∈ Pζ,ξ . From the definition of the cone P and 2.18, we see that u∗ t > 0
for all t ∈ a, b. Thus, Proposition 3.2 implies that u∗ is a solution of BVP 1.1. So we obtain
the desired result.
Adopting the same argument as in Theorem 3.3 , we obtain the following results.
ζ
Corollary 3.4. Let ρ, ζ, r, fρ be as in Theorem 3.3. Suppose that (ii) of Theorem 3.3 holds and
ξ
limξ → ∞ fμξ
∞. If (C1), (C2), and (C3) holds , then boundary value problem 1.1 has at least
one positive solution u
∈ Pζ,η such that
u
t ⎧
⎨ψt,
if t ∈ a − c, a,
⎩u∗∗ t,
if t ∈ a, b,
ζ ≤ u∗∗ ≤ η,
ζ < μη.
3.37
16
Boundary Value Problems
Theorem 3.5. Assume that there exist positive constants ρi , ζi , ξi , r with ζi < μξi , r < κ and ζi ≥
κSρi /κ − r, i 1, 2, . . . , m such that
ξ
ζ
iii fμξi i > R and fρii < r;
iv ft, ψt − c/ut < r, for all t ∈ Y1 and u ∈ ρi , ζi .
If (C1), (C2), and (C3) hold, then boundary value problem 1.1 has at least m positive solutions
u
i ∈ Pζi ,ξi such that for i 1, 2, . . . , m
u
i t ⎧
⎨ψt,
if t ∈ a − c, a,
⎩u∗ t,
if t ∈ a, b
ζ ≤ u∗ ≤ ξ.
3.38
i
i
Example 3.6. Let T R. Consider the following singular three-point boundary value problems
for delay four-order dynamic equations:
u4 t ft, ut − 1 0,
1
u1,
2
1
u 0 u 1,
2
ut et ,
u0 t ∈ 0, 4,
2u1 u4,
3.39
2u 1 u 4,
t ∈ −1, 0,
where, for any t ∈ 0, 4, ρ 1, ζ 1480, μ 0.112, ξ 13500, M1 1 and M2 1/502 ,
⎧
⎪
2M1 ut,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
π ut − ϑφ
π ut − ϑφ
⎪
⎪
⎪
,
cos ⎪M1 ut 1sin ⎪
⎪
2 ϑ − ϑφ
2 ϑ − ϑφ
⎪
⎪
⎪
⎪
⎪
⎪
⎪
π ut − μ
π ut − μ
⎪
1
⎪
⎪
2M1 ϑφ sin ,
⎨ M2 ut cos 2
2 ϑφ − μ
2 ϑφ − μ
ft, ut − 1 ⎪
⎪
⎪
⎪
π ut − π ut − 1
⎪
⎪
⎪ M2 ut 2 − sin − cos ,
⎪
⎪
2
2 μ−
2 μ−
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1
⎪
⎪
ρut−1/2 − ut1/2 M2 ρ,
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
⎩ 1 M2 ut,
2
t, u ∈ 1, 4×ξ, ∞,
t, u ∈ 1, 4× μξ, ξ ,
t, u ∈ 1, 4× ζ, μξ ,
t, u ∈ 1, 4 × ρ, ζ ,
t, u ∈ 1, 4 × 0, ρ ,
t, u ∈ 0, 1 × R.
3.40
Boundary Value Problems
17
Clearly, we know that
α
1
,
2
β 0,
3
p ,
2
γ 2,
δ
t 4−t
hi t min ,
,
5 8
7
q ,
2
G4, s 12s
η 1,
s ∈ 0, 1,
5
,
4
d
1
,
2
i 1, 2,
3.41
G4, s 44 − s s ∈ 1, 3,
Gs, s 4 − s1 s s ∈ 3, 4.
Simple computations yield
K
4
G1 ·, sds 1
0
L
4
12sds 3
0
44 − sds 1
4
1 s4 − sds 24.17,
3
G1 ·, sh2 sds
0
1
s
12s ds 5
0
20/13
1
s
44 − s ds 5
3
4−s
ds 44 − s
8
20/13
4
4 − s2 1 s
ds
8
3
4.695,
3.42
h1 p L h1 q L
μ min
,
0.112,
K
K
R
7/2
μK
−1
G2 ·, sds
0.282,
3/2
4
−1
κ K G2 ·, sds
0
1
.
24.172
Obviously,
G2 ·, s fs, us − cds 0,
lim sup K
m → ∞ u∈P
ζ,η
∀ 0 < ζ < η.
Ym
3.43
If t, u ∈ 1, 4 × 0, 1, then we have
ht htρ ≤ ut ≤ ρ 1.
3.44
Therefore, we get
1
ft, ut − 1 ≤ ht−1/2 − ht1/2 M2 ,
2
for t, u ∈ 1, 4 × 0, 1.
3.45
18
Boundary Value Problems
From 3.25, it follows that
S1 sup K
4
≤K
G2 ·, sfs, us − 1ds
1
u∈∂P1
20/13
12s
1
K
5
s
!1/2
3
44 − s
20/13
K
4
−
s
5
8
4−s
1 s4 − s
3
!1/2
!1/2
8
4−s
1
M2 ds
2
−
!1/2
1
M2 ds
2
!
4 − s 1/2 1
M2 ds
8
2
4−s
8
−
!1/2
3.46
≤ 1120.
Thus,
ζ 1480 ≥
κS1
≈ 1461.37,
κ−r
ξ 13500,
ζ < μξ.
3.47
Therefore, by Theorem 3.3 , the BVP 3.39 has at least one positive solution u
such that
u
t ⎧
⎨et ,
if t ∈ −1, 0,
⎩u∗ t,
if t ∈ 0, 4,
3.48
1480 ≤ u∗ ≤ 13500.
Acknowledgments
The authors would like to thank the referees for helpful comments and suggestions. The work
was supported partly by the NSF of China 10771202, the Research Fund for Shanghai Key
Laboratory of Modern Applied Mathematics 08DZ2271900, and the Specialized Research
Fund for the Doctoral Program of Higher Education of China 2007035805.
References
1 F. M. Atici and G. Sh. Guseinov, “On Green’s functions and positive solutions for boundary value
problems on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 75–99,
2002.
2 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application,
Birkhäuser, Boston, Mass, USA, 2001.
3 R. P. Agarwal, M. Bohner, and P. Rehák, “Half-linear dynamic equations,” in Nonlinear Analysis
and Applications: To V. Lakshmikantham on His 80th Birthday. Vol. 1, 2, pp. 1–57, Kluwer Academic
Publishers, Dordrecht, The Netherlands, 2003.
4 R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Multiple positive solutions of singular
Dirichlet problems on time scales via variational methods,” Nonlinear Analysis: Theory, Methods &
Applications, vol. 67, no. 2, pp. 368–381, 2007.
Boundary Value Problems
19
5 R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Multiple positive solutions in the sense
of distributions of singular BVPs on time scales and an application to Emden-Fowler equations,”
Advances in Difference Equations, vol. 2008, Article ID 796851, 13 pages, 2008.
6 B. Ahmad and J. J. Nieto, “The monotone iterative technique for three-point second-order
integrodifferential boundary value problems with p-Laplacian,” Boundary Value Problems, vol. 2007,
Article ID 57481, 9 pages, 2007.
7 D. R. Anderson, “Solutions to second-order three-point problems on time scales,” Journal of Difference
Equations and Applications, vol. 8, no. 8, pp. 673–688, 2002.
8 D. R. Anderson and I. Y. Karaca, “Higher-order three-point boundary value problem on time scales,”
Computers & Mathematics with Applications, vol. 56, no. 9, pp. 2429–2443, 2008.
9 D. R. Anderson and G. Smyrlis, “Solvability for a third-order three-point BVP on time scales,”
Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1994–2001, 2009.
10 B. Aulbach and L. Neidhart, “Integration on measure chains,” in Proceedings of the 6th International
Conference on Difference Equations, pp. 239–252, CRC Press, Boca Raton, Fla, USA, 2004.
11 K. L. Boey and P. J. Y. Wong, “Positive solutions of two-point right focal boundary value problems on
time scales,” Computers & Mathematics with Applications, vol. 52, no. 3-4, pp. 555–576, 2006.
12 A. Cabada and J. Á. Cid, “Existence of a solution for a singular differential equation with nonlinear
functional boundary conditions,” Glasgow Mathematical Journal, vol. 49, no. 2, pp. 213–224, 2007.
13 J. J. DaCunha, J. M. Davis, and P. K. Singh, “Existence results for singular three point boundary value
problems on time scales,” Journal of Mathematical Analysis and Applications, vol. 295, no. 2, pp. 378–391,
2004.
14 J. A. Gatica, V. Oliker, and P. Waltman, “Singular nonlinear boundary value problems for secondorder ordinary differential equations,” Journal of Differential Equations, vol. 79, no. 1, pp. 62–78, 1989.
15 J. Henderson, C. C. Tisdell, and W. K. C. Yin, “Uniqueness implies existence for three-point boundary
value problems for dynamic equations,” Applied Mathematics Letters, vol. 17, no. 12, pp. 1391–1395,
2004.
16 E. R. Kaufmann and Y. N. Raffoul, “Positive solutions for a nonlinear functional dynamic equation on
a time scale,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 7, pp. 1267–1276, 2005.
17 R. A. Khan, J. J. Nieto, and V. Otero-Espinar, “Existence and approximation of solution of three-point
boundary value problems on time scales,” Journal of Difference Equations and Applications, vol. 14, no.
7, pp. 723–736, 2008.
18 J. Liang, T.-J. Xiao, and Z.-C. Hao, “Positive solutions of singular differential equations on measure
chains,” Computers & Mathematics with Applications, vol. 49, no. 5-6, pp. 651–663, 2005.
19 İ. Yaslan, “Multiple positive solutions for nonlinear three-point boundary value problems on time
scales,” Computers & Mathematics with Applications, vol. 55, no. 8, pp. 1861–1869, 2008.
20 D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in
Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.