Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 261741, 19 pages
doi:10.1155/2010/261741
Research Article
Multiple Positive Solutions of a Second Order
Nonlinear Semipositone m-Point Boundary Value
Problem on Time Scales
Chengjun Yuan1, 2 and Yongming Liu1
1
2
Department of Mathematics, East China Normal University, Shanghai 200241, China
School of Mathematics and Computer, Harbin University, Harbin, Heilongjiang 150086, China
Correspondence should be addressed to Chengjun Yuan, ycj7102@163.com
Received 28 April 2010; Revised 26 June 2010; Accepted 7 September 2010
Academic Editor: Allan C Peterson
Copyright q 2010 C. Yuan and Y. Liu. 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.
In this paper, we study a general second-order m-point boundary value problem for nonlinear
singular dynamic equation on time scales uΔ∇ tatuΔ tbtutλqtft, ut 0, t ∈ 0, 1T ,
uρ0 0, uσ1 m−2
i1 αi uηi . This paper shows the existence of multiple positive solutions
if f is semipositone and superlinear. The arguments are based upon fixed-point theorems in a cone.
1. Introduction
In this paper, we consider the following dynamic equation on time scales:
uΔ∇ t atuΔ t btut λqtft, ut 0,
u ρ0 0,
uσ1 t ∈ 0, 1T ,
m−2
αi u ηi ,
1.1
i1
where αi ≥ 0, 0 < ηi < ηi1 < 1; for all i 1, 2, . . . , m − 2; f, q, a and b satisfy
C1 q ∈ L is continuously and nonnegative function and there exists t0 ∈ ρ0, σ1 s.t.
qt0 > 0, qt may be singular at t ρ0, σ1;
C2 a ∈ C0, 1, 0, ∞, b ∈ C0, 1, −∞, 0.
In the past few years, the boundary value problems of dynamic equations on time
scales have been studied by many authors see 1–15 and references therein. Recently,
2
Abstract and Applied Analysis
multiple-point boundary value problems on time scale have been studied, for instance, see
1–9.
In 2008, Lin and Du 2 studied the m-point boundary value problem for second-order
dynamic equations on time scales:
uΔ∇ t ft, u 0,
u0 0,
t ∈ 0, T ∈ T,
uT m−2
ki uξi ,
1.2
i1
where T is a time scale. This paper deals with the existence of multiple positive solutions for
second-order dynamic equations on time scales. By using Green’s function and the LeggettWilliams fixed point theorem in an appropriate cone, the existence of at least three positive
solutions of the problem is obtained.
In 2009, Topal and Yantir 1 studied the general second-order nonlinear m-point
boundary value problems 1.1 with no singularities and the case. The authors deal with the
determining the value of λ; the existences of multiple positive solutions of 1.1 are obtained
by using the Krasnosel’skii and Legget-William fixed point theorems.
Motivated by the abovementioned results, we continue to study the general secondorder nonlinear m-point boundary value problem 1.1, but the nonlinear term may be
singularity and semipositone.
In this paper, the nonlinear term f of 1.1 is suit to and semipositone and the
superlinear case, we will prove our two existence results for problem 1.1 by using
Krasnosel’skii fixed point theorem. This paper is organized as follows. In Section 2, starting
with some preliminary lemmas, we state the Krasnosel’skii fixed point theorem. In Section 3,
we give the main result which state the sufficient conditions for the m-point boundary value
problem 1.1 to have existence of positive solutions.
2. Preliminaries
In this section, we state the preliminary information that we need to prove the main results.
From Lemmas 2.1 and 2.3 in 1, we have the following lemma.
Lemma 2.1 see 1. Assuming that (C2) holds. Then the equations
φ1Δ∇ t atφ1Δ t btφ1 t 0, t ∈ 0, 1T ,
φ1 ρ0 0,
φ1 σ1 1,
2.1
φ2Δ∇ t atφ2Δ t btφ2 t 0, t ∈ 0, 1T ,
φ2 ρ0 1,
φ2 σ1 0
2.2
have unique solutions φ1 and φ2 , respectively, and
a φ1 is strictly increasing on ρ0, σ1,
b φ2 is strictly decreasing on ρ0, σ1.
Abstract and Applied Analysis
3
For the rest of the paper we need the following assumption:
C3 0 <
m−2
i1
αi φ1 ηi < 1.
Lemma 2.2 see 1. Assuming that (C2) and (C3) hold. Let y ∈ Cρ0, σ1. Then boundary
value problem
xΔ∇ t atxΔ t btxt yt 0,
x ρ0 0,
xσ1 t ∈ 0, 1T ,
2.3
m−2
αi x ηi
i1
is equivalent to integral equation
xt σ1
ρ0
2.4
Ht, spsys∇s Aφ1 t,
where
pt ea ρt, ρ0 ,
m−2
σ1 αi
H ηi , s psys∇s,
m−2
1 − i1 αi φ1 ηi i1
ρ0
⎧
⎨φ1 sφ2 t, s ≤ t,
1
Ht, s Δ φ1 ρ0 ⎩φ1 tφ2 s, t ≤ s.
A
1
2.5
2.6
Proof. First we show that the unique solution of 2.3 can be represented by 2.4. From
Lemma 2.1, we know that the homogenous part of 2.3 has two linearly independent
solution φ1 and φ2 since
φ1 ρ0 φΔ ρ0
1
Δ
/ 0.
Δ
−φ1 ρ0 φ2 ρ0 φ ρ0 2
2.7
Now by the method of variations of constants, we can obtain the unique solution of
2.3 which can be represented by 2.4 where A and H are as in 2.5 and 2.6, respectively.
Next we check the function defined in 2.4 is the solution of the boundary value problem
2.3. For this purpose we first show that 2.4 satisfies 2.3. From the definition of Green’s
function 2.6, we get
1
xt Δ φ1 ρ0
t
ρ0
φ1 sφ2 tpsys∇s σ1
t
φ1 tφ2 spsys∇s
Aφ1 t.
2.8
4
Abstract and Applied Analysis
Hence, the derivatives xΔ and xΔ∇ are as follows:
1
x t Δ φ1 ρ0
Δ
x
Δ∇
1
t Δ φ1 ρ0
φ2Δ t
t
φ2Δ∇ t
ρ0
φ1 spsys∇s ρt
ρ0
φ1Δ∇ t
φ1Δ t
σ1
t
φ2 spsys∇s
Aφ1Δ t,
φ1 spsys∇s φ2Δ tφ1 tptyt
σ1
ρt
φ2 spsys∇s φ1Δ tφ2 tptyt
Aφ1Δ∇ t.
2.9
Replacing the derivatives in 2.3, we deduce that
xΔ∇ t atxΔ t btxt
A φ1Δ∇ t atφ1Δ t btφ1 t
1
Δ
φ1 ρ0
t
ρ0
φ1 spsys∇s
φ2Δ∇ t atφ2Δ t btφ2 t
1
Δ∇
Δ
φ
φ
atφ
btφ
spsys∇s
t
t
t
2
1
1
1
φ1Δ ρ0 t
ρt
t
1
Δ∇
Δ∇
Δ
φ1 spsys∇s φ1 t
φ2 spsys∇s
φ2 t
φ1 ρ0
t
ρt
σ1
1
φ2Δ tφ1 t − φ1Δ tφ2 t ptyt
ρ0
Δ
φ1
φ1Δ
φ1Δ
1
φ2Δ∇ t ρt − t φ1 tptyt − φ1Δ∇ tφ2 tptyt
ρ0
φ2Δ tφ1 tptyt − φ1Δ tφ2 tptyt
1
ptyt φ2Δ tφ1 t − φ1Δ tφ2 t
ρ0
1
ptyt ρt − t φ2Δ∇ tφ1 t − φ1Δ∇ tφ2 t
φ1 ρ0
∇ Δ
1
Δ
Δ
Δ
Δ
ptyt φ2 tφ1 t − φ1 tφ2 t ρt − t φ2 tφ1 t − φ1 tφ2 t
φ1 ρ0
Δ
1
ptyt φ2Δ ρt φ1 ρt − φ1Δ ρt φ2 ρt
ρ0
Δ
φ1
Abstract and Applied Analysis
5
1
ptyteΘa ρt, ρ0 −φ1Δ ρ0
ρ0
Δ
φ1
−yt.
2.10
Therefore the function defined in 2.4 satisfies 2.3. Further we obtain that the boundary
value conditions are satisfied by 2.4. The first condition follows from 2.5 and 2.6 and
Lemma 2.1. Now we verify the second boundary condition. Since
Hσ1, s φ1Δ
1
φ1 sφ2 σ1 0,
ρ0
2.11
we obtain that
xσ1 σ1
ρ0
2.12
Hσ1, spsys∇s Aφ1 σ1 A.
On the other hand, by using 2.5, we find that
m−2
m−2
αi x ηi αi
i1
i1
m−2
αi
σ1
ρ0
σ1
ρ0
i1
H ηi , s psys∇s Aφ1 ηi
H ηi , s psys∇s
1
1−
m−2
i1
αi φ1 ηi
m−2
1
αi
m−2
1 − i1 αi φ1 ηi i1
m−2
αi φ1 ηi
i1
σ1
ρ0
σ1
ρ0
2.13
H ηi , s psys∇s
H ηi , s psys∇s A.
Combining the two equations above finishes the proof.
Lemma 2.3. Green’s function Ht, s has the following properties:
Ht, s ≤ Ht, t,
φ1Δ ρ0
Ht, tHs, s ≤ Ht, s ≤ Hs, s,
φ1 φ2 Ht, t ≤ φ1 t
φ2 .
ρ0
2.14
φ1Δ
Lemma 2.4. Assume that (C2) and (C3) hold. Let u be a solution of boundary value problem 1.1 if
and only if u is a solution of the following integral equation:
ut σ1
ρ0
Gt, spsqsfs, us∇s,
2.15
6
Abstract and Applied Analysis
where
m−2
1
αi H ηi , s φ1 t.
m−2
1 − i1 αi φ1 ηi i1
Gt, s Ht, s 2.16
The proofs of the Lemmas 2.3 and 2.4 can be obtained easily by Lemmas 2.1 and 2.2.
Lemma 2.5. Green’s function Gt, s defined by 2.16 has the following properties:
C2 φ1 tHs, s ≤ Gt, s ≤ C1 Hs, s,
Gt, s ≤ C3 φ1 t,
2.17
where
C1 1 m−2
φ1 αi ,
m−2
1 − i1 αi φ1 ηi i1
m−2
φ1Δ ρ0
1
C2 αi H ηi , ηi ,
m−2
φ1 φ2 1 − i1 αi φ1 ηi i1
C3 2.18
m−2
φ2 1
αi H ηi , ηi .
m−2
Δ
φ1 ρ0
1 − i1 αi φ1 ηi i1
Proof. From Lemma 2.3, we have
m−2
1
αi Hs, sφ1 ≤ C1 Hs, s,
m−2
1 − i1 αi φ1 ηi i1
Gt, s ≤ Hs, s Gt, s ≤
Gt, s ≥
m−2
φ2 1
φ
αi H ηi , ηi φ1 t ≤ C3 φ1 t,
t
1
m−2
Δ
φ1 ρ0
1 − i1 αi φ1 ηi i1
1
1−
m−2
i1
αi φ1 ηi
m−2
αi H ηi , s φ1 t
i1
φ1Δ ρ0
H ηi , ηi Hs, sφ1 t
≥
αi
m−2
φ
φ
1
2
1 − i1 αi φ1 ηi i1
1
m−2
≥ C2 φ1 tHs, s.
The proof is complete.
The following theorems will play major role in our next analysis.
2.19
Abstract and Applied Analysis
7
Theorem 2.6 see 16. Let X be a Banach space, and let P ⊂ X be a cone in X. Let Ω1 , Ω2 be open
subsets of X with 0 ∈ Ω1 ⊂ Ω1 ⊂ Ω2 , and let S : P → P be a completely continuous operator, such
that, either
1 Sw ≤ w, w ∈ P ∩ ∂Ω1 , Sw ≥ w, w ∈ P ∩ ∂Ω2 , or
2 Sw ≥ w, w ∈ P ∩ ∂Ω1 , Sw ≤ ww ∈ P ∩ ∂Ω2 .
Then S has a fixed point in P ∩ Ω2 \ Ω1 .
3. Main Results
We make the following assumptions:
H1 ft, u ∈ Cρ0, σ1 × 0, ∞, −∞, ∞, moreover there exists a function
gt ∈ L1 ρ0, σ1, 0, ∞ such that ft, u ≥ −gt, for any t ∈ ρ0, σ1,
u ∈ 0, ∞.
H∗1 ft, u ∈ Cρ0, σ1 × 0, ∞, −∞, ∞ may be singular at t ρ0, σ1,
moreover there exists a function gt ∈ L1 ρ0, σ1, 0, ∞ such that ft, u ≥
−gt, for any t ∈ ρ0, σ1, u ∈ 0, ∞.
H2 ft, 0 > 0, for t ∈ ρ0, σ1.
H3 There exists θ1 , θ2 ∈ ρ0, σ1 such that limu↑∞ mint∈θ1 ,θ2 ft, u/u ∞.
σ1
H4 ρ0 Hs, spsqsfs, z∇s < ∞ for any z ∈ 0, m, m > 0 is any constant.
In fact, we only consider the boundary value problem
xΔ∇ t atxΔ t btxt λqt f t, xt − vt∗ gt 0,
x ρ0 0,
xσ1 λ > 0,
3.1
m−2
αi x ηi ,
i1
where
yt∗ and vt λ
σ1
ρ0
⎧
⎨yt, yt ≥ 0,
⎩0,
3.2
yt < 0,
Gt, spsqsgs∇s, which is the solution of the boundary value problem
vΔ∇ t atvΔ t btvt λqtgt 0,
v ρ0 0,
vσ1 3.3
m−2
αi v ηi .
i1
From Lemma 2.1, it is easy to verify that vt ≤ λC0 φ1 t and C0 C3
σ1
ρ0
qsgs∇s.
8
Abstract and Applied Analysis
We will show that there exists a solution x for boundary value problem 3.1 with
xt ≥ vt, t ∈ ρ0, σ1. If this is true, then ut xt − vt is a nonnegative solution
positive on ρ0, σ1 of boundary value problem 3.1. Since for any t ∈ ρ0, σ1, from
ut vtΔ∇ atut vtΔ btut vt −λqt ft, u gt ,
3.4
we have
uΔ∇ t atuΔ t btut −λqtft, u.
3.5
As a result, we will concentrate our study on boundary value problem 3.1.
We note that xt is a solution of 3.1 if and only if
xt λ
σ1
ρ0
Gt, spsqs f t, xt − vt∗ gt ∇s.
3.6
For our constructions, we will consider the Banach space X Cρ0, σ1 equipped
with standard norm x max0≤t≤1 |xt|, x ∈ X. We define a cone P by
P
x ∈ X | xt ≥
C2
φ1 tx, t ∈ ρ0, σ1 ,
C1
3.7
where φ1 is defined by Lemma 2.1 namely, φ1 is solution 2.1. Define an integral operator
T : P → X by
T xt λ
σ1
ρ0
Gt, spsqs f t, xt − vt∗ gt ∇s.
3.8
Notice, from 3.8 and Lemma 2.5, we have T xt ≥ 0 on 0, 1 for x ∈ P and
T xt λ
σ1
ρ0
≤ C1 λ
then T x ≤ C1 λ
σ1
ρ0
Gt, spsqs f t, xt − vt∗ gt ∇s
σ1
ρ0
Hs, spsqs f t, xt − vt∗ gt ∇s,
Hs, spsqsft, xt − vt∗ gt∇s.
3.9
Abstract and Applied Analysis
9
On the other hand, we have
T xt λ
≥λ
σ1
ρ0
σ1
ρ0
Gt, spsqs f t, xt − vt∗ gt ∇s
C2 φ1 tHs, spsqs f t, xt − vt∗ gt ∇s
σ1
≥
C2
φ1 tλ
C1
≥
C2
φ1 tT x.
C1
ρ0
3.10
C1 Hs, spsqs f t, xt − vt∗ gt ∇s
Thus, T P ⊂ P . In addition, standard arguments show that T P ⊂ P and T is a compact, and
completely continuous.
Theorem 3.1. Suppose that (H1 )-(H2 ) hold. Then there exists a constant λ > 0 such that, for any
0 < λ ≤ λ, boundary value problem 1.1 has at least one positive solution.
Proof. Fix δ ∈ 0, 1. From H2 , let 0 < ε < 1 be such that
ft, z ≥ δft, 0,
for ρ0 ≤ t ≤ σ1,
0 ≤ z ≤ ε.
3.11
Suppose that
0<λ<
ε
2cfε
: λ
where fε maxρ0≤t≤σ1,0≤z≤ε {ft, z gt} and c C1
lim
z↓0
fz
∞,
z
fε
1
<
,
ε
2cλ
3.12
σ1
ρ0
Hs, spsqs∇s. Since
3.13
there exists a R0 ∈ 0, ε such that
fR0 1
.
R0
2cλ
3.14
10
Abstract and Applied Analysis
Let x ∈ P and ν ∈ 0, 1 be such that x νT x, we claim that x /
R0 . In fact
T xt ≤ νλ
≤λ
≤λ
≤λ
σ1
ρ0
σ1
ρ0
σ1
ρ0
σ1
ρ0
C1 Hs, spsqs f s, xs − vs∗ gs ∇s
C1 Hs, spsqs f s, xs − vs∗ gs ∇s
C1 Hs, spsqs
max
0≤s≤1;0≤z≤R0
fs, z gs ∇s
3.15
C1 Hs, spsqsfR0 ∇s
≤ λcfR0 ,
that is,
fR0 fR0 1
1
>
≥
R0
cλ 2cλ
R0
3.16
which implies that x /
R0 . Let U {x ∈ P : x < R0 }. By nonlinear alternative of LeraySchauder type theorem, T has a fixed point x ∈ U. Moreover, combing 3.8, 3.28, and
R0 < ε, we obtain that
xt λ
≥λ
σ1
ρ0
σ1
ρ0
Gt, spsqs f s, xs − vs∗ gs ∇s
Gt, spsqs δfs, 0 gs ∇s
σ1
σ1
≥λ δ
Gt, spsqsfs, 0∇s Gt, spsqsgs∇s
ρ0
>λ
σ1
ρ0
vt
ρ0
3.17
Gt, spsqsgs∇s
for t ∈ ρ0, σ1 .
Let ut xt − vt > 0. Then 1.1 has a positive solution u and u ≤ x ≤ R0 < 1. This
completes the proof of Theorem 3.1.
Abstract and Applied Analysis
11
Theorem 3.2. Suppose that (H∗1 ) and (H3 )-(H4 ) hold. Then there exists a constant λ∗ > 0 such that,
for any 0 < λ ≤ λ∗ , boundary value problem 1.1 has at least one positive solution.
Proof. Let Ω1 {x ∈ P : x < R1 }, where R1 max{1, r} and r C1 C3 /C2 Choose
⎧
⎨
σ1
λ∗ min 1, R1 C1
⎩
ρ0
σ1
ρ0
⎫
−1 ⎬
.
Hs, spsqs max fs, z gs ∇s
⎭
0≤z≤R1
qsgs∇s.
3.18
Then for any x ∈ P ∩ ∂Ω1 , then x R1 and xs − vs ≤ xs ≤ x, we have
T xt ≤ λ
≤λ
σ1
ρ0
σ1
ρ0
≤ λC1
C1 Hs, spsqs f s, xs − vs∗ gs ∇s
C1 Hs, spsqs f s, xs − vs∗ gs ∇s
σ1
Hs, spsqs max fs, z gs ∇s
3.19
0≤z≤R1
ρ0
≤ R1 x.
This implies that
T x ≤ x,
x ∈ P ∩ ∂Ω1 .
3.20
On the other hand, choose a constant N > 0 such that
λC22 N
2C1
θ2
θ1
Hs, sφ1 spsqs∇s min φ1 t ≥ 1.
θ1 ≤t≤θ2
3.21
By assumption H3 , for any t ∈ θ1 , θ2 , there exists a constant B > 0 such that
ft, z
> N,
z
namely, ft, z > Nz,
for z > B.
3.22
12
Abstract and Applied Analysis
Choose R2 max{R1 1, 2λr, 2C1 B 1/C2 minθ1 ≤t≤θ2 φ1 t}, and let Ω2 {x ∈ P : x < R2 },
then for any x ∈ P ∩ ∂Ω2 , we have
xt − vt xt − λ
σ1
ρ0
Gt, spsqsgs∇s
≥ xt − λC3 φ1 t
σ1
ρ0
C1 xt
λC3
≥ xt −
C2 x
psqsgs∇s
σ1
ρ0
psqsgs∇s
3.23
xt
λr
R2
λr
≥ 1−
xt
R2
≥ xt −
≥
1
xt ≥ 0,
2
t ∈ 0, 1.
Then,
min {xt − vt} ≥ min
θ1 ≤t≤θ2
θ1 ≤t≤θ2
1
C2
xt ≥ min
φ1 tx
θ1 ≤t≤θ2 2C1
2
R2 C2 minθ1 ≤t≤θ2 φ1 t
≥ B 1 > B.
2C1
Now,
T xt ≥ max λ
0≤t≤1
σ1
ρ0
C2 φ1 tHs, spsqs f s, xs − vs∗ gs ∇s
≥ max λC2 φ1 t
σ1
0≤t≤1
≥ λC2 min φ1 t
θ1 ≤t≤θ2
≥ λC2 min φ1 t
θ1 ≤t≤θ2
≥ λC2 min φ1 t
θ1 ≤t≤θ2
ρ0
θ2
θ1
θ2
θ1
θ2
θ1
Hs, spsqsf s, xs − vs∗ ∇s
Hs, spsqsfs, xs − vs∇s
Hs, spsqsNxs − vs∇s
Hs, spsqs
N
xs∇s
2
3.24
Abstract and Applied Analysis
13
λC22 N
≥
min φ1 t
2C1 θ1 ≤t≤θ2
≥
λC22 N
min φ1 t
2C1 θ1 ≤t≤θ2
θ2
θ1
θ2
θ1
Hs, spsqsφ1 sx∇s
Hs, sφ1 spsqs∇sx
≥ x.
⇒ T x ≥ x,
x ∈ P ∩ ∂Ω2 .
3.25
Condition 2.1 of Krasnosel’skii’s fixed-point theorem is satisfied. So T has a fixed
point x with R1 ≤ x < R2 such that
xΔ∇ t atxΔ t btxt −λqs f s, xs − vs∗ gs ,
x ρ0 0,
xσ1 0 < t < 1,
m−2
αi x ηi .
3.26
i1
Since r < x,
xt − vt ≥
C2
φ1 tx − λ
C1
σ1
ρ0
Gt, spsqsgs∇s
C2
φ1 tx − φ1 tλC3
≥
C1
σ1
≥
C2
C2
φ1 tx −
φ1 tλr
C1
C1
≥
C2
C2
φ1 tr −
φ1 tλr
C1
C1
≥ 1 − λ
ρ0
psqsgs∇s
3.27
C2
rφ1 t > 0.
C1
Let ut xt − vt, then ut is a positive solution of boundary value problem 1.1. This
completes the proof of Theorem 3.2.
Since condition H1 implies conditions H∗1 and H4 , and from proof of Theorems
3.1 and 3.2, we immediately have the following theorem.
Theorem 3.3. Suppose that (H1 )–(H3 ) hold. Then boundary value problem 1.1 has at least two
positive solutions for λ > 0 sufficiently small.
14
Abstract and Applied Analysis
Proof. On the hand, fix δ ∈ 0, 1. From H2 , let 0 < ε < min{1, r/2} be such that
ft, z ≥ δft, 0,
for ρ0 ≤ t ≤ σ1, 0 ≤ z ≤ ε.
3.28
Choose
ε
2cfε
: λ,
3.29
σ1
where fε maxρ0≤t≤σ1,0≤z≤ε {ft, z gt}, c C1 ρ0 Hs, spsqs∇s, and r σ1
C1 C3 /C2 ρ0 qsgs∇s.
On the other hand, set Ω1 {x ∈ P : x < R1 }, where R1 1 r. Choose
⎧
⎨
λ∗ min 1, R1 C1
⎩
σ1
ρ0
⎫
−1 ⎬
.
Hs, spsqs max fs, z gs ∇s
⎭
0≤z≤R1
3.30
So, let
0 < λ < min λ, λ∗ .
3.31
From 0 < λ < min{λ, λ∗ }, we have 0 < λ < λ, from proof of Theorem 3.1, we know
that 1.1 has a positive solution u1 and u1 ≤ u1 ≤ R0 < r/2. Further, also from 0 < λ <
min{λ, λ∗ }, we have 0 < λ < λ∗ , from proof of Theorem 3.2, we know that 1.1 has a positive
solution u2 and u2 ≥ R1 /2 > r/2. Then 1.1 has at least two positive solutions u1 and u2 .
This completes the proof of Theorem 3.3.
4. Examples
To illustrate the usefulness of the results, we give some examples.
Example 4.1. Consider the boundary value problem
u
Δ∇
Δ
a
t atu t btut λ u t u ρ0 0,
1
t − t2 uσ1 1/2
m−2
cos2πut
αi u ηi ,
0,
λ > 0,
4.1
i1
where a > 1. Then, if λ > 0 is sufficiently small, 4.1 has a positive solution u with ut > 0
for t ∈ 0, 1.
Abstract and Applied Analysis
15
To see this, we will apply Theorem 3.2 with
ft, u ua t qt 1,
1
t −
t2 1/2
gt cos2πut,
1
t − t2 1/2
.
4.2
Clearly
ft, 0 1
t −
t2 1/2
ft, u gt ≥ ua t > 0,
> 0,
lim
u↑∞
for t ∈ ρ0, σ1, u > 0. Namely, H∗1 and H2 –H4 hold. From
2 1/2
s − ρ0 σ1
ρ0
ft, u
∞,
u
σ1
ρ0
4.3
1/σ1 − ρ0s −
∇s π, set R1 C1 π and m maxρ0≤t≤σ1 {pt} 1, we have
C1 Hs, sps max fs, z gs ∇s
0≤z≤R1
≤
≤
σ1
ρ0
⎤
⎡
mC1 φ1 φ2 ⎢
1
⎥
a
∇s
⎣C1 π 2 1/2 ⎦
φ1Δ ρ0
σ1 − ρ0 s − s − ρ0
4.4
mC1 φ1 φ2 C1 πa π
Δ
φ1 ρ0
and λ∗ min{1, φ1Δ ρ0/mφ1 φ2 C1a π a−1 1}. Now, if λ < λ∗ , Theorem 3.2 guarantees
that 4.1 has a positive solution u with u ≥ 2.
Example 4.2. Consider the boundary value problem
9
uΔ∇ t atuΔ t btut λ u2 t − ut 2 0,
2
u ρ0 0,
uσ1 0 < t < 1, λ > 0,
4.5
m−2
αi u ηi .
i1
Then, if λ > 0 is sufficiently small, 4.5 has two solutions ui with ui t > 0 for t ∈ 0, 1, i 1, 2.
To see this, we will apply Theorem 3.3 with
9
ft, u u2 t − ut 2,
2
4.6
gt 4.
Clearly
qt 0,
ft, 0 2 > 0,
ft, u gt ≥
15
> 0,
16
lim
u↑∞
ft, u
∞.
u
4.7
16
Abstract and Applied Analysis
Namely, H1 –H3 hold. Let δ 1/4, ε 1/4 and c have
λ
σ1
ρ0
C1 Hs, sps∇s, then we may
1
.
48c
8c max0≤z≤ε ft, z 4
1
4.8
Now, if λ < λ, Theorem 3.2 guarantees that 4.5 has a positive solution u1 with u1 ≤ 1/4.
Next, let R1 p, where p 4cC3 /C2 1 is a constant, then we have
σ1
ρ0
σ1
17c
9
C1 Hs, sps max fs, z gs ∇s C1 Hs, sps 4 ∇s 0≤z≤R1
2
2
ρ0
4.9
and λ∗ min{1, 2p/17c}. Now, if 0 < λ < λ∗ , Theorem 3.1 guarantees that 4.5 has a positive
solution u2 with u2 ≥ p.
So, since all the conditions of Theorem 3.3 are satisfied, if λ < min{λ, λ∗ }, Theorem 3.3
guarantees that 4.5 has two solutions ui with ui t > 0 i 1, 2.
Example 4.3. Consider the boundary value problem
uΔ∇ t atuΔ t btut λua t cos2πut 0,
u ρ0 0,
uσ1 0 < t < 1, λ > 0,
m−2
αi u ηi ,
4.10
i1
where a > 1. Then, if λ > 0 is sufficiently small, 4.10 has two solutions ui with ui t > 0 for
t ∈ 0, 1, i 1, 2.
To see this we will apply Theorem 3.3 with
ft, u ua t cos2πut,
gt 2.
4.11
Clearly
ft, 0 1 > 0,
ft, u gt ≥ ua t 1 > 0,
ft, u
∞,
u↑∞
u
lim
Namely, H1 –H3 hold. Let δ 1/2, ε 1/8 and c have
σ1
ρ0
for t ∈ ρ0, σ1 .
4.12
C1 Hs, sps∇s, then we may
ε
1
≥
: λ.
2c max0≤x≤ε ft, x 2
16c 1/8a 3
4.13
Now, if 0 < λ < λ then 0 < λ < ε/2cmax0≤x≤ε ft, x 2, Theorem 3.2 guarantees that 4.10
has a positive solution u1 with u1 ≤ 1/8.
Abstract and Applied Analysis
17
Next, let R1 p, where p 4cC3 /C2 1 is a constant, then we have
σ1
ρ0
C1 Hs, sps max fs, z gs ∇s ≤
σ1
0≤z≤R1
ρ0
C1 Hs, sps pa 2 ∇s pa 2 c
4.14
and λ∗ min{1, p/pa 2c}. Now, if 0 < λ < λ∗ , then 0 < λ < p ×
σ1
ρ0 C1 Hs, spsmax0≤z≤R1 fs, z gs∇s−1 , Theorem 3.1 guarantees that 4.10 has a
positive solution u2 with u2 ≥ 1.
So, if λ < min{λ, λ∗ }, Theorem 3.3 guarantees that 4.10 has two solutions ui with
ui t > 0 i 1, 2.
Example 4.4. Let T {0, 1/4, 2/4, 3/4, 1, 5/4, . . .}. We consider the following four point
boundary value problem:
uΔ∇ t 12 Δ
16
u t − ut λua t cos2πut 0,
5
5
1
1
1
1
5
u
u
,
u0 0,
u
4
2
4
4
2
λ > 0,
4.15
where at 12/5, bt −16/5, and qt 1. Then, if λ > 0 is sufficiently small, 4.15 has
two solutions ui with ui > 0 i 1, 2.
Let φ1 and φ2 be the solutions of the following linear boundary value problems,
respectively,
u
Δ∇
12
16
t uΔ t − ut 0,
5
5
u0 0,
12 Δ
16
u t − ut 0,
5
5
u0 1,
uΔ∇ t 5
u
1,
4
5
u
0.
4
4.16
It is evident form the Corollaries 4.24 and 4.25 and Theorem 4.28 of 17 that
φ1 t 5/44t − 1/24t
5/45 − 1/25
,
φ2 t 5/45 1/24t − 1/25 5/44t
5/45 − 1/25
.
4.17
Also φ1 satisfies C3. Green’s function is
⎧ 4s 4s 5 4t 5 4t
1
1
5
1
5
⎪
⎪ 5
−
−
,
1024 ⎨ 4
2
4
2
2
4
Ht, s 4t 4t 5 4s 5 4s
9279 ⎪
1
1
5
1
5
⎪
⎩ 5
−
−
,
4
2
4
2
2
4
and pt 2/54t−1 follows from eα t, t0 1 αht−t0 /h on T hN.
s ≤ t,
t ≤ s,
4.18
18
Abstract and Applied Analysis
To see this, we will apply Theorem 3.3 with
ft, u ua t cos2πut,
gt 2.
4.19
Clearly
ft, 0 1 > 0,
ft, u gt ≥ ua t 1 > 0,
ft, u
∞,
u↑∞
u
lim
Namely, H1 –H3 hold. Let δ 1/2, ε 1/8 and c have
σ1
ρ0
for t ∈ ρ0, σ1 .
4.20
C1 Hs, sps∇s, then we may
1
ε
≥
: λ.
2c max0≤x≤ε ft, x 2
16c 1/8a 3
4.21
Now, if 0 < λ < λ then 0 < λ < ε/2cmax0≤x≤ε ft, x 2, Theorem 3.2 guarantees that 4.15
has a positive solution u1 with u1 ≤ 1/8.
Next, let R1 4cC3 /C2 1 is a constant, then we have
σ1 ρ0
σ1
C1 Hs, sps Ra1 2 ∇s pa 2 c
max fs, z gs ∇s ≤
0≤z≤R1
ρ0
4.22
and λ∗ min{1, R1 /Ra1 2c}. Now, if 0 < λ < λ∗ , then 0 < λ < R1 ×
σ1
ρ0 C1 Hs, smax0≤z≤R1 fs, z gs∇s−1 , Theorem 3.1 guarantees that 4.15 has a
positive solution u2 with u2 ≥ 1.
So, if λ < min{λ, λ∗ }, Theorem 3.3 guarantees that 4.15 has two solutions ui with
ui t > 0 i 1, 2.
Acknowledgments
The work was supported by the Scientific Research Fund of Heilongjiang Provincial
Education Department no. 11544032 and the National Natural Science Foundation of China
no. 10871071.
References
1 S. G. Topal and A. Yantir, “Positive solutions of a second order m-point BVP on time scales,” Nonlinear
Dynamics and Systems Theory, vol. 9, no. 2, pp. 185–197, 2009.
2 X. Lin and Z. Du, “Positive solutions of m-point boundary value problem for second-order dynamic
equations on time scales,” Journal of Difference Equations and Applications, vol. 14, no. 8, pp. 851–864,
2008.
3 H.-R. Sun and W.-T. Li, “Positive solutions for nonlinear three-point boundary value problems on
time scales,” Journal of Mathematical Analysis and Applications, vol. 299, no. 2, pp. 508–524, 2004.
4 Y. Pang and Z. Bai, “Upper and lower solution method for a fourth-order four-point boundary value
problem on time scales,” Applied Mathematics and Computation, vol. 215, no. 6, pp. 2243–2247, 2009.
Abstract and Applied Analysis
19
5 F. M. Atici and S. G. Topal, “The generalized quasilinearization method and three point boundary
value problems on time scales,” Applied Mathematics Letters, vol. 18, no. 5, pp. 577–585, 2005.
6 S. Liang and J. Zhang, “The existence of countably many positive solutions for nonlinear singular
m-point boundary value problems on time scales,” Journal of Computational and Applied Mathematics,
vol. 223, no. 1, pp. 291–303, 2009.
7 S. Liang, J. Zhang, and Z. Wang, “The existence of three positive solutions of m-point boundary value
problems for some dynamic equations on time scales,” Mathematical and Computer Modelling, vol. 49,
no. 7-8, pp. 1386–1393, 2009.
8 L. Hu, “Positive solutions to singular third-order three-point boundary value problems on time
scales,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 606–615, 2010.
9 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.
10 J. Li and J. Shen, “Existence results for second-order impulsive boundary value problems on time
scales,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 4, pp. 1648–1655, 2009.
11 J.-P. Sun, “Twin positive solutions of nonlinear first-order boundary value problems on time scales,”
Nonlinear Analysis. Theory, Methods & Applications, vol. 68, no. 6, pp. 1754–1758, 2008.
12 R. P. Agarwal, M. Bohner, and D. O’Regan, “Time scale boundary value problems on infinite
intervals,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 27–34, 2002.
13 H. Chen, H. Wang, Q. Zhang, and T. Zhou, “Double positive solutions of boundary value problems
for p-Laplacian impulsive functional dynamic equations on time scales,” Computers & Mathematics
with Applications, vol. 53, no. 10, pp. 1473–1480, 2007.
14 R. Ma and H. Luo, “Existence of solutions for a two-point boundary value problem on time scales,”
Applied Mathematics and Computation, vol. 150, no. 1, pp. 139–147, 2004.
15 D. R. Anderson, G. S. Guseinov, and Joan Hoffacker, “Higher-order self-adjoint boundary-value
problems on time scales,” Journal of Computational and Applied Mathematics, vol. 194, no. 2, pp. 309–
342, 2006.
16 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.
17 Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2003.