Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 976406, 14 pages
doi:10.1155/2009/976406
Research Article
Solutions for m-Point BVP with
Sign Changing Nonlinearity
Hua Su
School of Statistics and Mathematic, Shandong University of Finance, Jinan, Shandong 250014, China
Correspondence should be addressed to Hua Su, jnsuhua@163.com
Received 24 October 2008; Accepted 31 January 2009
Recommended by Binggen Zhang
We study the existence of positive solutions for the following nonlinear m-point boundary value
problem for an increasing homeomorphism and homomorphism with sign changing nonlinearity:
k
s
{φu t atft, ut 0, 0 < t < 1, u 0 m−2
i1 bi uξi −
i1 ai u ξi , u1 ik1 bi uξi −
m−2
is1 bi u ξi , where φ : R → R is an increasing homeomorphism and homomorphism and
φ0 0. The nonlinear term f may change sign. As an application, an example to demonstrate our
results is given. The conclusions in this paper essentially extend and improve the known results.
Copyright q 2009 Hua Su. 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 study the existence of positive solutions of the following nonlinear m-point
boundary value problem with sign changing nonlinearity:
φ u t atft, ut 0,
u 0 m−2
ai u ξi ,
u1 i1
k
bi u ξi −
i1
0 < t < 1,
s
m−2
bi u ξi −
bi u ξi ,
ik1
is1
1.1
where φ : R → R is an increasing homeomorphism and homomorphism and φ0 0;
ξi ∈ 0, 1 with 0 < ξ1 < ξ2 < · · · < ξm−2 < 1 and ai , bi , a, f satisfy
H1 ai , bi ∈ 0, ∞, 0 <
k
i1
bi −
s
ik1
bi < 1, 0 <
m−2
i1
ai < 1;
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Discrete Dynamics in Nature and Society
H2 at : 0, 1 → 0, ∞ does not vanish identically on any subinterval of 0, 1 and
satisfies
1
0<
atdt < ∞;
1.2
0
H3 f ∈ C0, 1 × 0, ∞, −∞, ∞, ft, 0 ≥ 0 and ft, 0 /
0.
Definition 1.1. A projection φ : R → R is called an increasing homeomorphism and
homomorphism, if the following conditions are satisfied:
i if x ≤ y, then φx ≤ φy, for all x, y ∈ R;
ii φ is a continuous bijection and its inverse mapping is also continuous;
iii φxy φxφy, for all x, y ∈ R.
The study of multipoint boundary value problems for linear second-order ordinary
differential equations was initiated by Il’in and Moiseev 1, 2. Motivated by the study
of 1, 2, Gupta 3 studied certain three-point boundary value problems for nonlinear
ordinary differential equations. Since then, more general nonlinear multipoint boundary
value problems have been studied by several authors. We refer the reader to 4–12 for some
references along this line. Multipoint boundary value problems describe many phenomena
in the applied mathematical sciences. For example, the vibrations of a guy wire of a
uniform cross-section and composed of N parts of different densities can be set up as a
multipoint boundary value problems see Moshinsky 13; many problems in the theory
of elastic stability can be handled by the method of multipoint boundary value problems see
Timoshenko 14.
In 2001, Ma 6 studied m-point boundary value problem BVP:
u t htfu 0,
u0 0,
u1 0 ≤ t ≤ 1,
m−2
αi u ξi ,
1.3
i1
where αi > 0 i 1, 2, . . . , m − 2, m−2
i1 αi < 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, and
f ∈ C0, ∞, 0, ∞, h ∈ C0, 1, 0, ∞. Author established the existence of positive
solutions under the condition that f is either superlinear or sublinear.
In 11, we considered the existence of positive solutions for the following nonlinear
four-point singular boundary value problem with p-Laplacian:
φp u t atfut 0, 0 < t < 1,
αφp u0 − βφp u ξ 0,
γφp u1 δφp u η 0,
1.4
where φp s |s|p−2 s, p > 1, φq φp −1 , 1/p 1/q 1, α > 0, β ≥ 0, γ > 0, δ ≥ 0, ξ, η ∈
0, 1, ξ < η, a : 0, 1 → 0, ∞. By using the fixed-point theorem of cone, the existence of
positive solution and many positive solutions for nonlinear singular boundary value problem
p-Laplacian is obtained.
Discrete Dynamics in Nature and Society
3
Recently, Ma et al. 5 used the monotone iterative technique in cones to prove the
existence of at least one positive solution for m-point boundary value problem BVP:
φp u t atft, ut 0,
u 0 m−2
ai u ξi ,
u1 i1
0 < t < 1,
m−2
1.5
bi u ξi ,
i1
m−2
1
where 0 < m−2
i1 bi < 1, 0 <
i1 ai < 1, 0 < ξ1 < ξ2 < · · · < ξm−2 < 1, at ∈ L 0, 1,
f ∈ C0, 1 × 0, ∞, 0, ∞.
In 9, Wang and Hou investigated the following m-point BVP:
φp u t ft, ut 0,
n−2
φp u 0 ai φp u ξi ,
t ∈ 0, 1,
u1 i1
n−2
bi u ξi ,
1.6
i1
where φp u |u|p−2 u, p > 1, ξi ∈ 0, 1 with 0 < ξ1 < ξ2 < · · · < ξn−2 < 1 and ai , bi satisfy
n−2
ai , bi ∈ 0, ∞, 0 < n−2
i1 ai < 1, 0 <
i1 bi < 1.
However, in all the above-mentioned paper, the authors discuss the boundary
value problem BVP under the key conditions that the nonlinear term is positive
continuous function. Motivated by the results mentioned above, in this paper we study the
existence of positive solutions of m-point boundary value problem 1.1 for an increasing
homeomorphism and homomorphism with sign changing nonlinearity. We generalize the
results in 4–12.
By a positive solution of BVP 1.1, we understand a function u which is positive on
0, 1 and satisfies the differential equation as well as the boundary conditions in BVP 1.1.
2. The Preliminary Lemmas
In this section, we present some lemmas which are important to our main results.
Lemma 2.1. Let H1 and H2 hold. Then for u ≥ 0 ∈ C0, 1, the problem
φ u t atft, ut 0,
u 0 m−2
ai u ξi ,
u1 i1
k
i1
0 < t < 1,
s
m−2
bi u ξi −
bi u ξi −
bi u ξi
ik1
2.1
is1
has a unique solution ut if and only if ut can be express as the following equation:
ut −
1
t
ωf sds B,
2.2
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Discrete Dynamics in Nature and Society
where A, B satisfy
φ−1 A m−2
ξi
ai φ−1 A − asfs, usds ,
B−
1−
1
k
i1 bi
2.3
0
i1
s
ik1 bi
1
1
k
s
bi ωf sds −
bi ωf sds
ξi
i1
m−2
ik1
bi φ
−1
A−
ξi
asfs, usds ,
ξi
2.4
0
is1
where
ωf s φ
−1
−
s
arfr, urdr A .
2.5
0
1
m−2
Define l φ m−2
i1 ai /1 − φ i1 ai ∈ 0, 1, then there exists a unique A ∈ −l 0 asfs,
usds, 0 satisfying 2.3.
Proof. The method of the proof is similar to 5, Lemma 2.1, we omit the details.
Lemma 2.2. Let H1 and H2 hold. If u ∈ C 0, 1, the unique solution of the problem 2.1
satisfies
ut ≥ 0,
t ∈ 0, 1.
2.6
Proof. According to Lemma 2.1, we first have
−A s
arft, urdr ≥ 0.
2.7
0
So
u1 B
−
1−
k
1
i1
bi s
ik1
bi
1
1
k
s
bi ωf sds −
bi ωf sds
ξi
i1
m−2
ik1
ξi
bi φ
A − asft, usds
−1
0
is1
1−
k
i1
1
bi s
ik1
bi
ξi
1
1
k
s
bi
− ωf sds −
bi
− ωf sds
ξi
i1
m−2
is1
bi φ
ik1
−1
−A
ξi
0
ξi
asft, usds
Discrete Dynamics in Nature and Society
1
1
k
s
1
bi
− ωf sds −
bi
− ωf sds
≥
1 − ki1 bi sik1 bi i1
ξk
ξk
ik1
k
i1
5
1
bi − sik1 bi ξk − ωf sds
1 − ki1 bi sik1 bi
≥ 0.
2.8
If t ∈ 0, 1, we have
ut B −
1
t
u1 s
φ−1 A − arfr, urdr ds
1
0
φ−1
−A
t
s
arfr, urdr ds
0
2.9
≥ u1
≥ 0.
So ut ≥ 0, t ∈ 0, 1. The proof of Lemma 2.2 is completed.
Lemma 2.3. Let H1 and H2 hold. If u ∈ C 0, 1, the unique solution of the problem 2.1
satisfies
inf ut ≥ γu,
2.10
t∈0,1
where γ k
i1
bi −
s
ik1
bi 1−ξk /1−
k
i1
bi ξk s
ik1
bi ξk ∈ 0, 1, u maxt∈0,1 |ut|.
Proof. Clearly
t
u t φ−1 A − asfs, usds
0
−φ
−1
−A
t
asfs, usds
2.11
0
≤ 0.
This implies that
u u0,
min ut u1.
t∈0,1
2.12
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Discrete Dynamics in Nature and Society
It is easy to see that u t2 ≤ u t1 , for any t1 , t2 ∈ 0, 1 with t1 ≤ t2 . Hence u t is a decreasing
function on 0, 1. This means that the graph of ut is concave down on 0, 1. So we have
u ξk − u1ξk ≥ 1 − ξk u0.
Together with u1 get
u0 ≤
≤
k
i1
bi uξi −
s
ik1
bi uξi −
m−2
is1
bi u ξi and u t ≤ 0 on 0, 1, we
k
bi u ξk − u1 ki1 bi ξk − sik1 bi u ξk u1 sik1 bi ξk
s
k
b
1 − ξk
i
i1 bi −
ik1
k
bi u ξi − u1 ki1 bi ξk − sik1 bi u ξi u1 sik1 bi ξk
s
k
1 − ξk
i1 bi −
ik1 bi
i1
i1
k
s
≤
u1 1 − i1 bi ξk ik1 bi ξk
s
k
1 − ξk
i1 bi −
ik1 bi
u1
.
γ
2.13
2.14
The proof of Lemma 2.3 is completed.
Lemma 2.4 see 8. Let K be a cone in a Banach space X. Let D be an open bounded subset of X
φ and DK /
AK
K. Assume that A : DK → K is a compact map such that x /
with DK D ∩ K /
for x ∈ ∂DK . Then the following results hold.
1 If Ax ≤ x, x ∈ ∂DK , then iA, DK , K 1.
2 If there exists x0 ∈ K \ {θ} such that x / Ax λx0 , for all x ∈ ∂DK and all x > 0, then
iA, DK , K 0.
3 Let U be open in X such that U ⊂ DK . If iA, DK , K 1 and iA, DK , K 0, then A has
a fixed point in DK \ UK . The same results hold, if iA, DK , K 0 and iA, DK , K 1.
Let E C0, 1, then E is Banach space, with respect to the norm u supt∈0,1 |ut|.
Denote
K u | u ∈ C0, 1, ut ≥ 0, inf ut ≥ γu ,
t∈0,1
where γ is the same as in Lemma 2.3. It is obvious that K is a cone in C0, 1.
2.15
Discrete Dynamics in Nature and Society
7
We define ϕt min{t, 1 − t}, t ∈ 0, 1 and
Kρ {ut ∈ K : u < ρ},
Kρ∗ {ut ∈ K : ρϕt < ut < ρ},
Ωρ ut ∈ K : min ut < γρ
2.16
ξm−2 ≤t≤1
ut ∈ E : u ≥ 0, γu ≤ min ut < γρ .
ξm−2 ≤t≤1
Lemma 2.5 see 13. Ωρ defined above has the following properties:
a Kγρ ⊂ Ωρ ⊂ Kρ ;
b Ωρ is open relative to K;
c X ∈ ∂Ωρ if and only if minξm−2 ≤t≤1c xt γρ;
d If x ∈ ∂Ωρ , then γρ ≤ xt ≤ ρ for t ∈ ξm−2 , 1.
Now, for the convenience, one introduces the following notations:
ft, u
: u ∈ γρ, ρ ,
ξm−2 ≤t≤1 φρ
ft, u
ρ
f0 max max
: u ∈ 0, ρ ,
0≤t≤1 φρ
ft, u
ρ
fϕtρ max max
: u ∈ ϕtρ, ρ ,
0≤t≤1 φρ
ρ
fγρ min
min
f α lim sup max
ft, u
,
φu
fα lim inf min
ft, u
,
φu
u→α
0≤t≤1
u→α
ξm−2 ≤t≤1
2.17
α : ∞ or 0 ,
⎧
1
⎫−1
−1
⎨ 1 sik1 bi m−2
l 1 0 asds ⎬
is1 bi φ
,
m
⎭
⎩
1 − ki1 bi sik1 bi
k
M
1−
−1
s
1
s
i1 bi −
ik1 bi
−1
φ
ardr ds
k
s
0
i1 bi ik1 bi ξk
.
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Discrete Dynamics in Nature and Society
3. The Main Result
In the rest of the section, we also assume the following conditions.
A1 There exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < γρ2 such that
t ∈ 0, 1, u ∈ ρ1 ϕt, ∞ ,
1
ft, u > 0,
2
1
fϕtρ
≤ φm,
1
ρ
ρ
fγρ22 ≥ φMγ.
3.1
A2 There exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < ρ2 such that
t ∈ 0, 1, u ∈ min γρ1 , ρ2 ϕt , ∞ ,
3
ft, u > 0,
4
fγρ11 ≥ φMγ,
ρ
ρ
2
fϕtρ
≤ φm.
2
3.2
A3 There exist ρ1 , ρ2 , ρ3 ∈ 0, ∞ with ρ1 < γρ2 and ρ2 < ρ3 such that
t ∈ 0, 1, u ∈ ρ1 ϕt, ∞ ,
1
ft, u > 0,
2
1
fϕtρ
≤ φm,
1
ρ
ρ
ρ
fγρ22 ≥ φMγ,
3
fϕtρ
≤ φm.
3
3.3
A4 There exist ρ1 , ρ2 , ρ3 ∈ 0, ∞ with ρ1 < ρ2 < γρ3 such that
t ∈ 0, 1, u ∈ min γρ1 , ρ2 ϕt , ∞ ,
3
ft, u > 0,
4
fγρ11 ≥ φMγ,
ρ
ρ
ρ
2
fϕtρ
≤ φm,
2
fγρ33 ≥ φMγ.
3.4
A5 There exist ρ , ρ ∈ 0, ∞ with ρ < γρ such that
t ∈ 0, 1, u ∈ ρ ϕt, ∞ ,
1
ft, u > 0,
2
fϕtρ ≤ φm,
ρ
ρ
0 ≤ f ∞ < φm.
fγρ ≥ φMγ,
3.5
A6 There exist ρ , ρ ∈ 0, ∞ with ρ < ρ such that
t ∈ 0, 1, u ∈ min γρ , ρϕt , ∞ ,
3
ft, u > 0,
4
fγρ ≥ φMγ,
ρ
ρ
fϕtρ ≤ φm,
φM < f∞ ≤ ∞.
3.6
Discrete Dynamics in Nature and Society
9
Our main results are the following theorems.
Theorem 3.1. Assume that H1 , H2 , H3 , A3 hold. Then BVP 1.1 has at least three
positive solutions.
Theorem 3.2. Assume that H1 , H2 , H3 , A4 hold. Then BVP 1.1 has at least two positive
solutions.
Theorem 3.3. Assume that H1 , H2 , H3 hold and also assume that A1 or A2 hold. Then
BVP 1.1 has at least a positive solution.
Theorem 3.4. Assume that H1 , H2 , H3 hold and also assume that A5 or A6 hold. Then
BVP 1.1 has at least two positive solutions.
Proof of Theorem 3.1. Without loss of generality, we suppose that A3 hold. Denote
⎧
⎨ft, u,
u ≥ ρ1 ϕt,
f ∗ t, u ⎩f t, ρ ϕt , 0 ≤ u < ρ ϕt,
1
1
3.7
it is easy to check that f ∗ t, u ∈ C0, 1 × 0, ∞, 0, ∞.
Now define an operator T : K → C0, 1 by setting
T ut −
1−
k
i1
1
bi s
ik1
bi
1
1
k
s
bi ωsds −
bi ωsds
ξi
i1
m−2
ik1
bi φ
−1
A−
ξi
ξi
1
− ωsds,
asf s, usds
∗
t
0
is1
3.8
where
ωs φ
−1
−
s
arf ∗ τ, urdr A .
3.9
0
By Lemma 2.3, we have T K ⊂ K. So by applying Arzela-Ascoli’s theorem, we can obtain
that T K is relatively compact. In view of Lebesgue’s dominated convergence theorem, it is
easy to prove that T is continuous. Hence, T : K → K is completely continuous.
Now, we consider the following modified BVP 1.1:
atf ∗ t, ut 0,
φ u
u 0 m−2
i1
ai u ξi ,
u1 k
i1
0 < t < 1,
s
m−2
bi u ξi −
bi u ξi −
bi u ξi .
ik1
is1
3.10
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Discrete Dynamics in Nature and Society
Obviously, BVP 3.10 has a solution ut if and only if u is a fixed point of the operator T .
From the condition A3 2, we have
∗ρ
∗ρ
1
≤ φm,
fϕtρ
1
∗ρ
fγρ22 ≥ φMγ,
3
fϕtρ
≤ φm.
3
3.11
Next, we will show that iT, Kρ∗1 , K 1.
∗ρ1
In fact, by fϕtρ
≤ φm, for ∀u ∈ ∂Kρ∗1 , we have
1
T ut −
1−
1
k
i1
bi s
ik1
bi
1
1
ξi
k
s
m−2
−1
∗
×
bi ωsds −
bi ωsds bi φ
A − asf s, usds
ξi
i1
ik1
ξi
0
is1
−
1
ωsds
t
≤
1−
1
k
i1
bi s
ik1
bi
1
1
k
−1
∗
bi φ
l 1 arf r, ur dr ds
i1
0
m−2
0
bi φ
−1
l 1
φ
0
≤
−1
1
l 1
1
asf s, usds
∗
0
is1
1
1
∗
arf r, urdr ds
0
s
1
φ−1 l 1 0 asds
b
i
is1
φ−1 φ ρ1 φm
k
s
1 − i1 bi ik1 bi
ik1 bi m−2
ρ1 u.
3.12
This implies that T u ≤ u for u ∈ ∂Kρ∗ . By Lemma 2.41, we have
i T, Kρ∗1 , K 1.
3.13
Furthermore, we will show that iT, Kρ2 , K 1.
Let et ≡ 1, for t ∈ 0, 1, then e ∈ ∂K1 . We claim that
u/
T u λe,
u ∈ ∂Ωρ2 , λ > 0.
In fact, if not, there exist u0 ∈ ∂Ω2 and λ0 > 0 such that u0 T u0 λ0 e.
3.14
Discrete Dynamics in Nature and Society
11
By A3 and Lemma 2.1, we have for t ∈ 0, 1,
−
s
aτf τ, uτdτ A ≤ −φ ρ2 φMγ
∗
0
s
aτdτ ,
3.15
0
so that
−ωs φ
−1
−
s
aτf τ, uτdτ A
∗
0
≥ ρ2 Mγφ
−1
3.16
s
aτdτ .
0
Then, we have that
u0 t T u0 t λ0 et
≥
1−
k
i1
1
bi s
ik1
bi
1
1
s
k
bi −ωsds −
bi −ωsds λ0
i1
ξk
ik1
ξk
s
1
s
i1 bi −
ik1 bi
−1
≥
ρ
Mγ
φ
ardr
ds λ0
2
1 − ki1 bi sik1 bi
ξk
0
k
3.17
γρ2 λ0 .
This implies that γρ2 ≥ γρ2 λ0 , this is a contradiction. Hence, by Lemma 2.42, it follows
that
iT, Ωρ2 , K 0.
3.18
Finally, similar to the proof of iT, Kρ∗1 , K 1, we can show that iT, Kρ∗3 , K 1.
By Lemma 2.5a and ρ1 < γρ2 and ρ2 < ρ3 , we have K ρ1 ⊂ Kγρ2 ⊂ Ωρ2 ⊂
Kρ2 ⊂ Kρ3 . It follows from Lemma 2.43 that T has three positive fixed points u1 , u2 , u3 in
Kρ∗1 , Ωρ2 \ K ∗ ρ1 , Kρ∗3 , respectively. Therefore, BVP 3.10 has three positive solutions u1 , u2 , u3
in Kρ∗1 , Ωρ2 \ K ∗ ρ1 , Kρ∗3 , respectively.
Then, BVP 3.10 has three positive solutions u1 , u2 , u3 ∈ ρ1 ϕt, ∞, which means that
u1 , u2 , u3 are also the positive solutions of BVP 1.1.
Proof of Theorem 3.2. The proof of Theorem 3.2 is similar to that of Theorem 3.1, and so we
omit it here. The proof of Theorem 3.2 is completed.
Proof of Theorem 3.3. Theorem 3.3 is corollary of Theorem 3.1. The proof of Theorem 3.3 is
completed.
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Discrete Dynamics in Nature and Society
Proof of Theorem 3.4. We show that condition A5 implies condition A3 . Let k ∈ f ∞ , φm,
then there exists r > ρ such that maxt∈0,1 ft, u ≤ kφu, u ∈ r, ∞ since 0 ≤ f ∞ < φm.
Denote
β max max ft, u : ρ ϕt ≤ u ≤ r ,
t∈0,1
β
,ρ .
φm − k
3.19
u ∈ ρ ϕt, ∞ .
3.20
ρ3 > max φ−1
Then we have
max ft, u ≤ kφu β ≤ kφ ρ3 β ≤ φmφ ρ3 ,
t∈0,1
ρ
3
This implies that fϕtρ
≤ φm and A3 holds. Similarly condition A6 implies condition
3
A4 .
By an argument similar to that Theorem 3.1, we can obtain the result of Theorem 3.4.
The proof of Theorem 3.4 is completed.
4. Examples
Example 4.1. Consider the following five-point boundary value problem with p-Laplacian:
ft, u 0, 0 < t < 1,
φ u
1 1
1 3
1 1
u 0 u
u
u
,
128
4
256
2
64
4
1
1
1
1
u1 u
− u
,
8
4
64
2
4.1
where a1 1/128, a2 1/256, a3 1/64, b1 1/8, b2 1/64, b3 0, ξ1 1/4, ξ2 1, /2, ξ3 3/4:
φu ⎧
⎨−u2 ,
u ≤ 0,
⎩u2 ,
u > 0,
⎧
⎪
ϕt 30
1
⎪
⎪
1
t
ut
−
, t, u ∈ 0, 1 × 0, 2,
⎪
⎨5
2
ft, u ⎪
⎪1
ϕt 30
⎪
⎪
,
t, u ∈ 0, 1 × 2, ∞.
⎩ 1 t 2 −
5
2
4.2
Discrete Dynamics in Nature and Society
13
It is easy to check that f : 0, 1 × 0, ∞ → 0, ∞ is continuous. It follows from a
direct calculation that
⎧
1
⎫−1
⎨ 1 sik1 bi m−2
φ−1 l 1 0 asds ⎬
b
i
is1
0.96,
m
⎭
⎩
1 − ki1 bi sik1 bi
γ
k
i1
1−
bi 1 − ξk
21
,
s
250
i1 bi ξk ik1 bi ξk
bi −
k
s
k
M
1−
4.3
ik1
−1
s
1
s
i1 bi −
ik1 bi
−1
φ
ardr ds
k
s
0
i1 bi ik1 bi ξk
0.76.
Choose ρ1 1, ρ2 250, it is easy to check that ρ1 < γρ2 and
ft, u > 0,
t ∈ 0, 1, u ∈ ϕt, ∞,
ρ
fρ11ϕt
1/51 tut − ϕt/230
max max
0≤t≤1
12
< φm m2 0.92,
ρ
fγρ22
min
min
2502
2
1/51 1130 2
5
12
"
t ∈ 0, 1, u ∈ ϕtρ1 , ρ1 ,
1/51 t2 − ϕt/230
3/4≤t≤1
> φMγ Mγ 0.004,
t∈
1/51 3/42 − 1/230
"
3
, 1 , u ∈ γρ2 , ρ2 .
4
2502
1.0742
4.4
It follows that f satisfies the condition A1 of Theorem 3.3, then problems 1.1 have at least
two positive solutions.
Remark 4.2. Let ϕu u, the problem is second-order m-point boundary value problem.
Remark 4.3. Let φp s |s|p−2 s, p > 1, the problem is boundary value problem with pLaplacian operators.
Hence our results generalize boundary value problem with p-Laplacian operators.
Acknowledgment
This research is supported by the Doctor of Scientific Startup Foundation of Shandong
University of Finance of China 08BSJJ32.
14
Discrete Dynamics in Nature and Society
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