Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 123283, 15 pages
doi:10.1155/2009/123283
Research Article
Triple Positive Solutions for Third-Order m-Point
Boundary Value Problems on Time Scales
Jian Liu1 and Fuyi Xu2
1
School of Statistics and Mathematics Science, Shandong Economics University, Jinan,
Shandong 250014, China
2
School of Science, Shandong University of Technology, Zibo, Shandong 255049, China
Correspondence should be addressed to Jian Liu, liujiankiki@163.com and
Fuyi Xu, xfy 02@163.com
Received 25 February 2009; Accepted 25 May 2009
Recommended by Leonid Shaikhet
∇
We study the following third-order m-point boundary value problems on time scales ϕuΔ∇ Δ
Δ∇
Δ∇
atfut 0, t ∈ 0, T T , u0 m−2
0 m−2
ξi , where
i1 bi uξi , u T 0, ϕu
i1 ci ϕu
ϕ : R → R is an increasing homeomorphism and homomorphism and ϕ0 0, 0 < ξ1 < · · · <
ξm−2 < ρT . We obtain the existence of three positive solutions by using fixed-point theorem in
cones. The conclusions in this paper essentially extend and improve the known results.
Copyright q 2009 J. Liu and F. Xu. 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
The theory of time scales was initiated by Hilger 1 as a mean of unifying and extending
theories from differential and difference equations. The study of time scales has led to
several important applications in the study of insect population models, neural networks,
heat transfer, and epidemic models; see, for example 2–6. Recently, the boundary value
problems with p-Laplacian operator have also been discussed extensively in the literature,
for example, see 7–15.
A time scale T is a nonempty closed subset of R. We make the blanket assumption that
0, T are points in T. By an interval 0, T , we always mean the intersection of the real interval
0, T with the given time scale; that is, 0, T ∩ T.
In 16, Anderson considered the following third-order nonlinear boundary value
problem BVP:
x t ft, xt,
xt1 x t2 0,
t1 ≤ t ≤ t3 ,
γxt3 δx t3 0.
1.1
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Discrete Dynamics in Nature and Society
He used the Krasnoselskii and Leggett-Williams fixed-point theorems to prove the existence
of solutions to the nonlinear boundary value problem.
In 9, 10, He considered the existence of positive solutions of the p-Laplacian dynamic
equations on time scales
∇
atfut 0,
φp uΔ
t ∈ 0, T T ,
1.2
satisfying the boundary conditions
u0 − B0 uΔ η 0,
uΔ T 0,
1.3
uT − B1 uΔ η 0,
1.4
or
uΔ 0 0,
where η ∈ 0, ρT . He obtained the existence of at least double and triple positive solutions
of the problems by using a new double fixed point theorem and triple fixed point theorem,
respectively.
In 15, Zhou and Ma firstly studied the existence and iteration of positive solutions for
the following third-order generalized right-focal boundary value problem with p-Laplacian
operator
φp u t qtft, ut,
u0 m
αi uξi ,
u η 0,
i1
0 ≤ t ≤ 1,
u 1 n
βi u θi .
1.5
i1
They established a corresponding iterative scheme for the problem by using the monotone
iterative technique.
However, to the best of our knowledge, little work has been done on the existence of
positive solutions for the increasing homeomorphism and positive homomorphism operator
on time scales. So the goal of the present paper is to improve and generate p-Laplacian
operator and establish some criteria for the existence of multiple positive solutions for the
following third-order m-point boundary value problems on time scales
∇
atfut 0,
ϕ uΔ∇
u0 m−2
i1
bi uξi ,
uΔ T 0,
t ∈ 0, T T ,
m−2
ci ϕ uΔ∇ ξi ,
ϕ uΔ∇ 0 1.6
i1
where ϕ : R → R is an increasing homeomorphism and homomorphism and ϕ0 0, and
bi , ci , a, f satisfy
m−2
H1 bi , ci ∈ 0, ∞, 0 < ξ1 < · · · < ξm−2 < ρT , 0 < m−2
i1 bi < 1, 0 <
i1 ci < 1;
Discrete Dynamics in Nature and Society
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H2 f : 0, ∞ → R is continuous, a ∈ Cld 0, T T , R and there exits t0 ∈ 0, T T such
that at0 > 0, where R 0, ∞.
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, ∀x, y ∈ R;
ii ϕ is continuous bijection and its inverse mapping is also continuous;
iii ϕxy ϕxϕy, ∀x, y ∈ R.
2. Preliminaries and Lemmas
For convenience, we list the following definitions which can be found in 1–5.
Definition 2.1. A time scale T is a nonempty closed subset of real numbers R. For t < sup T and
r > inf T, define the forward jump operator σ and backward jump operator ρ, respectively,
by
σt inf{τ ∈ T | τ > t} ∈ T,
ρr sup{τ ∈ Tτ < r} ∈ T,
2.1
for all t, r ∈ T. If σt > t, t is said to be right scattered, and if ρr < r, r is said to be left
scattered; if σt t, t is said to be right dense, and if ρr r, r is said to be left dense. If T
has a right scattered minimum m, define Tk T − {m}; otherwise set Tk T. If T has a left
scattered maximum M, define Tk T − {M}; otherwise set Tk T.
Definition 2.2. For f : T → R and t ∈ Tk , the delta derivative of f at the point t is defined
to be the number f Δ t provided it exists with the property that for each > 0, there is a
neighborhood U of t such that
fσt − fs − f Δ tσt − s ≤ |σt − s|,
2.2
for all s ∈ U.
For f : T → R and t ∈ Tk , the nabla derivative of f at t, denoted by f ∇ t provided it
exists with the property that for each > 0, there is a neighborhood U of t such that
f ρt − fs − f ∇ t ρt − s ≤ ρt − s,
2.3
for all s ∈ U.
Definition 2.3. A function f is left-dense continuous i.e., ld-continuous, if f is continuous at
each left-dense point in T and its right-sided limit exists at each right-dense point in T.
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Discrete Dynamics in Nature and Society
Definition 2.4. If φΔ t ft, then we define the delta integral by
b
ftΔt φb − φa.
2.4
a
If F ∇ t ft, then we define the nabla integral by
b
ft∇t Fb − Fa.
2.5
a
Definition 2.5. Let E be a real Banach space over R. A nonempty closed set P ⊂ E is said to be
a cone provided that
i u ∈ P , a ≥ 0 implies au ∈ P ;
ii u, −u ∈ P implies u 0.
Definition 2.6. Given a cone P in a real Banach space E, a functional ψ : P → P is said to be
increasing on P , provided ψx ≤ ψy, for all x, y ∈ P with x ≤ y.
Definition 2.7. Given a cone P in a real Banach space E, we define for each a > 0 the set
Pa {x ∈ P | x < a}.
2.6
Definition 2.8. A map α is called nonnegative continuous concave functional on a cone P of a
real Banach space X if α : P → 0, ∞ is continuous and
α λx 1 − λy ≥ λαx 1 − λα y
2.7
for all x, y ∈ P and λ ∈ 0, 1. Similarly we say that the map β is called nonnegative continuous
concave functional on a cone P of a real Banach space X if β : P → 0, ∞ is continuous and
β λx 1 − λy ≤ λβx 1 − λβ y
2.8
for all x, y ∈ P and λ ∈ 0, 1.
Let γ, θ be nonnegative continuous convex functionals on P , let α be a nonnegative
continuous concave functional on P , and let ψ be a nonnegative continuous functional on P .
For nonnegative real numbers a, b, k, and c we define the following convex set:
P γ, c x ∈ P : γx < c ,
P α, b; γ, c x ∈ P : b ≤ αx, γx ≤ c ,
R ψ, a; γ, c x ∈ P : a ≤ ψx, γx ≤ c ,
P α, b; θ, k; γ, c x ∈ P : b ≤ αx, θx ≤ k, γx ≤ c .
2.9
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Theorem 2.9 17. Let P be a cone in a real Banach space X. Let γ and θ be nonnegative continuous
convex functionals on P , let α be a nonnegative continuous concave functional on P , and let ψ be a
nonnegative continuous functional on P satisfying ψλx ≤ λψx for 0 ≤ λ ≤ 1, such that for
some positive numbers M and c, αx ≤ ψx and x ≤ Mγx for all x ∈ P γ, c. Suppose that
Φ : P γ, c → P γ, c is a completely continuous operator and there exist nonnegative numbers a, b,
and k with 0 < a < b such that
i {x ∈ P α, b; θ, k; γ, c : αx > b} /
∅ and αΦx > b for x ∈ P α, b; θ, k; γ, c;
ii αΦx > b for x ∈ P α, b; γ, c with θΦx > k;
iii 0 ∈ Rψ, a; γ, c and ψΦx < a for x ∈ Rψ, a; γ, c with ψx a.
Then Φ has at least three fixed points x1 , x2 , x3 ∈ P γ, c satisfying
γxi ≤ c,
b < αx1 ,
i 1, 2, 3,
αx2 < b,
a < ψx2 ,
ψx3 < a.
2.10
Theorem 2.10 18. Let A be a bounded closed convex subset of a Banach space E. Assume that
A1 , A2 are disjoint closed convex subsets of A and U1 , U2 are nonempty open subsets of A with
U1 ⊂ A1 and U2 ⊂ A2 . Suppose that Φ : A → A is completely continuous and the following
conditions hold:
i ΦA1 ⊂ A1 , ΦA2 ⊂ A2 ;
ii Φ has no fixed points in A1 \ U1 ∪ A2 \ U2 .
Then Φ has at least three points x1 , x2 , x3 such that x1 ∈ U1 , x2 ∈ U2 , and x3 ∈ A \ A1 ∪ A2 .
Lemma 2.11. If condition H1 holds, then for h ∈ Cld 0, T T , the boundary value problem (BVP)
uΔ∇ ht 0,
u0 m−2
t ∈ 0, T ,
2.11
uΔ T 0
bi uξi ,
i1
has the unique solution
m−2 ξi
i1 bi 0 T − shs∇s
.
ut T − shs∇s 1 − m−2
0
i1 bi
t
2.12
Proof. By caculating, we can easily get 2.12. So we omit it.
Lemma 2.12. If condition H1 holds, then for h ∈ Cld 0, T T , the boundary value problem (BVP)
∇
ϕ uΔ∇
ht 0,
u0 m−2
i1
bi uξi ,
Δ
u T 0,
t ∈ 0, T ,
m−2
ci ϕ uΔ∇ ξi ϕ uΔ∇ 0 i1
2.13
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Discrete Dynamics in Nature and Society
has the unique solution
ut t
−1
T − sϕ
0
where C m−2 ξi
−1 s
hr∇r C ∇s
i1 bi 0 T − sϕ
0
hr∇r C ∇s ,
1 − m−2
0
i1 bi
s
2.14
m−2 ξi
m−2
−1
i1 ci 0 hr∇r/1 −
i1 ci , ϕ s is the inverse function to ϕs.
Proof. Integrating both sides of equation in 2.13 on 0, t, we have
t
ϕ uΔ∇ t ϕ uΔ∇ 0 − hr∇r.
2.15
0
So,
ξi
Δ∇
Δ∇
ϕ u ξi ϕ u 0 − hr∇r.
2.16
0
By boundary value condition ϕuΔ∇ 0 m−2
i1
ci ϕuΔ∇ ξi , we have
m−2 ξi
i1 ci 0 hr∇r
Δ∇
.
ϕ u 0 −
1 − m−2
i1 ci
2.17
By 2.15 and 2.17 we know
⎛
uΔ∇ t −ϕ−1 ⎝
ξi
m−2
i1 ci 0 hr∇r
1 − m−2
i1 ci
t
⎞
hr∇r ⎠.
2.18
0
This together with Lemma 2.11 implies that
ut t
0
where C −1
T − sϕ
m−2 ξi
−1 s
hr∇r C ∇s
i1 bi 0 T − sϕ
0
hr∇r C ∇s ,
1 − m−2
0
i1 bi
s
2.19
m−2 ξi
m−2
i1 ci 0 hr∇r/1 −
i1 ci . The proof is complete.
Lemma 2.13. Let condition H1 hold. If h ∈ Cld 0, T T and ht ≥ 0, then the unique solution ut
of 2.13 satisfies
ut ≥ 0,
t ∈ 0, T T .
2.20
t
ξi
m−2
Proof. By uΔ∇ t −ϕ−1 m−2
i1 ci 0 hr∇r/1 −
i1 ci 0 hr∇r ≤ 0, we can know
that the graph of ut is concave down on 0, T T and uΔ t is nonincreasing on 0, T T . This
Discrete Dynamics in Nature and Society
7
together with the assumption that the boundary condition is uΔ T 0 implies that uΔ t ≥ 0
for t ∈ 0, T T . This implies that
u uT ,
min ut u0.
t∈0,T T
2.21
So we only prove u0 ≥ 0. By condition H1 we have
m−2 ξi
−1 s
hr∇r C ∇s
i1 bi 0 T − sϕ
0
u0 1 − m−2
i1 bi
2.22
≥ 0.
The proof is completed.
3. Triple Positive Solutions
In this section, some existence results of positive solutions to BVP 1.6 are established by
imposing some conditions on f and defining a suitable Banach space and a cone.
Let E Cld 0, T T be endowed with the ordering x ≤ y if xt ≤ yt for all t ∈ 0, T T ,
and u maxt∈0,T T |ut| is defined as usual by maximum norm. Clearly, it follows that
E, u is a Banach space.
We define a cone by
P u : u ∈ E, ut is concave, nondecreasing, and nonnegative on 0, T T , uΔ T 0 .
3.1
Let
T
η max t ∈ T : t ≥
,
2
3.2
0 < η < l < T,
3.3
and fix l ∈ T such that
and define the nonnegative continuous convex functionals γ and θ, the nonnegative
continuous concave functional α, and the nonnegative continuous functional ψ on the cone P
by
γu θu max ut ul,
t∈0,lT
αu min ut u η ,
t∈η,T T
ψu max u η .
t∈0,ηT
3.4
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For notational convenience, denote
m−2 ξi
i1 ci 0 ar∇r
,
1 − m−2
i1 ci
m−2 ξi
s
−1
η
s
∇s
ar∇r C
i1 bi 0 T − sϕ
0
∇s, m mη T − sϕ−1
ar∇r C
,
1 − m−2
0
0
i1 bi
m−2 ξi
s
−1
l
s
∇s
b
ar∇r
C
−
sϕ
T
i
i1
0
0
∇s Ml T − sϕ−1
ar∇r C
,
m−2
1 − i1 bi
0
0
m−2 ξi
s
−1
η
s
∇s
ar∇r C
i1 bi 0 T − sϕ
0
∇s Mη T − sϕ−1
ar∇r C
,
1 − m−2
0
0
i1 bi
m−2 ξi
s
−1
T
s
∇s
b
ar∇r
C
−
sϕ
T
i
i1
0
0
∇s M T − sϕ−1
ar∇r C
.
m−2
1 − i1 bi
0
0
3.5
C
Lemma 3.1 9. If u ∈ P , then
1 ut ≥ t/T ||u|| for all t ∈ 0, T T ;
2 us/s ≥ ut/t for t, s ∈ 0, T T with s ≤ t.
Define an operator Φ : P → E by
Φut t
T − sϕ−1
s
0
0
m−2
∇s
arfur∇r C
i1
bi
ξi
0 T
− sϕ−1
1−
s
arfur∇r
0
m−2
i1
bi
∇s
C
3.6
,
m−2 ci ξi arfur∇r/1 − m−2 ci . Then, u is a solution of boundary value problem
where C
i1
i1
0
1.6 if and only if u is a fixed point of operator Φ. Obviously, for u ∈ P one has Φut ≥ 0 for
t ∈ 0, T T . In addition, ΦuΔ∇ t ≤ 0 for t ∈ 0, T T and ΦuΔ T 0, this implies ΦP ⊂ P . With
standard argument one may show that Φ : P → P is completely continuous.
Theorem 3.2. Suppose conditions H1 and H2 hold, and there exist positive numbers a <
η/T b < b < l/T c, Ml b < mc such that
B1 fu ≤ ϕc/Ml , u ∈ 0, T c/l;
B2 fu > ϕb/mη , u ∈ b, T 2 b/l2 ;
B3 fu < ϕa/Mη , u ∈ 0, T a/η.
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Then, the BVP 1.6 has at least three positive solutions u1 , u2 , u3 ∈ P γ, c satisfying
γui ≤ c,
b < αu1 ,
αu2 < b,
i 1, 2, 3,
a < ψu2 ,
ψu3 < a.
3.7
Proof. Based on Lemma 3.1, it is clear that for u ∈ P and λ ∈ 0, 1, there are αu ψu, ψλu λψu and u ≤ T/lul T/lγu. Furthermore, ψ0 0 < a and
therefore 0 ∈ Rψ, a; γ, c.
Take u ∈ P γ, c, then 0 ≤ u ≤ u ≤ T/lγu ≤ T/lc. By means of B1 one derives
γΦu Φul
l
s
∇s
arfur∇r C
0
0
m−2 ξi
s
−1
∇s
arfur∇r
C
i1 bi 0 T − sϕ
0
m−2
1 − i1 bi
⎛
⎞
m−2 ξi
s
−1
l
s
∇s
b
ar∇r
C
−
sϕ
T
i
i1
0
0
c ⎜
⎟
∇s ≤
ar∇r C
⎝ T − sϕ−1
⎠
m−2
Ml
1 − i1 bi
0
0
T − sϕ−1
c.
3.8
Thus Φ : P γ, c → P γ, c.
Set u ≡ T b/l and k T b/l, it follows that
Tb
> b,
αu u η l
θu ul Tb
,
l
γu Tb
< c,
l
3.9
which means {u ∈ P α, b; θ, T b/l; γ, c : αu > b} /
∅.
For u ∈ P α, b; θ, T b/l; γ, c, we have b ≤ ut ≤ T 2 b/l2 for t ∈ η, T T . By condition B2 we have
αΦu Φu η
η
s
∇s
T − sϕ−1
arfur∇r C
0
0
m−2 ξi
s
−1
∇s
arfur∇r C
i1 bi 0 T − sϕ
0
1 − m−2
i1 bi
η
s
b
−1
≥
ar∇r C ∇s
T − sϕ
mη 0
0
b.
So, i of Theorem 2.9 is fulfilled.
3.10
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Discrete Dynamics in Nature and Society
If u ∈ P α, b; γ, c and θΦu > c, then due to 2 of Lemma 3.1
η
ηc T ηb
η
αΦu Φu η ≥ Φul θΦu >
> 2 > b.
l
l
l
l
3.11
Therefore, ii of Theorem 2.9 is fulfilled.
Take u ∈ Rψ, a; γ, c and ψu a, then 0 ≤ u ≤ u ≤ T/ηuη T/ηψu T a/η,
it then follows from B3 that
ψΦu Φu η
η
s
−1
T − sϕ
arfur∇r C ∇s
0
0
m−2 ξi
s
−1
∇s
b
arfur∇r
C
−
sϕ
T
i
i1
0
0
m−2
1 − i1 bi
⎛
⎞
m−2 ξi
s
−1
η
s
∇s
b
ar∇r
C
−
sϕ
T
i
i1
0
0
a ⎜
⎟
∇s ≤
ar∇r C
⎝ T − sϕ−1
⎠
m−2
Mη
1
−
b
0
0
i
i1
a.
3.12
As a result, all the conditions of Theorem 2.9 are verified. This completes the proof.
Theorem 3.3. Suppose that conditions (H1 ) and (H2 ) hold. Let 0 < a < b < c, Mb < mc and
assume that the following conditions are satisfied:
C1 fu < ϕa/M, u ∈ 0, a;
C2 there exists a number d > c such that fu < ϕd/M, u ∈ 0, d;
C3 ϕb/m < fu < ϕc/M, u ∈ b, c.
Then, the BVP 1.6 has at least three positive solutions u1 , u2 , and u3 such that
b < u1 t < c,
u2 < a,
u3 > a.
Where for real number b, φb : 0, T T → 0, ∞ is continuous, φb t b, for t ∈ 0, T T .
3.13
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11
Proof. We first show that AP a ⊆ Pa ⊂ P a if condition C1 holds. If u ∈ P a , then 0 ≤ u ≤
u ≤ a, which implies fu < ϕa/M. We have
Φu ΦuT ≤
T
T − sϕ−1
0
s
∇s
arfur∇r C
0
m−2 ξi
s
−1
∇s
b
arfur∇r
C
−
sϕ
T
i
i1
0
0
m−2
1 − i1 bi
⎛
⎞
m−2 ξi
s
−1
T
s
∇s
b
ar∇r
C
−
sϕ
T
i
i1
0
0
a ⎜
⎟
∇s <
ar∇r C
⎝ T − sϕ−1
⎠
m−2
M
1
−
b
0
0
i
i1
a.
3.14
This implies that ΦP a ⊆ Pa ⊂ P a .
Next, condition C2 indicates that there exists d > c such that ΦP d ⊂ P d . Now we
let
A P d,
A1 ϕb , ϕc ,
U1 intA1 ,
A2 P a ,
U2 Pa ,
3.15
where intA1 is the interior of A1 . Then we have ΦA ⊂ A, ΦA2 ⊂ A2 . Moreover, ΦP a ⊆
Pa ⊂ P a means ΦA2 ⊆ U2 ⊂ A2 . Thus Φ has no fixed point in A2 \ U2 .
To show ΦA1 ⊂ A1 and Φ has no fixed point in A1 \ U1 , set u ∈ A1 , following the
definition of ϕb , we can know b ≤ ut ≤ c, for t ∈ 0, T T . Condition C3 then gives rise to
ϕb/m < fu < ϕc/M, which in turn produces
Φut ≥ Φu0
m−2 ξi
s
−1
∇s
arfur∇r C
i1 bi 0 T − sϕ
0
1 − m−2
i1 bi
b
>
m
b,
m−2 ξi
s
−1
∇s
ar∇r C
i1 bi 0 T − sϕ
0
1 − m−2
i1 bi
3.16
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Discrete Dynamics in Nature and Society
Φut ≤ ΦuT 2
T
s
∇s
arfur∇r C
≤ T − sϕ−1
0
0
m−2
i1
⎛
<
c ⎜
⎝
M
bi
ξi
0 T
− sϕ−1
1−
T
T − sϕ−1
0
s
arfur∇r
0
m−2
i1
∇s
C
bi
⎞
m−2 ξi
s
−1
b
ar∇r
C
∇s
−
sϕ
T
i 0
i1
0
⎟
∇s ar∇r C
⎠
m−2
1 − i1 bi
0
s
c.
3.17
Combining the above two inequalities one achieves φb t b < Φut < c φc t, for
t ∈ 0, T T . That is, Φu ∈ U1 . So ΦA1 ⊆ U1 ⊂ A1 and Φ has no fixed point in A1 \ U1 .
Therefore, all conditions of Theorem 2.10 are fulfilled, and the BVP 1.6 has at least three
positive solutions u1 , u2 , and u3 such that
b < u1 t < c,
u2 < a,
u3 > a.
3.18
4. Some Examples
In the section, we present some simple examples to explain our results. We only study the
case T R, 0, T T 0, 1.
Example 4.1. Consider the following third-order three-point boundary value problem:
atfu 0, 0 < t < 1,
ϕ u
1
1
1
1
,
u 1 0,
,
u0 u
ϕ u 0 ϕ u
3
2
4
2
4.1
where ϕx x, at ≡ 1, b1 1/3, c1 1/4, ξ1 1/2.
We choose η 1/2, by computing we can know mη 11/24, Ml 351/256, Mη 33/48. Let a 100, b 245, c 770, l 7/8, then a < ηb < b < lc. Obviously, Ml b < mc. We
define a nonlinearity f as follows:
⎧
⎪
140,
⎪
⎪
⎪
⎪
⎪
410
⎪
⎨140 u − 200,
45
fu ⎪
550,
⎪
⎪
⎪
⎪
⎪
⎪
⎩550 5 u − 320,
560
u ∈ 0, 200,
u ∈ 200, 245,
u ∈ 245, 320,
u ∈ 320, ∞.
4.2
Discrete Dynamics in Nature and Society
13
Then, by the definition of f, we have
i fu ≤ ϕc/Ml ≈ 557.2, u ∈ 0, 880;
ii fu > ϕb/mη ≈ 534.5, u ∈ 245, 320;
iii fu < ϕa/Mη ≈ 145.4, u ∈ 0, 200.
By Theorem 3.2, BVP 4.1 has at least three positive solutions.
Example 4.2. Consider the following third-order three-point boundary value problem:
ϕ u
atfu 0, 0 < t < 1,
1
1
1
1
u0 u
ϕ u 0 ϕ u
,
u 1 0,
,
3
2
4
2
4.3
where ϕx x, at ≡ 1, b1 1/3, c1 1/4, ξ1 1/2, η 1/2.
By computing, we can know m 11/48, M 83/48. Let a 7, b 12, c 336, l 7/8, then a < b < c. Obviously, Mb < mc. We define a nonlinearity f as follows:
⎧
⎪
3,
⎪
⎪
⎪
⎪
⎪
97
⎪
⎪
⎨3 u − 72 ,
25
fu ⎪
⎪100,
⎪
⎪
⎪
⎪
⎪
⎪
⎩100 1100 u − 336,
1764
u ∈ 0, 7,
u ∈ 7, 12,
u ∈ 12, 336,
4.4
u ∈ 336, ∞.
Then, by the definition of f, we have
i fu < ϕa/M ≈ 4.2, u ∈ 0, 7;
ii and there exists d 2100 > c such that fu ≤ ϕd/M ≈ 1214.4, u ∈ 0, 2100;
iii ϕb/m ≈ 52.4 < fu < ϕc/M ≈ 194.3, u ∈ 12, 336.
By Theorem 3.3, BVP 4.3 has at least three positive solutions.
Remark 4.3. Consider following nonlinear m-point boundary value problem:
∇
ϕ uΔ∇
atfut 0,
u0 m−2
0 < t < T,
bi uξi ,
i1
uΔ T 0,
m−2
ci ϕ uΔ∇ ξi ,
ϕ uΔ∇ 0 i1
4.5
14
Discrete Dynamics in Nature and Society
where
ϕu ⎧
⎨u3 , u ≤ 0,
4.6
⎩u2 , u > 0,
f and a satisfy the conditions H1 and H2 . It is clear that ϕ : R → R is an increasing
homeomorphism and homomorphism and ϕ0 0. Because p-Laplacian operators are odd,
they do not apply to our example. Hence we generalize boundary value problem with pLaplacian operator, and the results 8–11, 13–15 do not apply to the example.
Remark 4.4. In a similar way, we can get the corresponding results for the following boundary
value problem:
∇
atfut 0,
ϕ uΔ∇
uT m−2
i1
ai uξi ,
uΔ 0 0,
t ∈ 0, T T ,
m−2
ci ϕ uΔ∇ ξi .
ϕ uΔ∇ 0 4.7
i1
Acknowledgment
The project was supported by the National Natural Science Foundation of China 10471075.
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