Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2008, Article ID 252876, 13 pages
doi:10.1155/2008/252876
Research Article
Several Existence Theorems of Nonlinear
m-Point BVP for an Increasing Homeomorphism
and Homomorphism on Time Scales
Yanbin Sang,1 Hua Su,2 and Yafeng Xiao1
1
Department of Mathematics, North University of China, Taiyuan,
Shanxi 030051, China
2
School of Information Science and Technology, Shandong University of Science and Technology,
Qingdao, Shandong 266510, China
Correspondence should be addressed to Yanbin Sang, sangyanbin@126.com
Received 20 January 2008; Accepted 26 March 2008
Recommended by Bing Zhang
Several existence theorems of positive solutions are established for nonlinear m-point boundary
∇
value problem for the following dynamic equations on time scales φuΔ atft, ut 0,
m−2
m−2
Δ
Δ
t ∈ 0, T , φu 0 i1 ai φu ξi , uT i1 bi uξi , where φ : R→R is an increasing
homeomorphism and homomorphism and φ0 0. As an application, an example to demonstrate
our results is given.
Copyright q 2008 Yanbin Sang 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 study the existence of positive solutions of the following dynamic equations
on time scales:
Δ ∇
φ u
atf t, ut 0,
m−2
φ uΔ 0 ai φ uΔ ξi ,
i1
t ∈ 0, T ,
uT m−2
bi u ξi ,
i1
where φ : R→R is an increasing homeomorphism and homomorphism and φ0 0.
1.1
2
Discrete Dynamics in Nature and Society
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.
We will assume that the following conditions are satisfied throughout this paper:
m−2
H1 0 < ξ1 < · · · < ξm−2 < ρT , ai , bi ∈ 0, ∞ satisfy 0 < m−2
i1 ai < 1, and
i1 bi <
m−2
m−2
1, T i1 bi ≥ i1 bi ξi ;
H2 at ∈ Cld 0, T , 0, ∞ and there exists t0 ∈ ξm−2 , T , such that at0 > 0;
H3 f ∈ C0, T × 0, ∞, 0, ∞.
Recently, there is much attention focused on the existence of positive solutions for
second-order, three-point boundary value problem on time scales. On the other hand, threepoint and m-point boundary value problems with p-Laplacian operators on time scales have
also been studied extensively, for details, see 1–11 and references therein. But with an
increasing homeomorphism and homomorphism, few works were done as far as we know.
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.
We would like to mention some results of Anderson et al. 2, He 4, 5, Sun and Li
9, Ma et al. 12, Wang and Hou 13, Wang and Ge 14, which motivate us to consider our
problem.
In 2, Anderson et al. considered the following problem:
∇
φp uΔ
atf ut 0, t ∈ a, b,
ua − B0 uΔ υ 0,
uΔ b 0,
1.2
where φp u |u|p−2 u, p > 1, υ ∈ a, b, f ∈ Cld 0, ∞, 0, ∞, at ∈ Cld a, b, 0, ∞,
and Km x ≤ B0 x ≤ KM x for some positive constants Km , KM . They established the existence
results of at least one positive solution by using a fixed point theorem of cone expansion and
compression of functional type.
In 4, 5, He considered the existence of positive solutions of the p-Laplacian dynamic
equations on time scales:
Δ ∇
φp u
atf ut 0,
t ∈ 0, T 1.3
satisfying the boundary conditions
u0 − B0 uΔ η 0,
uΔ T 0
1.4
uT − B1 uΔ η 0,
1.5
or
uΔ 0 0,
Yanbin Sang et al.
3
where φp u |u|p−2 u, p > 1, η ∈ 0, ρT , at ∈ Cld 0, T , 0, ∞, f ∈ C0, ∞, 0, ∞,
and Ax ≤ Bi x ≤ Bx i 0, 1 for some positive constants A, B. He obtained the existence of at
least double and triple positive solutions of the problem 1.3, 1.4, and 1.5 by using a new
double fixed point theorem and triple fixed point theorem, respectively.
In recent papers, Ma et al. 12 have obtained the existence of monotone positive
solutions for the following BVP:
φp u
atft, ut 0, t ∈ 0, 1,
m−2
u 0 ai u ξi ,
u1 1.6
m−2
i1
bi u ξi .
i1
The main tool is the monotone iterative technique.
In 9, Sun and Li studied the following p-Laplacian, m-point BVP on time scales:
Δ ∇
ϕp u
atf t, ut 0, t ∈ 0, T ,
1.7
m−2
ϕp uΔ T ai ϕp uΔ ξi ,
u0 0,
i1
where ϕp u |u|p−2 u, p > 1, ai ≥ 0 for i 1, . . . , m − 2, 0 < ξ1 < · · · < ξm−2 < ρT , m−2
i1 ai <
1, at ∈ Cld 0, T , 0, ∞, f ∈ Cld 0, T × 0, ∞, 0, ∞. Some new results are obtained
for the existence of at least twin or triple positive solutions of the problem 1.7 by applying
Avery-Henderson and Leggett-Williams fixed point theorems, respectively.
In 15, Sang and xi investigated the existence of positive solutions of the p-Laplacian
dynamic equations on time scales:
Δ ∇
φp u
atf t, ut 0, t ∈ 0, T ,
m−2
φp uΔ 0 ai φp uΔ ξi ,
uT i1
m−2
bi u ξi ,
1.8
i1
where φp s is p-Laplacian operator, that is, φp s |s|p−2 s, p > 1, φp−1 φq , 1/p 1/q 1, 0 <
ξ1 < · · · < ξm−2 < ρT . Let
ft, u
ρ
: u ∈ γρ, ρ ,
fγρ min min
ξm−2 ≤t≤T φp ρ
ρ
f0
ft, u
max max
: u ∈ 0, ρ ,
0≤t≤T φp ρ
f α lim sup max
u→α
m
1−
0≤t≤T
1
m−2
i1
ft, u
,
φp u
φq
0
i1
fα lim inf max
u→α
ξm−2 ≤t≤T
ft, u
,
φp u
α : ∞ or 0 ,
−1
m−2 ξi
i1 ai 0 aτ∇τ
Δs
aτ∇τ ,
1 − m−2
0
i1 ai
s
T
bi
bi T − ξi
,
γ
T − m−2
i1 bi ξi
m−2
m−2
−1
m−2 ξi
T s
T i1 bi − m−2
i1 ai 0 aτ∇τ
i1 bi ξi
Δs
φq
aτ∇τ ,
M
T 1 − m−2
1 − m−2
0
0
i1 bi
i1 ai
they mainly obtained the following results.
1.9
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Discrete Dynamics in Nature and Society
Theorem 1.1. Assume (H1 ), (H2 ), and (H3 ) hold, and assume that one of the following conditions holds:
H4 there exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < γρ2 such that
ρ
f0 1 ≤ φp m,
ρ
fγρ22 ≥ φp Mγ ;
1.10
H5 there exist ρ1 , ρ2 ∈ 0, ∞ with ρ1 < ρ2 such that
ρ
f0 2 ≤ φp m,
ρ
fγρ11 ≥ φp Mγ .
1.11
Then, 1.8 have a positive solution.
In this paper, we will establish two new theorems of positive solution of 1.8, our
work concentrates on the case when the nonlinear term does not satisfy the conditions of
Theorem 1.1. At the end of the paper, we will give an example which illustrates that our work
is true.
2. Preliminaries and some lemmas
For convenience, we list the following definitions which can be found in 16–19.
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 ∈ T | τ < 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
f σt − fs − f Δ t σt − s ≤ σt − s
2.2
for all s ∈ U.
For f : T→R and t ∈ Tk , the nabla derivative of f at t is the number 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.
Yanbin Sang et al.
5
Definition 2.4. If GΔ t ft, then we define the delta integral by
b
ftΔt Gb − Ga.
2.4
a
If F ∇ t ft, then we define the nabla integral by
b
ft∇t Fb − Fa.
2.5
a
To prove the main results in this paper, we will employ several lemmas. These lemmas are
based on the linear BVP:
Δ ∇
φ u
ht 0,
m−2
φ uΔ 0 ai φ uΔ ξi ,
t ∈ 0, T ,
uT m−2
i1
bi u ξi .
2.6
i1
We can prove the following lemmas by the methods of 15.
Lemma 2.5. For h ∈ Cld 0, T , the BVP 2.6 has the unique solution:
t
ut − φ
−1
s
0
hτ∇τ − A Δs B,
2.7
0
where
m−2 ξi
i1 ai 0 hτ∇τ
A−
,
1 − m−2
i1 ai
T
B
φ−1
0
ξi −1 s
hτ∇τ − A Δs − m−2
hτ∇τ − A Δs
i1 bi 0 φ
0
0
.
1 − m−2
i1 bi
s
2.8
Lemma 2.6. Assume (H1 ) holds, For h ∈ Cld 0, T and h ≥ 0, then the unique solution u of 2.6
satisfies
ut ≥ 0,
for t ∈ 0, T .
2.9
Lemma 2.7. Assume (H1 ) holds, if h ∈ Cld 0, T and h ≥ 0, then the unique solution u of 2.6
satisfies
inf ut ≥ γ
u
,
2.10
t∈0,T where
γ
m−2 i1 bi T − ξi
,
T − m−2
i1 bi ξi
u
max ut.
t∈0,T 2.11
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Discrete Dynamics in Nature and Society
Let the norm on Cld 0, T be the maximum norm. Then, the Cld 0, T is a Banach space.
It is easy to see that the BVP 1.1 has a solution u ut if and only if u is a fixed point of the
operator equation:
t
s
Δs B,
Aut − φ−1
aτf τ, uτ ∇τ − A
0
2.12
0
where
−
A
B m−2 ξi
i1 ai 0 aτf τ, uτ ∇τ
,
1 − m−2
i1 ai
T
0
φ
−1
s
Δs − m−2 bi ξi φ−1 s aτf τ, uτ ∇τ − A
Δs
aτf τ, uτ ∇τ − A
i1
0
0
0
.
1 − m−2
i1 bi
2.13
Throughout this paper, we will assume that 0 ≤ μ ≤ ν ≤ T . Denote
K
u | u ∈ Cld 0, T , ut ≥ 0, inf ut ≥ γ
u
,
t∈μ,ν
2.14
where γ is the same as in Lemma 2.7. It is obvious that K is a cone in Cld 0, T . By Lemma 2.7,
AK ⊂ K. So by applying Arzela-Ascoli theorem on time scales 20, we can obtain that AK
is relatively compact. In view of Lebesgue’s dominated convergence theorem on time scales
21, it is easy to prove that A is continuous. Hence, A : K→K is completely continuous.
Lemma 2.8. Let
ϕs φ−1
s
,
aτf τ, uτ ∇τ − A
2.15
0
for ξi i 1, . . . , m − 2, then
ξi
0
ξi
ϕsΔs ≤
T
T
ϕsΔs.
2.16
0
Lemma 2.9 22. Let E be a Banach space, and let K ⊂ E be a cone. Assume Ω1 , Ω2 are open bounded
subset of E with 0 ∈ Ω1 , Ω1 ⊂ Ω2 , and let
F : K ∩ Ω2 \ Ω1 −→ K
be a completely continuous operator such that
i Fu
≤ u
, u ∈ K ∩ ∂Ω1 , and Fu
≥ u
, u ∈ K ∩ ∂Ω2 ; or
ii Fu
≥ u
, u ∈ K ∩ ∂Ω1 , and Fu
≤ u
, u ∈ K ∩ ∂Ω2 .
Then, F has a fixed point in K ∩ Ω2 \ Ω1 .
2.17
Yanbin Sang et al.
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Now, we introduce the following notations. Let
A0 1−
1
m−2
i1
T
bi
φ
0
−1
−1
m−2 ξi
i1 ai 0 aτ∇τ
Δs
aτ∇τ ,
1 − m−2
0
i1 ai
s
m−2
s
−1
m−2 ξi
ν
T i1 bi − m−2
i1 ai 0 aτ∇τ
i1 bi ξi
−1
Δs
φ
aτ∇τ .
B0 T 1 − m−2
1 − m−2
μ
0
i1 bi
i1 ai
2.18
For l > 0, Ωl {u ∈ K : u
< l}, ∂Ωl {u ∈ K : u
l},
αl sup Au
: u ∈ ∂Ωl ,
βl inf Au
: u ∈ ∂Ωl .
2.19
By Lemma 2.6, α and β are well defined.
3. Existence theorems of positive solution
Theorem 3.1. Assume (H1 ), (H2 ), (H3 ) hold, and assume that the following conditions hold:
A1 pi ∈ C0, ∞, 0, ∞, i 1, 2, and
p2 l
B0
;
lim
>φ
φl
γ
l→∞
p1 l
lim
< φ A0 ,
l→0 φl
3.1
A2 k1 ∈ L1 0, T , 0, ∞, k2 ∈ L1 0, T , 0, ∞;
A3 there exist 0 < c1 ≤ c2 , 0 ≤ λ2 < 1 < λ1 , such that
λ
ft, l ≤ p1 l k1 t φl 1 ,
λ2
ft, l ≥ p2 l − k2 t φl
,
t, l ∈ 0, T × 0, c1 ,
t, l ∈ μ, ν × c2 , ∞ .
3.2
Then, the problems 1.1 have at least one positive solution.
Theorem 3.2. Assume (H1 ), (H2 ), (H3 ) hold, and assume that the following conditions hold:
B1 pi ∈ C0, ∞, 0, ∞, i 3, 4, and
p3 l
lim
< φ A0 ,
l→∞ φl
p4 l
B0
;
lim
>φ
φl
γ
l→0
3.3
B2 k3 ∈ L1 0, T , 0, ∞, k4 ∈ L1 0, T , 0, ∞;
B3 there exist 0 < c3 ≤ c4 , 0 ≤ λ4 < 1 < λ3 , such that
λ
ft, l ≤ p3 l k3 t φl 4 ,
t, l ∈ 0, T × c4 , ∞ ,
λ
ft, l ≥ p4 l − k4 t φl 3 ,
t, l ∈ μ, ν × 0, c3 .
Then, the problems 1.1 have at least one positive solution.
3.4
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Discrete Dynamics in Nature and Society
Proof of Theorem 3.1. By liml→0 p1 l/φl < φA0 , then we can get that there exist 0 < a1 ≤
c1 , 0 < < φA0 such that
p1 l ≤ φ A0 − φl, 0 ≤ l ≤ a1 .
3.5
If 0 ≤ l ≤ a1 , u ∈ ∂Ωl , then 0 ≤ ut ≤ l, 0 ≤ t ≤ T . By condition A3 , we have
λ
f t, ut ≤ p1 ut k1 t φ ut 1
λ
≤ φ A0 − φ ut k1 t φut 1
λ
≤ φ A0 − φ u
k1 t φ u
1
3.6
λ
φ A0 − φl k1 t φl 1 .
Let Ml, τ φA0 − φl k1 τφlλ1 so that
m−2 ξi
s
s
i1 ai 0 aτfτ, uτ∇τ
aτf τ, uτ ∇τ − A aτf τ, uτ ∇τ 1 − m−2
0
0
i1 ai
m−2 ξi
i1 ai 0 aτMl, τ∇τ
≤ aτMl, τ∇τ .
1 − m−2
0
i1 ai
s
Therefore,
Au
≤ B ≤
≤
1−
1−
1−
1
m−2
i1
1
m−2
i1
1
m−2
It follows that
αl
1
≤
m−2
l
1 − i1 bi
i1
T
T
bi
ξi
i1
0
bi
ϕsΔs
T
bi
T
bi
φ−1
φ
−1
m−2 ξi
i1 ai 0 aτMl, τ∇τ
Δs.
aτMl, τ∇τ 1 − m−2
0
i1 ai
s
0
s
Ml, τ
∇τ aτ
φl
0
Noticing λ1 > 1, we have
T
φ
0
3.8
ϕsΔs
0
0
αl
1
lim
≤
m−2
l→0 l
1 − i1 bi
ϕsΔs −
0
m−2
3.7
−1
s
m−2 ξi
i1 ai 0 aτ Ml, τ/φl ∇τ
Δs.
1 − m−2
i1 ai
aτ φ A0 − ∇τ 0
3.9
m−2 ξi
i1 ai 0 aτ φ A0 − ∇τ
Δs
1 − m−2
i1 ai
m−2 ξi
φ−1 φ A0 − T −1 s
i1 ai 0 aτ∇τ
Δs
φ
aτ∇τ 1 − m−2
1 − m−2
0
0
i1 bi
i1 ai
−1
≤ φ−1 φ A0 A−1
0 A0 · A0 1.
3.10
Therefore, there exist 0 < a1 < a1 such that αa1 < a1 . It implies that Au
< u
, u ∈ ∂Ωa1 .
Yanbin Sang et al.
9
On the other hand, by liml→∞ p2 l/φl > φB0 /γ, we can get that there exist > 0,
and c2 ≤ d < ∞ such that
B0
φl, d ≤ l ≤ ∞, for d ≤ γl < ∞ , u ∈ ∂Ωl ,
p2 l ≥ φ
3.11
γ
then d ≤ γl ≤ ut ≤ l, μ ≤ T ≤ ν.
Let ml, τ φB0 /γ φγl − k2 τφlλ2 . By Lemma 2.8 and condition A3 ,
applying Lemma 2.8, it follows that
T
T
m−2
ξi
1
Au
maxAut ≥ − ϕsΔs ϕsΔs −
bi ϕsΔs
0≤t≤T
1 − m−2
0
0
0
i1
i1 bi
≥
≥
m−2
1−
T
i1 bi
ϕsΔs
m−2
i1 bi 0
m−2
1−
T
T
T
i1 bi
ϕsΔs
m−2
i1 bi 0
m−2 ξi
i1 bi 0 ϕsΔs
−
1 − m−2
i1 bi
m−2
bi ξi
− m−2 T 1 − i1 bi
i1
T
ϕsΔs
3.12
0
m−2
T
bi − m−2
i1 bi ξi
ϕsΔs
m−2 T 1 − i1 bi
0
i1
s
m−2 ξi
ν
bi − m−2
i1 ai 0 aτml, τ∇τ
i1 bi ξi
−1
Δs.
φ
aτml, τ∇τ T 1 − m−2
1 − m−2
μ
0
i1 bi
i1 ai
m−2
i1
It follows that
s
m−2 ξi
m−2
ν
βl T m−2
ml, τ
i1 ai 0 aτ ml, τ/φl ∇τ
i1 bi −
i1 bi ξi
−1
≥
∇τ Δs.
φ
aτ
l
φl
T 1 − m−2
1 − m−2
μ
0
i1 bi
i1 ai
3.13
Noticing 0 ≤ λ2 < 1, we get
m−2
βl T m−2
i1 bi −
i1 bi ξi
lim
≥
m−2 l
l→∞
T 1 − i1 bi
×
ν
φ
μ
−1
m−2 ξi
φ B0 /γ φγl
i1 ai 0 aτ φ B0 /γ φγl /φl ∇τ
Δs
aτ
∇τ φl
1 − m−2
0
i1 ai
s
φ−1 φB0 B0−1 B0 B0−1 1.
3.14
Therefore, there exists b with b > a1 > 0 such that βb > b. It implies that Au
> u
for
u ∈ ∂Ωb .
By Lemma 2.9, we assert that the operator A has one fixed point u∗ ∈ K such that a1 ≤
∗
u ≤ b. Therefore, u∗ is positive solution of the problems 1.1.
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Discrete Dynamics in Nature and Society
Proof of Theorem 3.2. By liml→∞ p3 l/φl < φA0 , then we can get that there exist d ≥ λ4 , 0 <
< φA0 such that
p3 l ≤ φ A0 − φl, l ≥ d.
3.15
For l ≥ d, u ∈ ∂Ωl , then 0 ≤ ut ≤ l, 0 ≤ t ≤ T . By condition B3 , we have
λ
f t, ut ≤ p3 ut k3 t φ ut 4
λ
≤ φ A0 − φ ut k3 t φ ut 4
λ
≤ φ A0 − φ u
k3 t φ u
4
3.16
λ
φ A0 − φl k3 t φl 4 .
Let M1 l, τ φA0 − φl k3 τφlλ4 so that
m−2 ξi
s
s
a aτM1 l, τ∇τ
≤ aτM1 l, τ∇τ i1 i 0 aτf τ, uτ ∇τ − A
.
1 − m−2
0
0
i1 ai
Denote
m1 max fτ, uτ : τ ∈ 0, s 0, ξi , uτ ≤ d
3.18
m1 φ−1 φ A0 − ≤ A0 .
3.19
satisfying
Therefore,
Au
≤
m1 A−1
0
3.17
1−
1
m−2
i1
It follows that
αl
1
≤ m1 A−1
m−2
0 l
1 − i1 bi
T
φ
bi
T
φ
0
−1
0
−1
m−2 ξi
i1 ai 0 aτM1 l, τ∇τ
Δs.
aτM1 l, τ∇τ 1 − m−2
0
i1 ai
3.20
s
s
M1 l, τ
∇τ aτ
φl
0
m−2 ξi
i1 ai 0 aτ M1 l, τ/φl ∇τ
Δs.
1 − m−2
i1 ai
3.21
Noticing 0 ≤ λ4 < 1, we have
lim
l→∞
αl
l
≤
m1 A−1
0
m1 A−1
0
1−
1
m−2
i1
T
bi
φ
0
−1
s
aτ φ A0 − ∇τ 0
m−2 ξi
i1 ai 0 aτ φ A0 − ∇τ
Δs
1 − m−2
i1 ai
m−2 ξi
φ−1 φ A0 − T −1 s
i1 ai 0 aτ∇τ
Δs
φ
aτ∇τ 1 − m−2
1 − m−2
0
0
i1 bi
i1 ai
−1
−1
m1 A−1
φ A0 − A−1
0 φ
0 ≤ A0 · A0 1.
Therefore, there exists a1 ≥ d such that αa1 < a1 . It implies that Au
< u
, u ∈ ∂Ωa1 .
3.22
Yanbin Sang et al.
11
On the other hand, by liml→0 p4 l/φl > φB0 /γ, we can get that there exist > 0,
and 0 < e < c3 such that
B0
φl,
for 0 ≤ l ≤ e, u ∈ ∂Ωl ,
3.23
p4 l ≥ φ
γ
then γl ≤ ut ≤ l, μ ≤ T ≤ ν.
Let m1 l, τ φB0 /γ φγl − k4 τφlλ3 . By Lemma 2.8 and condition B3 ,
applying Lemma 2.8, it follows that
s
m−2 ξi
m−2
ν
T m−2
i1 ai 0 aτm1 l, τ∇τ
i1 bi −
i1 bi ξi
−1
Au
≥
Δs. 3.24
φ
aτm1 l, τ∇τ T 1 − m−2
1 − m−2
μ
0
i1 bi
i1 ai
It follows that
s
m−2 ξi
m−2
ν
βl T m−2
m1 l, τ
i1 ai 0 aτ m1 l, τ/φl ∇τ
i1 bi −
i1 bi ξi
−1
≥
∇τ Δs.
φ
aτ
l
φl
T 1 − m−2
1 − m−2
μ
0
i1 bi
i1 ai
3.25
Noticing λ3 > 1, we can get
lim
l→0
βl
≥ 1.
l
3.26
Therefore, there exists b with 0 < b < a1 such that βb > b. It implies that Au
> u
for
u ∈ ∂Ωb .
By Lemma 2.9, we assert that the operator A has one fixed point u∗ ∈ K such that b ≤
∗
u ≤ a1 . Therefore, u∗ is positive solution of the problems 1.1.
4. Example
In this section, we present a simple example to explain our results.
Let ft, 0 ≡ 0, T R, T 1. Consider the following BVP:
Δ ∇
φ u
f t, ut 0, t ∈ 0, T ,
1
1
1
1
,
uT u
,
φ uΔ 0 φ uΔ
4
3
2
3
where
φu 4.1
⎧
⎨−u2 , u ≤ 0,
⎩u2 ,
u > 0,
⎧
√
⎪
1
4 √ 5
5 2
⎪
3
⎪
,
u , t, u ∈ 0, 1 × 0, 2,
min 45u ⎪
⎪
2
⎪
t1 − t u
⎪
⎨
t, u ∈ 0, 1 × 2, 5,
ft, u 400,
⎪
⎪
√
⎪
⎪
1
17 3
2u √ 3
5
2 1
⎪
⎪
⎪
u
min
,
× 5, ∞.
,
−
u
,
t,
u
∈
⎩5
2
5 2
t1 − t 5
4.2
12
Discrete Dynamics in Nature and Society
It is easy to check that f : 0, 1 × 0, ∞→0, ∞ is continuous. In this case, at ≡ 1, m 3, a1 1/4, b1 1/2, ξ1 1/3, it follows from a direct calculation that
1
−1
−1 1
3 · 27
a1 ξ1
1/4 · 1/3 1/2
φ
Δs
s
Δs
s
√
,
1
−
a
1
−
1/2
1
−
1/4
1
410 10 − 1
0
0
1/21 − 1/3
b1 T − ξ1
2
,
γ
T − b1 ξ1
1 − 1/2 · 1/3
5
1
A0 1 − b1
B0 T
−1
T b1 − b1 ξ1
T 1 − b1
−1 −1
a1 ξ1
1/4 · 1/3 1/2
1/2 − 1/2 · 1/3 1/2
Δs
s
φ−1 s Δs
1 − a1
1 − 1/2
1 − 1/4
2/5
2/5
1/2
√
√
4 33 × 25 22 − 69 × 4 115 −1
.
9
8100
4.3
Choose
c1 2, c2 5, b 5, √
λ1 5/4,λ2 3/4, p1 u 45u3 , p2 u 17/5u3 , k1 t √
5 2/21/ t1 − t, k2 t 5/21/ t1 − t, it is easy to check that
5/4
ft, u ≤ p1 u k1 t u2
,
t, u ∈ 0, 1 × 0, 2,
3/4
ft, u ≥ p2 u − k2 t u2
,
2 1
× 5, ∞,
t, u ∈
,
5 2
2
p1 u
45u3
3 · 27
,
lim 2 0 < A20 √
u→0 φu
u→0 u
410 10 − 1
p2 u
17/5u3
B0
.
lim
∞
>
φ
lim
u→∞ φu
u→∞
γ
u2
4.4
lim
It follows that f satisfies the conditions A1 –A3 of Theorem 3.1, then problems 1.1 have at
least one positive solution.
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