Hindawi Publishing Corporation
Boundary Value Problems
Volume 2011, Article ID 567054, 14 pages
doi:10.1155/2011/567054
Research Article
New Fixed Point Theorems of Mixed Monotone
Operators and Applications to Singular Boundary
Value Problems on Time Scales
Huiye Xu
College of Economics and Management, North University of China, Taiyuan, Shanxi 030051, China
Correspondence should be addressed to Huiye Xu, silviahsu2005@yahoo.com.cn
Received 3 July 2010; Accepted 13 December 2010
Academic Editor: Daniel Franco
Copyright q 2011 Huiye 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.
Some new existence and uniqueness theorems of fixed points of mixed monotone operators are
obtained, and then they are applied to a nonlinear singular second-order three-point boundary
value problem on time scales. We prove the existence and uniqueness of a positive solution for the
above problem which cannot be solved by using previously available methods.
1. Introduction
The study of mixed monotone operators has been a matter of discussion since they were
introduced by Guo and Lakshmikantham 1 in 1987, because it has not only important
theoretical meaning but also wide applications in microeconomics, the nuclear industry, and
so on see 1–4. Recently, some new and interesting results about these kinds of operators
have emerged, and they are used extensively in nonlinear differential and integral equations
see 5–9.
In this paper, we extend the main results of 9 to mixed monotone operators. Without
demanding compactness and continuity conditions and the existence of upper and lower
solutions, we study the existence, uniqueness, and iterative convergence of fixed points of a
class of mixed monotone operators. Then, we apply these results to the following singular
second-order three-point boundary value problem on time scales:
−x t wt f1 xt f2 xt , t ∈ a, bÌ,
xa 0, x σ 2 b δx η ,
1.1
2
Boundary Value Problems
where a, b ∈ Ì with a < b, η ∈ a, bÌ, and 0 < δ < σ 2 b − a/η − a. The functions
w : a, b → 0, ∞ and fi : 0, ∞ → 0, ∞ i 1, 2 are continuous. Our nonlinearity w
may have singularity at t a and/or t b and fi x i 1, 2 may have singularity at x 0.
To understand the notations used in 1.1, we recall that Ì is a time scales, that is, Ì is
an arbitrary nonempty closed subset of Ê. For each interval I of Ê, we define IÌ I ∩ Ì. For
more details on time scales, one can refer to 10–12.
In recent years, there is much attention paid to the existence of positive solutions
for nonlocal boundary value problems on time scales, see 13–18 and references therein.
Dynamic equations have been applied in the study of insect population models, stock market
and heat transfer and so on 19–22. Time scales can be used in microeconomics models to
study behavior which is sometimes continuous and sometimes discrete. A simple example
of this continuous-discrete behavior is seen in suppliers short-run decisions and long-run
decisions. Unifying both continuous and discrete model can avoid repeat research and has the
capacity to get some different types of models which neither continuous models nor discrete
models can effectively describe.
On the other hand, singular boundary value problems on time scales have also been
investigated extensively see 23–27. We would like to mention some results of DaCunha
et al. 23, Hao et al. 25, Luo 26, and Hu 27, which motivated us to consider problem
1.1.
In 23, DaCunha et al. considered the following singular second-order three-point
dynamic boundary value problem:
−x t ft, x,
x0 0,
t ∈ 0, 1Ì,
1.2
x σ 1 x η ,
2
where η ∈ 0, 1Ì is fixed and ft, x is singular at x 0 and possible at t 0, x ∞. The
authors claimed that “we note that this is the first work to our knowledge that deals with
singular boundary value problems in a general time scales setting.” The results on existence
of positive solutions were obtained by means of a fixed point theorem due to Gatica, Oliker
and Waltman for mappings that are decreasing with respect to a cone.
In 25, Hao et al. were concerned with the following singular boundary value problem
of nonlinear dynamic equation
Δ
ϕtxΔ t λmtft, xσt 0,
αxa − βxΔ a 0,
t ∈ a, bÌ,
1.3
γ xσb δxΔ σb 0,
where m· : a, σbÌ → 0, ∞ is rl-continuous and may be singular at t a and/or
t σb. With suitable growth and limit conditions, an existence theorem of positive solutions
was established by using the Krasnoselskii fixed point theorem.
Boundary Value Problems
3
In 26, Luo studied the following singular m-point dynamic eigenvalue problem with
mixed derivatives:
∇
− ptuΔ t λft, ut,
u0 m−2
ai uξi ,
t ∈ 0, 1Ì,
γ u1 δp1uΔ 1 i1
1.4
m−2
bi pξi uΔ ξi ,
i1
where ft, w is singular at t 0 and w 0. The author obtained eigenvalue intervals in
which there exists at least one positive solution of problem 1.4 by making use of the fixed
point index theory.
In 27, Hu were concerned with the following singular third-order three-point
boundary value problem on time scales:
∇
uΔΔ t wtft, ut 0,
u ρa − βuΔ ρa αu η ,
t ∈ a, bÌ,
γ u η ub,
uΔΔ ρa 0,
1.5
where w : a, bÌ → 0, ∞ and f : a, bÌ × 0, ∞ → 0, ∞ are continuous. The
nonlinearity w may have singularity at t a and/or t b and ft, u may have singularity
at u 0. With the aid of the fixed point theorem of cone expansion and compression type,
results on the existence of positive solutions to 1.5 were obtained in the bounded set.
From the above research, we note that there is no result on the uniqueness of solutions
and convergence of the iterative sequences for singular boundary value problems on time
scales. As we know, completely continuity condition is crucial for the above discussion.
However, it is difficult to verify for singular problems on time scales, in particular, in order
to remove the singularity in fu at u 0, more restricted conditions are required. For
instance, condition A1 of Theorem 2.3 in 23 and condition C2 of Theorem 3.1 in 27.
In our abstract results on mixed monotone operators, since the compactness and continuity
conditions are not required, they can be directly applied to singular boundary value problem
1.1.
The purpose of this paper is to present some conditions for problem 1.1 that have a
unique solution, the iterative sequences yielding approximate solutions are also given. Our
main result generalizes and improves Theorem 2.3 in 18.
2. Preliminaries and Abstract Theorems
Let the real Banach space E be partially ordered by a cone P of E, that is, x ≤ y if and only if
y − x ∈ P . A : P × P → P is said to be a mixed monotone operator if Ax, y is increasing in x
and decreasing in y, that is, ui , vi i 1, 2 ∈ P , u1 ≤ u2 , v1 ≥ v2 implies Au1 , v1 ≤ Au2 , v2 .
Element x ∈ P is called a fixed point of A if Ax, x x.
Recall that cone P is said to be solid if the interior P ◦ is nonempty and we denote x θ
if x ∈ P ◦ . P is said to be normal if there exists a positive constant N, such that θ ≤ x ≤ y ⇒
x ≤ Ny, the smallest N is called the normal constant of P . For all x, y ∈ E, the notation
4
Boundary Value Problems
x ∼ y means that there exist λ > 0 and μ > 0 such that λx ≤ y ≤ μx. Clearly, ∼ is an equivalence
relation. Given h > θ i.e., h ≥ θ and h /
θ, we denote by Ph the set Ph {x ∈ E | x ∼ h}. It is
∅ and h ∈ P ◦ , it is clear that
easy to see that Ph ⊂ P is convex and λPh Ph for all λ > 0. If P ◦ /
◦
Ph P .
All the concepts discussed above can be found in 1, 2, 4. For more results about
mixed monotone operators and other related concepts, the reader is referred to 3, 5–9 and
some of the references therein.
In 9, Zhai and Cao introduced the following definition of τ-ϕ-concave operators.
Definition 2.1 see 9. Let E be a real Banach space and P be a cone in E. We say an operator
A : P → P is τ-ϕ-concave if there exist two positive-valued functions τt, ϕt on interval
a, b such that
H1 τ : a, b → 0, 1 is a surjection;
H2 ϕt > τt, for all t ∈ a, b;
H3 Aτtx ≥ ϕtAx, for all t ∈ a, b, x ∈ P .
They obtained the following result.
Theorem 2.2 see 9. Let E be a real Banach space and P be a normal cone in E. Suppose that an
operator A : P → P is increasing and τ-ϕ-concave. In addition, suppose that there exists h ∈ P \ {θ}
such that Ah ∈ Ph . Then
i there are u0 , v0 ∈ Ph and r ∈ 0, 1 such that rv0 ≤ u0 ≤ v0 , u0 ≤ Au0 ≤ Av0 ≤ v0 ;
ii operator A has a unique fixed point x∗ in u0 , v0 ;
iii for any initial x0 ∈ Ph , constructing successively the sequence xn Axn−1 , n 1, 2, . . .,
we have xn − x∗ → 0 n → ∞.
We can extend Theorem 2.2 to mixed monotone operators, our main results can be
stated as follows.
Theorem 2.3. Let P be a normal cone in a real Banach space E, and A : P × P → P a mixed
monotone operator. Assume that for all a < t < b, there exist two positive-valued functions τt, ϕt
on interval a, b such that
C1 τ : a, b → 0, 1 is a surjection;
C2 ϕt > τt, for all t ∈ a, b;
C3 Aτtx, 1/τty ≥ ϕtAx, y, for all t ∈ a, b, x, y ∈ P .
In addition, suppose that there exists h ∈ P \ {θ} such that Ah, h ∈ Ph . Then
i there are u0 , v0 ∈ Ph and r ∈ 0, 1 such that rv0 ≤ u0 ≤ v0 , u0 ≤ Au0 , v0 ≤ Av0 , u0 ≤
v0 ;
ii operator A has a unique fixed point x∗ in u0 , v0 ;
iii for any initial x0 , y0 ∈ Ph , constructing successively the sequences xn Axn−1 , yn−1 ,
yn Ayn−1 , xn−1 , n 1, 2, . . ., we have xn − x∗ → 0 and yn − x∗ → 0 as n → ∞.
Boundary Value Problems
5
Corollary 2.4. Let E be a real Banach space, P a normal, solid cone in E. Suppose A : P ◦ × P ◦ → P ◦
is a mixed monotone operator and satisfies the conditions C1 –C3 of Theorem 2.3. Then
i there are u0 , v0 ∈ P ◦ and r ∈ 0, 1 such that rv0 ≤ u0 ≤ v0 , u0 ≤ Au0 , v0 ≤ Av0 , u0 ≤
v0 ;
ii operator A has a unique fixed point x∗ in P ◦ ;
iii for any initial x0 , y0 ∈ P ◦ , constructing successively the sequences xn Axn−1 , yn−1 ,
yn Ayn−1 , xn−1 , n 1, 2, . . ., we have xn − x∗ → 0 and yn − x∗ → 0 as n → ∞.
Remark 2.5. In Theorem 2.3, if A : P ◦ × P ◦ → P ◦ with P is a solid cone, we can know that
Ah, h ∈ Ph is automatically satisfied. Therefore, we can deduce that Corollary 2.4 holds
from Theorem 2.3. For simplicity, we only present the proof of Theorem 2.3.
Proof of Theorem 2.3. Note that Ah, h ∈ Ph , we can find a sufficiently small number e0 ∈ 0, 1
such that
h
.
e0
e0 h ≤ Ah, h ≤
2.1
According to C1 , we can obtain that there exists t1 ∈ a, b such that τt1 e0 , thus
h
.
τt1 τt1 h ≤ Ah, h ≤
2.2
Since ϕt1 > τt1 , we can find a positive integer k such that
ϕt1 τt1 k
≥
1
.
τt1 2.3
Let u0 τt1 k h, v0 1/τt1 k h, and construct successively the sequences
un Aun−1 , vn−1 ,
vn Avn−1 , un−1 ,
n 1, 2, . . . .
2.4
It is clear that u0 , v0 ∈ Ph and u0 < v0 , u1 Au0 , v0 ≤ Av0 , u0 v1 . In general, we obtain
un ≤ vn , n 1, 2, . . ..
It follows from C3 , 2.2, and 2.3 that
u1 Au0 , v0 A τt1 h,
A τt1 τt1 k−1
τt1 k
1
h
h,
τt1 τt1 k−1
≥ ϕt1 A τt1 h
k
k−1
h,
h
τt1 k−1
6
Boundary Value Problems
ϕt1 A τt1 τt1 k−2 h,
2
≥ ϕt1 A τt1 k−2 h,
1
h
τt1 τt1 k−2
h
τt1 k−2
k
≥ · · · ≥ ϕt1 Ah, h
k
≥ ϕt1 τt1 h
≥ τt1 k h u0 .
2.5
From C3 , we have
A
x
, τty
τt
≤
1
A x, y ,
ϕt
∀t ∈ a, b, x, y ∈ P.
2.6
Combining 2.2 with 2.3 and 2.6, we have
v1 Av0 , u0 A
A
k
τt1 k
, τt1 h
1
h
k−1
, τt1 τt1 h
τt1 τt1 k−1
1
≤
A
ϕt1 1
A
ϕt1 1
≤
ϕt1 h
k−1
τt1 k−1
, τt1 2 A
h
τt1 k
h
1
h
k−2
, τt1 τt1 h
τt1 τt1 k−2
h
τt1 k−2
1 k
≤ ··· ≤
Ah, h
ϕt1 1 k h
≤
ϕt1 τt1 ≤
h
k−2
, τt1 h
v0 .
2.7
Boundary Value Problems
7
Thus, we obtain
u0 ≤ u1 ≤ v1 ≤ v0 .
2.8
u0 ≤ u1 ≤ · · · ≤ un ≤ · · · ≤ vn ≤ · · · ≤ v1 ≤ v0 .
2.9
By induction, it is easy to obtain that
Take any r ∈ 0, τt1 2k , then r ∈ 0, 1 and u0 ≥ rv0 . So we can know that
un ≥ u0 ≥ rv0 ≥ rvn ,
n 1, 2, . . . .
2.10
Let
rn sup{r > 0 | un ≥ rvn },
n 1, 2, . . . .
2.11
n 1, 2, . . . .
2.12
Thus, we have un ≥ rn vn , n 1, 2, . . ., and then
un1 ≥ un ≥ rn vn ≥ rn vn1 ,
Therefore, rn1 ≥ rn ; that is,
0 < r0 ≤ r1 ≤ · · · ≤ rn ≤ · · · ≤ 1.
2.13
Set r ∗ limn → ∞ rn , we will show that r ∗ 1. In fact, if 0 < r ∗ < 1, by C1 , there exists
t2 ∈ a, b such that τt2 r ∗ . Consider the following two cases.
i There exists an integer N such that rN r ∗ . In this case, we have rn r ∗ and
∗
un ≥ r vn for all n ≥ N hold. Hence
un1
1
1
∗
Aun , vn ≥ A r vn , ∗ un A τt2 vn ,
un
r
τt2 ≥ ϕt2 Avn , un ϕt2 vn1 ,
2.14
n ≥ N.
By the definition of rn , we have
rn1 r ∗ ≥ ϕt2 > τt2 r ∗ ,
which is a contradiction.
n ≥ N,
2.15
8
Boundary Value Problems
ii For all integers n, rn < r ∗ . Then, we obtain 0 < rn /r ∗ < 1. By C1 , there exist
zn ∈ a, b such that τzn rn /r ∗ . Hence
un1
rn ∗
1
1
1
∗
Aun , vn ≥ A rn vn , un A ∗ r vn ,
un A τzn r vn ,
un
rn
r
τzn r ∗
rn /r ∗ r ∗
1
1
≥ ϕzn A r ∗ vn , ∗ un ϕzn A τt2 vn ,
un
r
τt2 ≥ ϕzn ϕt2 Avn , un ϕzn ϕt2 vn1 .
2.16
By the definition of rn , we have
rn1 ≥ ϕzn ϕt2 > τzn ϕt2 rn
ϕt2 .
r∗
2.17
Let n → ∞, we have
r ∗ ≥ ϕt2 > τt2 r ∗ ,
2.18
which is also a contradiction. Thus, limn → ∞ rn 1.
Furthermore, similarly to the proof of Theorem 2.1 in 9, there exits x∗ ∈ u0 , v0 such
that limn → ∞ un limn → ∞ vn x∗ , and x∗ is the fixed point of operator A.
In the following, we prove that x∗ is the unique fixed point of A in Ph . In fact, suppose
that x∗ ∈ Ph is another fixed point of operator A. Let
1
c1 sup 0 < c ≤ 1 | cx∗ ≤ x ≤ x∗ .
c
∗
2.19
Clearly, 0 < c1 ≤ 1 and c1 x∗ ≤ x∗ ≤ 1/c1 x∗ . If 0 < c1 < 1, according to C1 , there exists
t3 ∈ a, b such that τt3 c1 . Then
1
1
x Ax , x ≥ A c1 x∗ , x∗ A τt3 x∗ ,
x∗
c1
τt3 ∗
∗
∗
≥ ϕt3 Ax∗ , x∗ ϕt3 x∗ ,
1
1
x∗ Ax∗ , x∗ ≤ A
x∗ , c1 x∗ A
x∗ , τt3 x∗
c1
τt3 ≤
2.20
1
1
Ax∗ , x∗ x∗ .
ϕt3 ϕt3 It follows that
ϕt3 x∗ ≤ x∗ ≤
1
x∗ .
ϕt3 2.21
Boundary Value Problems
9
Hence, c1 ≥ ϕt3 > τt3 c1 , which is a contradiction. Thus we have c1 1, that is, x∗ x∗ .
Therefore, A has a unique fixed point x∗ in Ph . Note that u0 , v0 ⊂ Ph , so we know that x∗
is the unique fixed point of A in u0 , v0 . For any initial x0 , y0 ∈ Ph , we can choose a small
number e ∈ 0, 1 such that
eh ≤ x0 ≤
1
h,
e
eh ≤ y0 ≤
1
h.
e
2.22
From C1 , there is t4 ∈ a, b such that τt4 e, thus
τt4 h ≤ x0 ≤
1
h,
τt4 τt4 h ≤ y0 ≤
1
h.
τt4 2.23
We can choose a sufficiently large positive integer q such that
ϕt4 τt4 q
≥
1
.
τt4 2.24
Take u
0 τt4 q h, v0 1/τt4 q h. We can find that
u
0 ≤ x0 ≤ v0 ,
u
0 ≤ y0 ≤ v0 ,
2.25
constructing successively the sequences
xn A xn−1 , yn−1 ,
yn A yn−1 , xn−1 ,
n 1, 2, . . . ,
u
n A
un−1 , vn−1 ,
vn−1 , u
n−1 ,
vn A
n 1, 2, . . . .
2.26
By using the mixed monotone properties of operator A, we have
u
n ≤ xn ≤ vn ,
u
n ≤ yn ≤ vn ,
n 1, 2, . . . .
2.27
Similarly to the above proof, we can know that there exists y∗ ∈ Ph such that
A y∗ , y∗ y∗ ,
lim u
n lim vn y∗ .
n→∞
2.28
n→∞
By the uniqueness of fixed points of operator A in Ph , we have y∗ x∗ . Taking into account
that P is normal, we deduce that limn → ∞ xn limn → ∞ yn x∗ . This completes the proof.
3. Applications to Singular BVP 1.1 on Time Scales
A Banach space E Ca, σ 2 bÌ is the set of real-valued continuous in the topology of Ì
function ut defined on a, σ 2 bÌ with the norm u maxt∈a,σ 2 b |ut|.
Ì
10
Boundary Value Problems
Define a cone by
P
u∈E|
min
t∈a,σ 2 b
ut ≥ 0 .
3.1
Ì
It is clear that P is a normal cone of which the normality constant is 1.
In order to obtain our main result, we need the following lemmas.
Lemma 3.1 see 18. The Green function corresponding to the following problem
−xΔΔ t 0,
xa 0,
t ∈ a, bÌ,
3.2
x σ 2 b δx η
is given by
δG η, s
Kt, s Gt, s 2
t − a,
σ b − a − δ η − a
3.3
where
Gt, s ⎧
⎪
⎨t − a σ 2 b − σs ,
1
σ 2 b − a ⎪
⎩σs − aσ 2 b − t,
t ≤ s,
3.4
t ≥ σs,
is Green’s function for the BVP:
−xΔΔ t 0,
t ∈ a, bÌ,
3.5
xa x σ b 0.
2
Lemma 3.2 see 18. For any t, s ∈ a, σ 2 bÌ × a, σbÌ, we have
δG η, s
δG η, s
t − a ≤ Kt, s ≤ 1 2
t − a.
σ 2 b − a − δ η − a
σ b − a − δ η − a
3.6
Our main result is the following theorem.
Theorem 3.3. Assume that
E1 f1 is nondecreasing, f2 is nonincreasing and there exist τt, ϕt on interval a, bÌ such
that τ : a, bÌ → 0, 1 is a surjection and ϕt > τt, for all t ∈ a, bÌ which satisfy
f1 τ μ x1 f2
1
x2
τ μ
≥ ϕ μ f1 x1 f2 x2 ,
∀μ ∈ a, bÌ, x1 , x2 ∈ P ;
3.7
Boundary Value Problems
11
E2 there exist two constants N1 , N2 > 0 and h ∈ P \ {0} such that
N1 ht ≤
σb
∀t ∈ a, σ 2 b .
Kt, sws f1 hs f2 hs Δs ≤ N2 ht,
Ì
a
3.8
Then problem 1.1 has a unique positive solution x∗ in Ph . Moreover, for any initial
x0 , y0 ∈ Ph , constructing successively the sequences
xn t σb
Kt, sws f1 xn−1 s f2 yn−1 s Δs,
t ∈ a, σ 2 b , n 1, 2, . . . ,
Ì
a
yn t σb
Kt, sws f1 yn−1 s f2 xn−1 s Δs,
t ∈ a, σ 2 b , n 1, 2, . . . ,
3.9
Ì
a
we have xn − x∗ → 0 and yn − x∗ → 0 as n → ∞.
Proof of Theorem 3.3. Define an operator A : P × P → E
Ax1 , x2 t σb
Kt, sws f1 x1 s f2 x2 s Δs,
t ∈ a, σ 2 b .
Ì
a
3.10
It is easy to check that x is a solution of problem 1.1 if and only if x is a fixed point of
operator A. Clearly, we can know that A : P × P → P is a mixed monotone operator. For any
μ ∈ a, bÌ and x1 , x2 ∈ P , according to E1 , we obtain
1
A τ μ x1 , x2
τ μ
σb
Kt, sws f1 τ μ x1 s f2
a
≥ϕ μ
σb
1
x2 s Δs
τ μ
Kt, sws f1 x1 s f2 x2 s Δs
3.11
a
ϕ μ Ax1 , x2 .
Hence,
1
A τ μ x1 , x2
τ μ
≥ ϕ μ Ax1 , x2 ,
for μ ∈ a, bÌ, x1 , x2 ∈ P.
3.12
In addition, from E2 , we know that
N1 ht ≤ Ah, h σb
a
Kt, sws f1 hs f2 hs Δs ≤ N2 ht,
∀t ∈ a, σ 2 b .
Ì
3.13
Thus Ah, h ∈ Ph . Therefore, all the conditions of Theorem 2.3 are satisfied. By Theorem 2.3,
we can obtain the conclusions of Theorem 3.3.
12
Boundary Value Problems
Now, let us end this paper by the following example.
Example 3.4. Let Ì 0, 1/2 ∪ 2/3, 1, consider the following BVP on time scales
−xΔΔ t 1
1
3 x1/2 t 5 1/2
,
t
x t
x0 0,
t ∈ 0, 1Ì,
3.14
1
x1 2x
.
3
Set wt 1/t, f1 x 3x1/2 , f2 x 51/x1/2 , τt t, ϕt t2/3 . Then τ : 0, 1Ì → 0, 1
is a surjection and ϕt > τt for t ∈ 0, 1Ì.
For any t ∈ 0, 1, x1 , x2 ∈ P , it is easy to check that
f1 tx1 f2
1
x2
t
3 tx1 1/2 5 1
x2 /t1/2
1
1/2
2/3
3 x1 5 1/2
≥t
x2
t2/3 f1 x1 f2 x2 .
3.15
It follows from Lemma 3.1 that
1
G ,s
3
Let
d6
1
0
1/3
0
6
⎧
1
⎪
⎪
⎨ 3 1 − σs,
⎪
⎪
⎩ 2 σs,
3
1
≤ s,
3
1
≥ σs.
3
3.16
6G1/3, s1/s3 s1/2 5 1/s1/2 Δs d, since
1/2
1
2 1
1
1
1
s 8 s1/2 1/2 ds 6
1 − s 8 s1/2 1/2 ds
3 s
s
s
s
1/3 3
2/3
1/2
1
1
1
1
1
1
1
1/2
1/2
1 − σs 8 s 1/2 Δs 6
1 − σs 8 s 1/2 ds
3
s
s
s
s
2/3 3
32ln 3 − ln 2 −
16
−1/2
1
34 1 −1/2 4 1 3/2
−
4
9 2
3 2
3
5/2
−1/2
1/2
1
2 1 1/2
2
1
8
4
8
−
3
9 2
3
3
9
> 0.
3.17
Boundary Value Problems
13
We choose ht t − a t, according to Lemma 3.2, we have
1
1
f1 s f2 s Δs ≥ td,
s
0
1 1
1
1
1
1/2
Kt, s f1 s f2 s Δs ≤ t d 8 s 1/2 Δs .
s
s
0
0 s
Kt, s
3.18
By Theorem 3.3, problem 3.14 has a unique positive solution x∗ in Ph Pt . For any initial
x0 , y0 ∈ Pt , constructing successively the sequences
1
1
1/2
Δs,
Kt, s 8 xn−1 s 1/2
xn t s
yn−1 s
0
1
1
1
1/2
Δs,
Kt, s 8 yn−1 s 1/2
yn t s
x s
0
1
t ∈ 0, 1Ì, n 1, 2, . . . ,
3.19
t ∈ 0, 1Ì, n 1, 2, . . . ,
n−1
we have xn − x∗ → 0, yn − x∗ → 0 as n → ∞.
Remark 3.5. Example 3.4 indicates that Theorem 3.3 generalizes and complements Theorem
2.3 in 18 at the following aspects. Firstly, in our proof, we only need to check the conditions
“there exists h ∈ P \ {θ} such that Ah, h ∈ Ph ”, in fact, the author has shown that
“A : Ph → Ph ” in the proof of Theorem 2.3 in 18. It is clear that our hypotheses are
weaker than those imposed in Theorem 2.3 in 18. According to Lemma 3.2, we can know
that the condition E2 is automatically satisfied. Secondly, we have considered the case that
the condition “τt t and ϕt tq q ∈ 0, 1” is not satisfied, therefore, the condition E1 incorporates the more comprehensive functions than the condition H3 in Theorem 2.3 in
18. Thirdly, the more general conditions are imposed on our nonlinear term, they can be the
sum of nondecreasing functions and nonincreasing functions.
Acknowledgment
H. Xu was supported financially by the Science Foundation of North University of China.
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