Báo cáo hóa học: " Research Article The Iterative Method of Generalized u0 -Concave Operators" docx
Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (497.81 KB, 11 trang )
Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 979261, 11 pages
doi:10.1155/2011/979261
Research Article
The Iterative Method of Generalized
u
0
-Concave Operators
Yanqiu Zhou, Jingxian Sun, and Jie Sun
Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China
Correspondence should be addressed to Jingxian Sun,
Received 16 November 2010; Accepted 12 January 2011
Academic Editor: N. J. Huang
Copyright q 2011 Yanqiu Zhou 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.
We define the concept of the generalized u
0
-concave operators, which generalize the definition
of the u
0
-concave operators. By using the iterative method and the partial ordering method, we
prove the existence and uniqueness of fixed points of this class of the operators. As an example of
the application of our results, we show the existence and uniqueness of solutions to a class of the
Hammerstein integral equations.
1. Introduction and Preliminary
In 1, 2, C ollatz divided the typical problems in computation mathematics into five classes,
and the first class is how to solve the operator equation
Ax x 1.1
by the iterative method, that is, construct successively the sequence
x
n1
Ax
n
1.2
for some initial x
0
to solve 1.1.
Let P be a cone in real Banach space E and the partial ordering ≤ defined by P ,that
is, x ≤ y if and only if y − x ∈ P. The concept and properties of the cone can be found in 3–
5. People studied how to solve 1.1 by using the iterative method and the partial ordering
method see 1–11.
2 Fixed Point Theory and Applications
In 7, Krasnosel’ski
˘
ı gave the concept of u
0
-concave operators and studied the
existence and uniqueness of the fixed point for the operator by the iterative method. The
concept of u
0
-concave operators was defined by Krasnosel’ski
˘
ı as follows.
Let operator A : P → P and u
0
>θ. Suppose that
i for any x>θ, there exist α αx > 0andβ βx > 0, such that
αu
0
≤ Ax ≤ βu
0
; 1.3
ii for any x ∈ P satisfying α
1
u
0
≤ x ≤ β
1
u
0
α
1
α
1
x > 0, β
1
β
1
x > 0 and any
0 <t<1, there exists η ηx, t > 0, such that
A
tx
≥
1 η
tAx. 1.4
Then A is called an u
0
-concave operator.
In many papers, the authors studied u
0
-concave operators and obtained some results
see 3–5, 8–15. In this paper, we generalize the concept of u
0
-concave operators, give a
concept of the generalized u
0
-concave operators, and study the existence and uniqueness of
fixed points for this class of operators by the iterative method. Our results generalize the
results in 3, 4, 7, 15.
2. Main Result
In this paper, we always let P be a cone in real Banach space E and the partial ordering ≤
defined by P .Givenw
0
∈ E,letPw
0
{x ∈ E | x ≥ w
0
}.
Definition 2.1. Let operator A : P w
0
→ Pw
0
and u
0
>θ. Suppose that
i for any x>w
0
, there exist α αx > 0andβ βx > 0, such that
αu
0
w
0
≤ Ax ≤ βu
0
w
0
; 2.1
ii for any x ∈ Pw
0
satisfying α
1
u
0
w
0
≤ x ≤ β
1
u
0
w
0
α
1
α
1
x > 0, β
1
β
1
x >
0 and any 0 <t<1, there exists η ηx, t > 0, such that
A
tx
1 − t
w
0
≥
1 η
tAx
1 −
1 η
t
w
0
. 2.2
Then A is called a generalized u
0
-concave operator.
Remark 2.2. In Definition 2.1,letw
0
θ, we get the definition of the u
0
-concave operator.
Theorem 2.3. Let operator A : Pw
0
→ Pw
0
be generalized u
0
-concave and increasing (i.e.,
x ≤ y ⇒ Ax ≤ Ay), then A has at most one fixed point in Pw
0
\{w
0
}.
Fixed Point Theory and Applications 3
Proof. Let x
1
>w
0
, x
2
>w
0
be two different fixed points of A,thatis,Ax
1
x
1
, Ax
2
x
2
,andx
1
/
x
2
.ByDefinition 2.1, there exist real numbers α
1
α
1
x
1
> 0, β
1
β
1
x
1
>
0, α
2
α
2
x
2
> 0, β
2
β
2
x
2
> 0, such that
α
1
u
0
w
0
≤ x
1
≤ β
1
u
0
w
0
,α
2
u
0
w
0
≤ x
2
≤ β
2
u
0
w
0
.
2.3
Hence α
1
/β
2
x
2
− w
0
w
0
≤ α
1
u
0
w
0
≤ x
1
≤ β
1
u
0
w
0
≤ β
1
/α
2
x
2
− w
0
w
0
.
Let α α
1
/β
2
, β β
1
/α
2
,wegetthatαx
2
− w
0
w
0
≤ x
1
≤ βx
2
− w
0
w
0
,that
is, αx
2
1 − αw
0
≤ x
1
≤ βx
2
1 − βw
0
.Let
t
0
sup
t>0 | tx
2
1 − t
w
0
≤ x
1
≤ t
−1
x
2
1 − t
−1
w
0
, 2.4
hence 0 <t≤ t
−1
,thatis,0<t≤ 1, then t
0
∈ 0, 1.
Next we will show that t
0
1. Assume that t
0
< 1; by 2.2 and 2.4, there exists
η
1
η
1
x
2
,t
0
> 0, such that
x
1
Ax
1
≥ A
t
0
x
2
1 − t
0
w
0
≥
1 η
1
t
0
Ax
2
1 −
1 η
1
t
0
w
0
1 η
1
t
0
x
2
1 −
1 η
1
t
0
w
0
.
2.5
By 2.2, there exists η
2
η
2
x
2
,t
0
> 0, such that
x
2
Ax
2
A
t
0
t
−1
0
x
2
1 − t
−1
0
w
0
1 − t
0
w
0
≥
1 η
2
t
0
A
t
−1
0
x
2
1 − t
−1
0
w
0
1 −
1 η
2
t
0
w
0
,
2.6
hence,
A
t
−1
0
x
2
1 − t
−1
0
w
0
≤
1 η
2
−1
t
−1
0
Ax
2
1 −
1 η
2
−1
t
−1
0
w
0
. 2.7
Therefore,
x
1
Ax
1
≤ A
t
−1
0
x
2
1 − t
−1
0
w
0
≤
1 η
2
−1
t
−1
0
Ax
2
1 −
1 η
2
−1
t
−1
0
w
0
≤
1 η
2
−1
t
−1
0
x
2
1 −
1 η
2
−1
t
−1
0
w
0
.
2.8
Obviously, by 2.5 and 2.8,weget
1 η
1
t
0
x
2
1 −
1 η
1
t
0
w
0
≤ x
1
≤
1 η
2
−1
t
−1
0
x
2
1 −
1 η
2
−1
t
−1
0
w
0
. 2.9
4 Fixed Point Theory and Applications
Let η min{η
1
,η
2
}, we have
1 η
t
0
x
2
1 −
1 η
t
0
w
0
≤ x
1
≤
1 η
−1
t
−1
0
x
2
1 −
1 η
−1
t
−1
0
w
0
, 2.10
in contradiction to the definition of t
0
. Therefore, t
0
1.
By 2.4, x
1
x
2
. The proof is completed.
To prove the following Theorem 2.4, we will use the definition of the u
0
-norm as
follows.
Given u
0
>θ,set
E
u
0
{
x ∈ E | there exists a real number λ>0, such that − λu
0
≤ x ≤ λu
0
}
,
x
u
0
inf
{
λ>0 |−λu
0
≤ x ≤ λu
0
}
, ∀x ∈ E
u
0
.
2.11
It is easy to see that E
u
0
becomes a normed linear space under t he norm ·
u
0
. x
u
0
is called
the u
0
- norm of the element x ∈ E
u
0
see 3, 4.
Theorem 2.4. Let operator A : Pw
0
→ P w
0
be increasing and generalized u
0
-concave. Suppose
that A has a fixed point x
∗
in Pw
0
\{w
0
}, then, constructing successively the sequence x
n1
Ax
n
n 0, 1, 2, for any initial x
0
∈ Pw
0
\{w
0
}, we have x
n
− x
∗
u
0
→ 0 n →∞.
Proof. Suppose that {x
n
} is generated from x
n1
Ax
n
n 0, 1, 2, . Take 0 <ε
0
< 1, such
that ε
0
x
∗
1−ε
0
w
0
≤ x
1
≤ ε
−1
0
x
∗
1−ε
−1
0
w
0
.Lety
0
ε
0
x
∗
1−ε
0
w
0
,z
0
ε
−1
0
x
∗
1−ε
−1
0
w
0
,
and constructing successively the sequences y
n1
Ay
n
, z
n1
Az
n
n 0, 1, 2, . Since A
is a generalized u
0
-concave operator, we know that there exists η
1
η
1
x
∗
,ε
0
> 0, such that
x
∗
Ax
∗
A
ε
0
ε
−1
0
x
∗
1 − ε
−1
0
w
0
1 − ε
0
w
0
≥
1 η
1
ε
0
A
ε
−1
0
x
∗
1 − ε
−1
0
w
0
1 −
1 η
1
ε
0
w
0
,
2.12
hence, Aε
−1
0
x
∗
1 − ε
−1
0
w
0
≤ 1 η
1
−1
ε
−1
0
Ax
∗
1 − 1 η
1
−1
ε
−1
0
w
0
, then
z
1
A
z
0
A
ε
−1
0
x
∗
1 − ε
−1
0
w
0
≤
1 η
1
−1
ε
−1
0
Ax
∗
1 −
1 η
1
−1
ε
−1
0
w
0
1 η
1
−1
ε
−1
0
Ax
∗
− w
0
w
0
<ε
−1
0
Ax
∗
− w
0
w
0
ε
−1
0
Ax
∗
1 − ε
−1
0
w
0
ε
−1
0
x
∗
1 − ε
−1
0
w
0
z
0
.
2.13
By 2.2, we can easily get y
1
>y
0
.Soitiseasytoshowthat
y
0
≤ y
1
≤···≤y
n
≤···≤ x
∗
≤···≤ z
n
≤···≤ z
1
≤ z
0
. 2.14
Fixed Point Theory and Applications 5
Let
t
n
sup
t>0 | tx
∗
1 − t
w
0
≤ y
n
,z
n
≤ t
−1
x
∗
1 − t
−1
w
0
n 0, 1, 2,
, 2.15
then,
0 ≤ t
0
≤ t
1
≤···≤ t
n
≤···≤ 1, 2.16
which implies that the limit of {t
n
} exists. Let lim
n →∞
t
n
t
∗
, then 0 <t
n
≤ t
∗
≤ 1.
Next we will show that t
∗
1. Suppose that 0 <t
∗
< 1. Since A is a generalized
u
0
-concave operator, then there exists η
2
η
2
x
∗
,t
∗
> 0, such that
A
t
∗
x
∗
1 − t
∗
w
0
≥
1 η
2
t
∗
Ax
∗
1 −
1 η
2
t
∗
w
0
1 η
2
t
∗
x
∗
1 −
1 η
2
t
∗
w
0
.
2.17
Moreover,
x
∗
Ax
∗
A
t
∗
t
∗
−1
x
∗
1 −
t
∗
−1
w
0
1 − t
∗
w
0
≥
1 η
2
t
∗
A
t
∗
−1
x
∗
1 −
t
∗
−1
w
0
1 −
1 η
2
t
∗
w
0
.
2.18
Therefore,
A
t
∗
−1
x
∗
1 −
t
∗
−1
w
0
≤
1 η
2
−1
t
∗
−1
x
∗
1 −
1 η
2
−1
t
∗
−1
w
0
. 2.19
By 2.17 and 2.19, for any 0 <t≤ t
∗
, there exists η
3
η
3
x
∗
,t > 0, such that
A
tx
∗
1 − t
w
0
≥
1 η
3
tx
∗
1 −
1 η
3
t
w
0
,
A
t
−1
x
∗
1 − t
−1
w
0
≤
1 η
3
−1
t
−1
x
∗
1 −
1 η
3
−1
t
−1
w
0
.
2.20
Particularly, for any 0 <t
n
≤ t
∗
n 0, 1, 2, , we have
A
t
n
x
∗
1 − t
n
w
0
≥
1 η
t
n
x
∗
1 −
1 η
t
n
w
0
,
A
t
−1
n
x
∗
1 − t
−1
n
w
0
≤
1 η
−1
t
−1
n
x
∗
1 −
1 η
−1
t
−1
n
w
0
,
2.21
where η ηt
n
,x
∗
> 0.
Hence,
y
n1
Ay
n
≥ A
t
n
x
∗
1 − t
n
w
0
≥
1 η
t
n
x
∗
1 −
1 η
t
n
w
0
,
z
n1
Az
n
≤ A
t
−1
n
x
∗
1 − t
−1
n
w
0
≤
1 η
−1
t
−1
n
x
∗
1 −
1 η
−1
t
−1
n
w
0
.
2.22
6 Fixed Point Theory and Applications
By 2.15,and2.22,wegett
n1
≥ 1 ηt
n
n 0, 1, 2, therefore, t
n1
≥ 1 η
n1
t
0
n
0, 1, 2, , in contradiction to 0 <t
n
≤ 1 n 1, 2, . Hence,
t
∗
1. 2.23
Since A is a generalized u
0
-concave operator, thus there exist real numbers α αx
∗
> 0,
β βx
∗
> 0, such that αu
0
w
0
≤ x
∗
≤ βu
0
w
0
,andt
n
x
∗
1 − t
n
w
0
≤ y
n
≤ x
n1
≤ z
n
≤
t
−1
n
x
∗
1 − t
−1
n
w
0
n 0, 1, 2, , we have
t
n
− 1
x
∗
1 − t
n
w
0
≤ x
n1
− x
∗
≤
t
−1
n
− 1
x
∗
1 − t
−1
n
w
0
. 2.24
Moreover
t
n
− 1
x
∗
1 − t
n
w
0
≥
t
n
− 1
βu
0
w
0
1 − t
n
w
0
t
n
− 1
βu
0
,
t
−1
n
− 1
x
∗
1 − t
−1
n
w
0
≤
t
−1
n
− 1
βu
0
w
0
1 − t
−1
n
w
0
t
−1
n
− 1
βu
0
.
2.25
Hence,
1 − t
−1
n
βu
0
≤
t
n
− 1
βu
0
≤ x
n1
− x
∗
≤
t
−1
n
− 1
βu
0
n 0, 1, 2,
. 2.26
Consequently, by 2.23,wegetx
n
− x
∗
u
0
→ 0 n →∞.
To prove the following Theorem 2.5, we will use the definition of the normal cone as
follows.
Let P be a cone in E. Suppose that there exist constants N>0, such that
θ ≤ x ≤ y ⇒
x
≤ N
y
, 2.27
then P is said to be normal, and the smallest N is called the normal constant of P see
3–5.
Theorem 2.5. vLetP be a normal cone of E. If operator A : Pw
0
−→ Pw
0
is increasing and
generalized u
0
-concave, and η ηt, x is irrelevant to x in 2.2,thenA has the only one fixed point
x
∗
∈ Pw
0
\{w
0
}. Moreover, constructing successively the sequence x
n1
Ax
n
n 0, 1, 2,
for any initial x
0
>w
0
, we have x
n
− x
∗
→0 n →∞.
Proof. Since A is a generalized u
0
-concave operator, hence there exist real numbers β>α>0,
such that αu
0
w
0
≤ Au
0
w
0
≤ βu
0
w
0
. Take t
0
∈ 0, 1 small enough, then t
0
u
0
w
0
≤
Au
0
w
0
≤ 1/t
0
u
0
w
0
.
Therefore, t
n1
≥ t
n
,thatis,{t
n
} is an increasing sequence and 0 <t
n
≤ 1, hence, the
limit of {t
n
} exists. Set lim
n →∞
t
n
t
∗
, then 0 <t
∗
≤ 1.
Fixed Point Theory and Applications 7
Let γt1 ηtt, where ηt which is irrelevant to x is ηt, x in 2.2,andA is
increasing, so t<γt ≤ 1,Atx 1 − tw
0
≥ γtAx 1 − γtw
0
, for all t ∈ 0, 1.By
γt
0
/t
0
> 1, we can choose a natural number k>0 big enough, such that
γ
t
0
t
0
k
>
1
t
0
.
2.28
Let
y
0
t
k
0
u
0
w
0
,z
0
1
t
k
0
u
0
w
0
; y
n
Ay
n−1
,z
n
Az
n−1
n 1, 2,
.
2.29
Obviously, y
0
,z
0
∈ Pw
0
,y
0
<z
0
. Since A is increasing, we have
y
1
Ay
0
A
t
k
0
u
0
w
0
A
t
0
t
k−1
0
u
0
w
0
1 − t
0
w
0
≥ γ
t
0
A
t
k−1
0
u
0
w
0
1 − γ
t
0
w
0
γ
t
0
A
t
0
t
k−2
0
u
0
w
0
1 − t
0
w
0
1 − γ
t
0
w
0
≥ γ
t
0
γ
t
0
A
t
k−2
0
u
0
w
0
1 − γ
t
0
w
0
1 − γ
t
0
w
0
γ
2
t
0
A
t
k−2
0
u
0
w
0
1 − γ
2
t
0
w
0
≥···≥ γ
k
t
0
A
u
0
w
0
1 − γ
k
t
0
w
0
>t
k−1
0
t
0
u
0
w
0
1 − t
k−1
0
w
0
t
k
0
u
0
w
0
y
0
.
2.30
Since Ax A{t
0
t
−1
0
x 1 − t
−1
0
w
0
1 − t
0
w
0
}≥γt
0
At
−1
0
x 1 − t
−1
0
w
0
1 − γt
0
w
0
,
we get At
−1
0
x 1 − t
−1
0
w
0
≤ 1/γt
0
Ax 1 − 1/γt
0
w
0
. Hence
z
1
A
1
t
k
0
u
0
w
0
A
1
t
0
1
t
k−1
0
u
0
w
0
1 −
1
t
0
w
0
≤
1
γ
t
0
A
1
t
k−1
0
u
0
w
0
1 −
1
γ
t
0
w
0
≤···≤
1
γ
k
t
0
A
u
0
w
0
1 −
1
γ
k
t
0
w
0
≤
1
t
0
γ
k
t
0
u
0
w
0
<
1
t
k
0
u
0
w
0
z
0
,
2.31
then y
0
≤ y
1
≤ z
1
≤ z
0
.Itiseasytosee
y
0
≤ y
1
≤···≤ y
n
≤···≤ z
n
≤···≤ z
1
≤ z
0
. 2.32
8 Fixed Point Theory and Applications
Let
t
n
sup
t>0 | y
n
≥ tz
n
1 − t
w
0
. 2.33
Obviously, y
n
≥ t
n
z
n
1 − t
n
w
0
.Soy
n1
≥ y
n
≥ t
n
z
n
1 − t
n
w
0
≥ t
n
z
n1
1 − t
n
w
0
.
Therefore, t
n1
≥ t
n
,thatis,{t
n
} is an increasing sequence and 0 <t
n
≤ 1, hence, the
limit of {t
n
} exists. Set lim
n →∞
t
n
t
∗
, then 0 <t
∗
≤ 1.
Next we will show that t
∗
1. Suppose that 0 <t
∗
< 1, we have the following.
i If for any natural number n, t
n
<t
∗
< 1, then
y
n1
Ay
n
≥ A
t
n
z
n
1 − t
n
w
0
A
t
n
t
∗
t
∗
z
n
1 − t
∗
w
0
1 −
t
n
t
∗
w
0
≥ γ
t
n
t
∗
A
t
∗
z
n
1 − t
∗
w
0
1 − γ
t
n
t
∗
w
0
≥ γ
t
n
t
∗
γ
t
∗
Az
n
1 − γ
t
∗
w
0
1 − γ
t
n
t
∗
w
0
γ
t
n
t
∗
γ
t
∗
Az
n
1 − γ
t
n
t
∗
γ
t
∗
w
0
γ
t
n
t
∗
γ
t
∗
z
n1
1 − γ
t
n
t
∗
γ
t
∗
w
0
,
2.34
hence,
t
n1
≥ γ
t
n
t
∗
γ
t
∗
1 η
t
n
t
∗
t
n
t
∗
1 η
t
∗
t
∗
≥ t
n
1 η
t
∗
. 2.35
Taking limits, we have t
∗
≥ t
∗
1 ηt
∗
>t
∗
, a contradiction.
ii Suppose that there exists a natural number N>0, such that t
n
t
∗
n>N.
When n>N, so we have
y
n1
Ay
n
≥ A
t
n
z
n
1 − t
n
w
0
A
t
∗
z
n
1 − t
∗
w
0
≥ γ
t
∗
Az
n
1 − γ
t
∗
w
0
γ
t
∗
z
n1
1 − γ
t
∗
w
0
,
2.36
then t
∗
t
n1
≥ γt
∗
1 ηt
∗
t
∗
>t
∗
, a contradiction.
Therefore, t
∗
1.
For any natural numbers n, p, we have
θ ≤ y
np
− y
n
≤ z
np
− y
n
≤ z
n
− y
n
≤ z
n
−
t
n
z
n
1 − t
n
w
0
1 − t
n
z
n
− w
0
. 2.37
Similarly, θ ≤ z
n
− z
np
≤ z
n
− y
n
≤ 1 − t
n
z
n
− w
0
. By the normality of P and lim
n →∞
t
n
1,
we get
y
np
− w
0
−
y
n
− w
0
y
np
− y
n
≤ N
1 − t
n
z
n
− w
0
→ 0
n →∞
,
z
np
− w
0
−
z
n
− w
0
z
n
− z
np
≤ N
1 − t
n
z
n
− w
0
→ 0
n →∞
,
2.38
Fixed Point Theory and Applications 9
where N is the normal constant of P. Hence the limits of {y
n
} and {z
n
} exist. Let lim
n →∞
y
n
y
∗
, and let lim
n →∞
z
n
z
∗
, then y
n
≤ y
∗
≤ z
∗
≤ z
n
n 0, 1, 2, , hence,
θ ≤ z
∗
− y
∗
≤ z
n
− y
n
≤
1 − t
n
z
n
− w
0
→ θ
n →∞
. 2.39
That is, y
∗
z
∗
.Letx
∗
y
∗
z
∗
, then y
n1
Ay
n
≤ Ax
∗
≤ Az
n
z
n1
.
Taking limits, we get x
∗
≤ Ax
∗
≤ x
∗
. Hence Ax
∗
x
∗
,thatis,x
∗
∈ P w
0
\{w
0
} is a fixed
point of A.ByTheorem 2.4, the conclusions of Theorem 2.5 hold. The proof is completed.
3. Examples
Example 3.1. Let I 0, 1,let CI{xt : I → R | xt is continuous},letx
sup{|xt||t ∈ I},letP {x ∈ CI | xt ≥ 0, ∀t ∈ I}, then CI is a real Banach space
and P is a normal and solid cone in CIP is called solid if it contains interior points,
i.e.,
◦
P
/
∅. Take a<0, let w
0
w
0
t ≡ a, Pw
0
{x ∈ CI | xt ≥ w
0
, ∀t ∈ I},and
◦
Pw
0
{x w
0
∈ Pw
0
| x ∈
◦
P}.
Considering the Hammerstein integral equation
x
t
1
0
k
t, s
f
s, x
s
ds, t ∈
0, 1
,
3.1
where kt, s : I × I → 0, ∞ is continuous, fs, u : I × a, ∞ → R is increasing for u.
Suppose that
1 there exist real numbers 0 ≤ m ≤ M ≤ 1, such that m ≤ kt, s ≤ M, for all t, s ∈
I × I,andfs, u ≥ a/M, for alls, u ∈ I × a, ∞
,
2 for any λ ∈ 0, 1 and u ∈ a, ∞, there exists η ηλ > 0, such that
mf
s, λu
1 − λ
a
≥
1 η
λmf
s, u
1 −
1 η
λ
a. 3.2
Then 3.1 has the only one solution x
∗
∈ P w
0
\{w
0
}. Moreover, constructing successively
the sequence:
x
n
t
1
0
k
t, s
f
s, x
n−1
s
ds, ∀t ∈ I, n 1, 2,
3.3
for any initial x
0
∈ Pw
0
\{w
0
}, we have x
n
− x
∗
→0 n →∞.
Proof. Considering the operator
Ax
t
1
0
k
t, s
f
s, x
s
ds, t ∈ I.
3.4
10 Fixed Point Theory and Applications
Obviously, A : Pw
0
\{w
0
} →
◦
Pw
0
is increasing. Therefore, i of Definition 2.1 is satisfied.
For any x ∈
◦
Pw
0
,by3.2, we have
A
λx
t
1 − λ
w
0
1
0
k
t, s
f
s, λx
s
1 − λ
w
0
ds
1
0
1
m
k
t, s
mf
s, λx
s
1 − λ
w
0
ds
≥
1 η
λ
1
0
1
m
k
t, s
mf
s, x
s
ds
1 −
1 η
λ
w
0
1
0
1
m
k
t, s
ds
≥
1 η
λAx
t
1 −
1 η
λ
w
0
.
3.5
Therefore, ii of Definition 2.1 is satisfied. Hence the operator A : Pw
0
→ P w
0
is
generalized u
0
-concave. Consequently, operator A satisfies all conditions of Theorem 2.5,thus
the conclusion of Example 3.1 holds.
Example 3.2. Let R be a real numbers set, and let P {x | x ≥ 0, x ∈ R}, then R is a real Banach
space and P is a normal and solid cone in R.LetAx x2
1/2
−2. Considering the equation:
x Ax. Obviously, A is a generalized u
0
-concave operator and satisfies all the conditions of
Theorem 2.5. Hence A has the only one fixed point x
∗
∈ P−2 \{−2} −2, ∞. Moreover,
we know x
∗
−1 by computing.
In Example 3.2, we know that operator A : −2, ∞ → −2, ∞ doesn’t satisfy the
definition of u
0
-concave operators. Therefore, we can’t obtain the fixed point of A by the
fixed point theorem of u
0
-concave operators. The u
0
-concave operators’ fixed points are all
positive, but here A’s fixed point is negative.
Acknowledgment
The project is supported by the National Science Foundation of China 10971179,the
College Graduate Research and Innovation Plan Project of Jiangsu CX10S−037Z,the
Graduate Research and Innovation Programs of Xuzhou Normal University Innovation Plan
2010YLA001.
References
1 L. Collatz, “The theoretical basis of numerical mathematics,” Mathematics Asian Studies, vol. 4, pp.
1–17, 1966 Chinese.
2 L. Collatz, “Function analysis as the assistant tool for Numerical Mathematics,” Mathematics Asian
Studies, vol. 4, pp. 53–60, 1966 Chinese.
3 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.
4 D. Guo, Nonlinear Functional Analysis, Science and Technology Press, Shandong, China, 1985.
5 Jingxian Sun, Nonlinear Functional Analysis and Applications, Science Press, Beijing, China, 2007.
6 M. A. Krasnosel’skii et al., Approximate Solution of Operator Equations, Wolters-Noordhoff, 1972.
7 M. A. Krasnosel’ski
˘
ı and P. P. Zabre
˘
ıko, Geometrical Methods of Nonlinear Analysis, vol. 263 of
Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1984.
Fixed Point Theory and Applications 11
8 D. Guo, The partial order in non-linear analysis, Shandong Science and Technology Press, Ji’nan, China,
2000.
9 Z. Zhao, “Uniqueness and existence of fixed point on some mixed monotone mappings in ordered
linear spaces,” Journal of Systems Science and Complexity, vol. 19, no. 4, pp. 217–224, 1999 Chinese.
10 J. X. Sun and L. S. Liu, “An iterative solution method for nonlinear operator equations and its
applications,” Acta Mathematica Scientia. Series A, vol. 13, no. 2, pp. 141–145, 1993.
11 J. X. Sun, “Some new fixed point theorems of increasing operators and applications,” Applicable
Analysis, vol. 42, no. 3-4, pp. 263–273, 1991.
12 W. X. Wang and Z. D. Liang, “Fixed point theorems for a class of nonlinear operators and their
applications,” Acta Mathematica Sinica. Chinese Series, vol. 48, no. 4, pp. 789–800, 2005.
13 Z. D. Liang, W. X. Wang, and S. J. Li, “On concave operators,” Acta Mathematica Sinica (English Series),
vol. 22, no. 2, pp. 577–582, 2006.
14 M. A. Krasnosel’skii, Positive Solution of Operators Equations, Noordoff, Groningen, The Netherlands,
1964.
15 C. B. Zhai and Y. J. Li, “Fixed point theorems for u
0
-concave operators and their applications,” Acta
Mathematica Scientia. Series A, vol. 28, no. 6, pp. 1023–1028, 2008.
