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, [email protected]
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
≤ 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
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
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
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
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
≤ 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
1 −
1 η
1
t
0
w
0
1 η
1
t
0
x
2
1 −
1 η
1
t
1 − t
−1
0
w
0
1 − t
0
w
0
≥
1 η
2
t
0
A
t
−1
0
x
2
2
1 − t
−1
0
w
0
≤
1 η
2
−1
t
−1
0
Ax
2
1 −
1 η
2
−1
t
2
−1
t
−1
0
Ax
2
1 −
1 η
2
−1
t
−1
0
w
0
≤
1 η
2
−1
t
−1
1 −
1 η
1
t
0
w
0
≤ x
1
≤
1 η
2
−1
t
−1
0
x
2
1 −
1 η
2
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
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 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
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 − 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
−1
0
x
∗
1 − ε
−1
0
w
0
≤
1 η
1
−1
ε
−1
0
Ax
∗
1 −
1 η
1
∗
− w
0
w
0
ε
−1
0
Ax
∗
1 − ε
−1
0
w
0
ε
−1
0
x
∗
1 − ε
−1
0
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
η
2
x
∗
,t
∗
> 0, such that
A
t
∗
x
∗
1 − t
∗
w
0
≥
1 η
2
t
∗
Ax
∗
w
0
.
2.17
Moreover,
x
∗
Ax
∗
A
t
∗
t
∗
−1
x
∗
1 −
t
∗
−1
1 −
t
∗
−1
w
0
1 −
1 η
2
t
∗
w
0
.
2.18
Therefore,
A
t
∗
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
−1
w
0
≤
1 η
3
−1
t
−1
x
∗
1 −
1 η
3
−1
t
−1
∗
1 −
1 η
t
n
w
0
,
A
t
−1
n
x
∗
1 − t
−1
n
w
0
≤
y
n1
Ay
n
≥ A
t
n
x
∗
1 − t
n
w
0
≥
1 η
t
n
x
∗
1 −
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
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
Moreover
t
n
− 1
x
∗
1 − t
n
w
0
≥
t
n
− 1
βu
0
w
0
1 − t
t
−1
n
− 1
βu
0
w
0
1 − t
−1
n
w
0
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
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
/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
<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
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
1 − γ
t
0
w
0
γ
2
t
0
A
0
1 − γ
k
t
0
w
0
>t
k−1
0
t
0
u
0
w
0
1 − t
k−1
0
−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
0
w
0
≤
1
γ
t
0
A
1
t
k−1
0
u
0
w
0
1 −
1
γ
t
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
≥ 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
≤ 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
∗
A
t
∗
z
n
1 − t
∗
w
0
1 − γ
t
n
t
∗
w
t
n
t
∗
w
0
γ
t
n
t
∗
γ
t
∗
Az
n
1 − γ
t
n
t
∗
∗
γ
t
∗
w
0
,
2.34
hence,
t
n1
≥ γ
t
n
t
∗
γ
t
∗
1 η
∗
≥ 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
w
0
γ
t
∗
z
n1
1 − γ
t
∗
w
0
,
2.36
then t
∗
t
n1
≥ γt
∗
1 ηt
∗
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
− y
n
≤ N
1 − t
n
z
n
− w
0
→ 0
n →∞
,
z
np
− w
0
−
,
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,
∗
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
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
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
\{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
s
1 − λ
w
0
ds
≥
1 η
λ
1
0
1
m
k
t, s
mf
s, x
s
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,
ı and P. P. Zabre
˘
ıko, Geometrical Methods of Nonlinear Analysis, vol. 263 of
Fundamental Principles of Mathematical Sciences, Springer, Berlin, Germany, 1984.
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