Fixed Point Theory and Applications Volume 2011, Article ID 979261,11pages 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
Copyrightq2011 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 generalizedu0-concave operators, which generalize the definition
of theu0-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
In1,2, Collatz divided the typical problems in computation mathematics into five classes, and the first class is how to solve the operator equation
Axx 1.1
by the iterative method, that is, construct successively the sequence
xn1Axn 1.2
for some initialx0to solve1.1.
LetP be a cone in real Banach spaceEand the partial ordering≤defined byP, that is,x≤yif and only ify−x∈P. The concept and properties of the cone can be found in3–
In 7, Krasnosel’ski˘ı gave the concept of u0-concave operators and studied the
existence and uniqueness of the fixed point for the operator by the iterative method. The concept ofu0-concave operators was defined by Krasnosel’ski˘ıas follows.
Let operatorA:P →Pandu0> θ. Suppose that
ifor anyx > θ, there existααx>0 andββx>0, such that
αu0 ≤Ax≤βu0; 1.3
iifor anyx∈P satisfyingα1u0 ≤ x≤β1u0α1 α1x> 0,β1 β1x> 0and any
0< t <1, there existsηηx, t>0, such that
Atx≥1ηtAx. 1.4
ThenAis called anu0-concave operator.
In many papers, the authors studiedu0-concave operators and obtained some results see3–5,8–15. In this paper, we generalize the concept of u0-concave operators, give a
concept of the generalizedu0-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 in3,4,7,15.
2. Main Result
In this paper, we always letP be a cone in real Banach space Eand the partial ordering≤ defined byP. Givenw0 ∈E, letPw0 {x∈E|x≥w0}.
Definition 2.1. Let operatorA:Pw0→Pw0andu0 > θ. Suppose that
ifor anyx > w0, there existααx>0 andββx>0, such that
αu0w0 ≤Ax≤βu0w0; 2.1
iifor anyx∈Pw0satisfyingα1u0w0≤x≤β1u0w0α1α1x>0,β1β1x>
0and any 0< t <1, there existsηηx, t>0, such that
Atx 1−tw0≥
1ηtAx1−1ηtw0. 2.2
ThenAis called a generalizedu0-concave operator.
Remark 2.2. InDefinition 2.1, letw0θ, we get the definition of theu0-concave operator.
Theorem 2.3. Let operator A : Pw0 → Pw0be generalized u0-concave and increasing (i.e.,
Proof. Letx1 > w
0,x2 > w0be two different fixed points ofA, that is,Ax1 x1,Ax2
x2, andx1/x2. ByDefinition 2.1, there exist real numbersα
1α1x1>0,β1β1x1>
0,α2α2x2>0,β2β2x2>0, such that
α1u0w0≤x1≤β1u0w0, α2u0w0≤x2≤β2u0w0. 2.3
Henceα1/β2x2−w0 w0≤α1u0w0≤x1≤β1u0w0 ≤β1/α2x2−w0 w0.
Letαα1/β2,ββ1/α2, we get thatαx2−w0 w0 ≤x1≤βx2−w0 w0, that
is,αx2 1−αw
0≤x1≤βx2 1−βw0. Let
t0sup
t >0|tx2 1−tw0≤x1≤t−1x2
1−t−1w0 , 2.4
hence 0< t≤t−1, that is, 0< t≤1, thent
0∈0,1.
Next we will show thatt0 1. Assume thatt0 < 1; by 2.2and 2.4, there exists
η1η1x2, t0>0, such that
x1Ax1≥At0x2 1−t0w0
≥1η1
t0Ax2
1−1η1
t0
w0
1η1
t0x2
1−1η1
t0
w0.
2.5
By2.2, there existsη2 η2x2, t0>0, such that
x2Ax2At 0
t−1 0 x2
1−t−1 0
w0
1−t0w0
≥1η2
t0A
t−01x2
1−t−01
w0
1−1η2
t0
w0,
2.6
hence,
At−1 0 x2
1−t−1 0
w0
≤1η2 −1
t−1 0 Ax2
1−1η2 −1
t−1 0
w0. 2.7
Therefore,
x1Ax1≤At0−1x21−t−01w0
≤1η2 −1
t−1 0 Ax2
1−1η2 −1
t−1 0
w0
≤1η2 −1
t−01x2
1−1η2 −1
t−01
w0.
2.8
Obviously, by2.5and2.8, we get
1η1
t0x2
1−1η1
t0
w0≤x1≤
1η2 −1
t−1 0 x2
1−1η2 −1
t−1 0
Letηmin{η1, η2}, we have
1ηt0x2
1−1ηt0
w0≤x1≤
1η−1t0−1x21−1η−1t0−1w0, 2.10
in contradiction to the definition oft0. Therefore,t0 1.
By2.4,x1 x2. The proof is completed.
To prove the following Theorem 2.4, we will use the definition of the u0-norm as
follows.
Givenu0 > θ, set
Eu0{x∈E|there exists a real numberλ >0, such that−λu0≤x≤λu0},
xu0inf{λ >0| −λu0≤x≤λu0}, ∀x∈Eu0.
2.11
It is easy to see thatEu0becomes a normed linear space under the norm · u0.xu0is called theu0- norm of the elementx∈Eu0see3,4.
Theorem 2.4. Let operatorA:Pw0→Pw0be increasing and generalizedu0-concave. Suppose thatAhas a fixed point x∗ inPw0\ {w0}, then, constructing successively the sequencexn1
Axn n0,1,2, . . .for any initialx0∈Pw0\ {w0}, we havexn−x∗u0 → 0n → ∞.
Proof. Suppose that{xn}is generated fromxn1 Axn n0,1,2, . . .. Take 0< ε0 <1, such
thatε0x∗1−ε0w0≤x1≤ε0−1x∗1−ε−01w0. Lety0ε0x∗1−ε0w0, z0 ε−01x∗1−ε−01w0,
and constructing successively the sequencesyn1Ayn,zn1 Azn n0,1,2, . . .. SinceA
is a generalizedu0-concave operator, we know that there existsη1η1x∗, ε0>0, such that
x∗Ax∗Aε0
ε−01x∗1−ε−01w0
1−ε0w0
≥1η1
ε0A
ε−01x∗1−ε−01w0
1−1η1
ε0
w0,
2.12
hence,Aε−01x∗ 1−ε0−1w0≤1η1−1ε0−1Ax∗ 1−1η1−1ε−01w0, then
z1Az0 A
ε0−1x∗1−ε−01w0
≤1η1 −1
ε−01Ax∗1−1η1 −1
ε0−1w0
1η1 −1
ε−01Ax∗−w0 w0< ε0−1Ax∗−w0 w0ε−01Ax∗
1−ε−01w0
ε0−1x∗1−ε0−1w0z0.
2.13
By2.2, we can easily gety1> y0. So it is easy to show that
Let
tnsup
t >0|tx∗ 1−tw0≤yn, zn≤t−1x∗
1−t−1w0 n0,1,2, . . ., 2.15
then,
0≤t0≤t1 ≤ · · · ≤tn≤ · · · ≤1, 2.16
which implies that the limit of{tn}exists. Let limn→ ∞tnt∗, then 0< tn≤t∗≤1.
Next we will show that t∗ 1. Suppose that 0 < t∗ < 1. SinceA is a generalized u0-concave operator, then there existsη2η2x∗, t∗>0, such that
At∗x∗ 1−t∗w0≥
1η2
t∗Ax∗1−1η2
t∗w0
1η2
t∗x∗1−1η2
t∗w0. 2.17
Moreover,
x∗Ax∗At∗t∗−1x∗1−t∗−1w0
1−t∗w0
≥1η2
t∗At∗−1x∗1−t∗−1w0
1−1η2
t∗w0.
2.18
Therefore,
At∗−1x∗1−t∗−1w0
≤1η2 −1
t∗−1x∗1−1η2 −1
t∗−1w0. 2.19
By2.17and2.19, for any 0< t≤t∗, there existsη3η3x∗, t>0, such that
Atx∗ 1−tw0≥
1η3
tx∗1−1η3
tw0,
At−1x∗1−t−1w0
≤1η3 −1
t−1x∗1−1η3 −1
t−1w0.
2.20
Particularly, for any 0< tn≤t∗n0,1,2, . . ., we have
Atnx∗ 1−tnw0≥
1ηtnx∗
1−1ηtn
w0,
At−n1x∗
1−t−n1
w0
≤1η−1t−n1x∗
1−1η−1t−n1
w0,
2.21
whereηηtn, x∗>0.
Hence,
yn1Ayn≥Atnx∗ 1−tnw0≥
1ηtnx∗
1−1ηtn
w0,
zn1Azn≤A
t−1
n x∗
1−t−1
n
w0
≤1η−1t−1
n x∗
1−1η−1t−1
n
w0.
By2.15, and2.22, we gettn1 ≥1ηtn n0,1,2, . . .therefore,tn1 ≥1ηn1t0 n
0,1,2, . . ., in contradiction to 0< tn≤1n1,2, . . .. Hence,
t∗1. 2.23
SinceAis a generalizedu0-concave operator, thus there exist real numbersα αx∗ > 0,
β βx∗ >0, such thatαu0w0 ≤x∗ ≤βu0w0, andtnx∗ 1−tnw0 ≤ yn ≤xn1 ≤zn ≤
t−1
n x∗ 1−t−n1w0 n0,1,2, . . ., we have
tn−1x∗ 1−tnw0≤xn1−x∗≤
t−n1−1
x∗1−t−n1
w0. 2.24
Moreover
tn−1x∗ 1−tnw0 ≥tn−1
βu0w0
1−tnw0 tn−1βu0,
t−n1−1
x∗1−t−n1
w0≤
t−n1−1
βu0w0
1−t−n1
w0
t−n1−1
βu0.
2.25
Hence,
1−t−n1
βu0 ≤tn−1βu0≤xn1−x∗≤
t−n1−1
βu0 n0,1,2, . . .. 2.26
Consequently, by2.23, we getxn−x∗u0 → 0n → ∞.
To prove the followingTheorem 2.5, we will use the definition of the normal cone as follows.
LetPbe a cone inE. Suppose that there exist constantsN >0, such that
θ≤x≤y⇒ x ≤Ny, 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. v Let P be a normal cone ofE. If operatorA : Pw0 −→ Pw0is increasing and generalizedu0-concave, andηηt, xis irrelevant toxin2.2, thenAhas the only one fixed point
x∗ ∈Pw0\ {w0}. Moreover, constructing successively the sequencexn1 Axn n0,1,2, . . .
for any initialx0 > w0, we havexn−x∗ → 0n → ∞.
Proof. SinceAis a generalizedu0-concave operator, hence there exist real numbersβ > α >0,
such thatαu0w0 ≤ Au0w0≤βu0w0. Taket0 ∈0,1small enough, thent0u0w0 ≤
Au0w0≤1/t0u0w0.
Therefore,tn1 ≥ tn, that is,{tn}is an increasing sequence and 0 < tn ≤1, hence, the
Letγt 1ηtt, whereηtwhich is irrelevant toxisηt, xin2.2, andAis increasing, sot < γt ≤ 1, Atx 1−tw0 ≥ γtAx 1−γtw0,for allt ∈ 0,1. By
γt0/t0>1, we can choose a natural numberk >0 big enough, such that
γt0
t0 k
> 1 t0.
2.28
Let
y0tk0u0w0, z0
1 tk
0
u0w0; ynAyn−1, zn Azn−1 n1,2, . . .. 2.29
Obviously,y0, z0∈Pw0, y0< z0. SinceAis increasing, we have
y1 Ay0A
tk0u0w0
At0
tk0−1u0w0
1−t0w0
≥γt0A
tk0−1u0w0
1−γt0
w0
γt0A
t0
tk0−2u0w0
1−t0w0
1−γt0
w0
≥γt0
γt0A
tk0−2u0w0
1−γt0
w0
1−γt0
w0
γ2t0A
tk0−2u0w0
1−γ2t0
w0 ≥ · · · ≥γkt0Au0w0
1−γkt0
w0
> tk0−1t0u0w0
1−tk0−1w0tk0u0w0y0.
2.30
SinceAxA{t0t−01x 1−t−01w0 1−t0w0} ≥γt0At−01x 1−t0−1w0 1−γt0w0,
we getAt−1
0 x 1−t−01w0≤1/γt0Ax 1−1/γt0w0. Hence
z1A
1
tk0u0w0
A 1 t0 1
tk0−1u0w0
1− 1 t0
w0
≤ 1
γt0A
1
tk0−1u0w0
1− 1 γt0
w0
≤ · · · ≤ 1
γkt0Au0w0
1− 1 γkt0
w0≤
1 t0γkt0
u0w0<
1 tk
0
u0w0z0,
2.31
theny0≤y1≤z1≤z0. It is easy to see
Let
tnsup
t >0|yn≥tzn 1−tw0
. 2.33
Obviously,yn≥tnzn 1−tnw0. Soyn1≥yn≥tnzn 1−tnw0≥tnzn1 1−tnw0.
Therefore,tn1 ≥ tn, that is,{tn}is an increasing sequence and 0 < tn ≤1, hence, the
limit of{tn}exists. Set limn→ ∞tnt∗, then 0< t∗≤1.
Next we will show thatt∗1. Suppose that 0< t∗<1, we have the following.
iIf for any natural number n,tn< t∗<1, then
yn1Ayn≥Atnzn 1−tnw0 A
tn
t∗t
∗z
n 1−t∗w0
1−tn t∗ w0 ≥γ tn t∗
At∗zn 1−t∗w0
1−γ
tn t∗ w0 ≥γ tn t∗
γt∗Azn
1−γt∗w0
1−γ
tn t∗ w0 γ tn t∗
γt∗Azn
1−γ
tn
t∗
γt∗
w0γ
tn
t∗
γt∗zn1
1−γ
tn
t∗
γt∗
w0, 2.34
hence,
tn1≥γ
tn
t∗
γt∗
1η
tn t∗ tn t∗
1ηt∗t∗≥tn
1ηt∗. 2.35
Taking limits, we havet∗≥t∗1ηt∗> t∗, a contradiction.
iiSuppose that there exists a natural numberN >0, such thattnt∗n > N.
Whenn > N, so we have
yn1Ayn≥Atnzn 1−tnw0 At∗zn 1−t∗w0
≥γt∗Azn
1−γt∗w0γt∗zn1
1−γt∗w0,
2.36
thent∗tn1≥γt∗ 1ηt∗t∗> t∗, a contradiction.
Therefore,t∗1.
For any natural numbersn, p, we have
θ≤ynp−yn≤znp−yn≤zn−yn≤zn−tnzn 1−tnw0 1−tnzn−w0. 2.37
Similarly,θ≤zn−znp≤zn−yn≤1−tnzn−w0. By the normality ofPand limn→ ∞tn1,
we get
ynp−w0
−yn−w0ynp−yn≤N1−tnzn−w0 → 0 n → ∞, znp−w0
−zn−w0zn−znp≤N1−tnzn−w0 → 0 n → ∞,
whereNis the normal constant ofP. Hence the limits of{yn}and{zn}exist. Let limn→ ∞yn
y∗, and let limn→ ∞znz∗, thenyn≤y∗≤z∗≤zn n0,1,2, . . ., hence,
θ≤z∗−y∗≤zn−yn≤1−tnzn−w0 → θ n → ∞. 2.39
That is,y∗z∗. Letx∗y∗z∗, thenyn1Ayn≤Ax∗≤Aznzn1.
Taking limits, we getx∗≤Ax∗≤x∗. HenceAx∗x∗, that is,x∗∈Pw0\{w0}is a fixed
point ofA. ByTheorem 2.4, the conclusions ofTheorem 2.5hold. The proof is completed.
3. Examples
Example 3.1. Let I 0,1, letCI {xt : I → R | xt is continuous}, letx sup{|xt||t ∈ I}, letP {x ∈ CI | xt ≥ 0,∀t ∈ I}, thenCI is a real Banach space and P is a normal and solid cone in CI P is called solid if it contains interior points, i.e.,P /◦∅. Takea < 0, letw0 w0t ≡ a,Pw0 {x ∈ CI | xt ≥ w0,∀t ∈ I}, and
◦
Pw0 {xw0∈Pw0|x∈
◦
P}.
Considering the Hammerstein integral equation
xt
1
0
kt, sfs, xsds, t∈0,1, 3.1
wherekt, s : I×I → 0,∞is continuous,fs, u : I×a,∞ → Ris increasing foru. Suppose that
1there exist real numbers 0≤m≤M≤1, such thatm≤kt, s≤M, for allt, s∈ I×I, andfs, u≥a/M, for alls, u∈I×a,∞,
2for anyλ∈0,1andu∈a,∞, there existsηηλ>0, such that
mfs, λu 1−λa≥1ηλmfs, u 1−1ηλa. 3.2
Then3.1has the only one solutionx∗∈Pw0\ {w0}. Moreover, constructing successively
the sequence:
xnt
1
0
kt, sfs, xn−1sds, ∀t∈I, n1,2, . . . 3.3
for any initialx0∈Pw0\ {w0}, we havexn−x∗ → 0 n → ∞.
Proof. Considering the operator
Axt
1
0
Obviously,A:Pw0\{w0} →
◦
Pw0is increasing. Therefore,iofDefinition 2.1is satisfied.
For anyx∈P◦w0, by3.2, we have
Aλxt 1−λw0 1
0
kt, sfs, λxs 1−λw0ds
1
0
1
mkt, smfs, λxs 1−λw0ds
≥1ηλ
1
0
1
mkt, smfs, xsds
1−1ηλw0 1
0
1
mkt, sds
≥1ηλAxt 1−1ηλw0.
3.5
Therefore, ii of Definition 2.1 is satisfied. Hence the operator A : Pw0 → Pw0 is
generalizedu0-concave. Consequently, operatorAsatisfies all conditions ofTheorem 2.5, thus
the conclusion ofExample 3.1holds.
Example 3.2. LetRbe a real numbers set, and letP {x|x≥0,x∈R}, thenRis a real Banach space andPis a normal and solid cone inR. LetAx x21/2−2. Considering the equation: xAx. Obviously,Ais a generalizedu0-concave operator and satisfies all the conditions of Theorem 2.5. HenceAhas the only one fixed pointx∗∈P−2\ {−2} −2,∞. Moreover, we knowx∗−1 by computing.
InExample 3.2, we know that operatorA : −2,∞ → −2,∞doesn’t satisfy the definition of u0-concave operators. Therefore, we can’t obtain the fixed point of A by the
fixed point theorem ofu0-concave operators. Theu0-concave operators’ fixed points are all
positive, but hereA’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.
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