Journal of Applied Mathematics and Physics, 2014, 2, 21-25
Published Online January 2014 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2014.21004
OPEN ACCESS JAMP
Approximative Method of Fixed Point for
Φ-Pseudocontractive Operator and an Application
to Equation with Accretive Operator
Yixin Wen, Aifang Feng, Yuguang Xu
Department of Mathematics, Kunming University, Kunming, China
Received October 2013
ABSTRACT
In this paper, Φ-pseudo-contractive operators and Φ-accretive operators, more general than the strongly pseu-
do-contractive operators and strongly accretive operators, are introduced. By setting up a new inequality, au-
thors proved that if
→:TX X
is a uniformly continuous Φ-pseudo-contractive operator then T has unique
fixed point q and the Mann iterative sequence with random errors approximates to q. As an application, the
iterative solution of nonlinear equation with Φ-accretive operator is obtained. The results presented in this paper
improve and generalize some corresponding results in recent literature.
KEYWORDS
Duality Mapping; Φ-Pseudo-Contractive Operator; Φ-Accretive Operator; Mann Iterative Sequence with
Random Error Terms
1. Introduction and Preliminaries
In 1994, Chidume [1] solved a problem dealt with the fixed point for the class of Lipschitz strictly (strongly)
pseudo-contractive operators in uniformly smooth Banach space
X
. That is, he proved that the Ishikawa itera-
tive sequence converges strongly to the unique fixed point of
T
in
K
where
KX⊂
and
T:K K→
is
Lipschitz strictly (strongly) pseudo-contractive. Chang [2], in 1998, improved the result, i.e., he proved that the
conclusion of Chidume holds if T is uniformly continuous and the fixed point set of T is nonempty (i.e.,
( )
FT ≠∅
). Recently, Liu [3] proved, if the strongly pseudocontractive operators are replaced by the more
general
φ
-strongly pseudo-contractive operators then the conclusion of Chidume still holds.
The objective of this paper is to introduce Φ-pseudo-contractive operators—a class of operators which are
more general than the
φ
-strongly pseudo-contractive operators and to study the problems of existence, unique-
ness and the iterative approximate method of fixed point by setting up a new inequality in arbitrary Banach
space. As an application, the iterative solution of nonlinear equation with Φ-accretive operator is obtained. The
results presented in this paper improve and generalize the conclusions of Chidame, Chang and Liu.
To set the framework, we recall some basic notations as follows.
Throughout this paper, we assume that
X
is a real Banach space with dual
X
∗
,
( )
,⋅⋅
denotes the genera-
lized duality pairing. The mapping
:2
x
JX
∗
→
defined by
( )
{ }
= :,,JxjXx jxjjxxX
∗
∈== ∀∈
(1)
is called the normalized duality mapping [4].
Now, we introduce Φ-pseudo-contractive operators as follows.
Definition 1. Let
K
be nonempty subset of X. An operator
:TK K→
is said to be Φ-pseudo-contractive,
if there exists a strictly increasing function
[
)
[
)
: 0,0, Φ ∞→ ∞
with
( )
00 Φ =
and
( )( )
jxy Jxy−∈ −
such that

Y. X. WEN ET AL.
OPEN ACCESS JAMP
22
( )
( ) ( )
2
,,TxTyjxyxyxy xyK Φ −−≤−−−∀∈
. (2)
An operator
:AKK→
is said to be Φ-accretive, if
( )
( ) ( )
,,Ax Ayjx yx yxyK Φ −− ≥−∀∈
. (3)
It is easy to verify that the operator
T
is Φ-pseudo-accretive if and only if
IT−
is Φ-accretive where
I
is an identity mapping on
X
. Hence, the mapping theory for accretive operators is intimately connected with
the fixed point theory for pseudo-contractive operators.
We like to point out: every strongly pseudo-contractive operator is
φ
-strongly pseudo-contractive with
[ ) [ )
: 0,0, φ ∞→ ∞
defined by
( )
s ks φ =
where
( )
0, 1k∈
, and every
φ
-strongly pseudo-contractive operator
must be the Φ-pseudo-contractive operator with
[ ) [ )
: 0,0, Φ ∞→ ∞
defined by
( )( )
s ss Φφ =
.
Obviously, if a Φ-pseudo-contractive operator has a fixed point then it is unique.
Definition 2. Let
:TK K→
be an operator. For any given
0
xK∈
the sequence
{ }
n
x
defined by
( )
( )
1
10
nnnnnnn
xxTxu n ααγ
+
=−++≥
(4)
is called Mann iteration sequence with random errors. Here
{ }
n
u
is a bounded sequence in
X
and is said
random error terms of iterative process, and the parameters
{ }
n
α
and
{ }
n
γ
both are sequences in
[ ]
0, 1
. By
the way, Xu introduced another definition of Mann iterative sequence with random errors in [5].
In particular, the parameters
0
n
γ =
for all
0n≥
in Equation (4) then
{ }
n
x
is called Mann [6] iteration
sequence.
2. Main Results
First, we have an existence theorem of fixed point as follow.
Theorem 1. If
:TX X→
is a continuous Φ-pseudo-contractive operator with bounded range then
T
has
an unique fixed point in
X
.
Proof. Define
:
n
TX X→
by
1
nn
Txxc TxxXand n= −∀∈≥
where
( )
0, 1
n
c∈
and
1
n
n
lim c
→∞
=
. Note that
T
is Φ-pseudo-contractive, thus
( ) ( ) ( ) ( )
( ) ( )
( )
2
2
,,
1
nn n
nn
n
TxTyjxyxy cTxTyjxy
c xycxy
c xy
Φ
Φ
−−=−− −−
≥−− +−
≥−
(5)
for all
, ,1xy Xn≥≥
and some
() ()
jx yJxy−∈ +
.
Clearly,
n
T
is a continuous strongly accretive operator. It follows from the Theorem 13.1 of Deimling [7]
that there exists an
n
xX∈
such that
0
nn
Tx =
for any
1n≥
. Next, the sequence
{ }
1
nn
x
−
is bounded. In fact,
if
Tx M≤
then
n
xM≤
for all
xX∈
and
( )
limlim 10
nn nn
nn
xTxc Tx
→∞ →∞
−= −=
(6)
Since
T
is Φ-pseudo-contractive, that is,
( )
IT
−
is Φ-accretive, so
( )
( )( )
( )
( )
( )
( )
,
2
mkmk mk
mk mkmk
mm kk
xxITxI Txjxx
x Txx Txxx
M xTxxTx
Φ −≤ −−−−
≤− +−−
≤− +−
(7)
for any
{ }
mn n
xx x−∈
. Equations of (6) and (7) ensure that
{ }
1
nn
x
∞
=
is a Cauchy sequence.
Consequently,
{ }
1
nn
x
∞
=
converges to some
qX∈
. By the continuity of
T
we have
1
nn
nn
n
Tqlim Txlimxq
c
→∞ →∞
= ==
.
Suppose that there exists a
qX
∗
∈
such that
Tq q
∗∗
∈
then

Y. X. WEN ET AL.
OPEN ACCESS JAMP
23
( ) ( )
( )
2, =0qq qq TqTqjqq
Φ
∗ ∗∗∗
−≤−−−−
which means that
qq
∗
=
. i.e.,
q
is unique fixed point of
T
. The proof is completed.
The following two Lemmas will play crucial roles in the proof of Theorem 2.
Lemma 1. [2] If
X
be a real Banach space then there exists
( )( )
jxyJxy+∈ +
such that
( )
22
2, ,xyxyjxy xyX+ ≤++∀∈
. (8)
Second, to set up a new inequality as follows.
Lemma 2. Let
[ ) [ )
: 0,0, Φ ∞→ ∞
be a strictly increasing function with
( )
00 Φ =
and let
{ }
n
b
and
{ }
n
c
be two nonnegative real sequences satisfying
,and lin
nn n
n
cb b
→∞
<∞ =∞
∑∑
. (9)
Suppose
{ }
n
a
is a nonnegative real sequences. If there exists a integer
0
0N>
satisfying
( )()
22
1 10nnnnnn
aaobcban N Φ
++
≤++ −∀≥
(10)
then
lim 0
n
n
a
→∞
=
.
Proof. Let
{ }
inf 2
n
a σ
=
. If
0 σ >
, then
( )
( )
1n
a Φ Φσ
+
>
for all
0n≥
. From the conditions of Equation
(9) there exists an integer
0N>
0 such that
( )
( )
2nn obbnN Φσ ≤ ∀≥
. (11)
So, we have
( )
22
1
1Φ
2
n nnn
aacb nN σ
+
≤+ −∀≥
.
By induction, we obtain
( )
2
==
1Φ
2
jN j
jN jN
ba c σ
+∞ +∞
≤+≤ +∞
∑∑
. (12)
Equation (12) is in contradiction with
=0 n N
b
+∞
= +∞
∑
. It implies that
0
σ =
. Therefore, there exists a subse-
quence
{ }
{ }
j
nn
aa
⊂
such that
lin 0
j
n
j
a
→∞
=
. So, for any given
0 ε >
there exists an integer
0
0j≥
such that
( )
( )
0
0
0
j
n
nnn j
ajj and
obcbn n
ε
ε
<∀≥
+≤Φ∀ ≥>
. (13)
If
0
J
is fixed, we will prove that
0j
nk
a ε
+
<
for all integers
1k≥
. The proof is by induction. For
1k=
,
suppose
0j
nk
a ε
+
<
. It follows from Equations (10) and (13) that
( )
( )
000 000
2222 2
1
Φ
jjj jjj
nnn nnn
aa obcba ε εε
+
≤≤++−≤<
.
It is a contradiction. Hence,
01
j
n
a ε
+≤
holds for
1k=
. Assume now that
0
j
np
a ε
+≤
for some integer
1p>
.
We prove that
0
+1
j
np
a ε
+
≤
. Again, assuming the contrary,
Using Equations (10) and (13), as above, it leads to a contradiction as follows
( )
( )
+1
0
00 00
0
22
2
22
np
j
jj jj
np
j
nnpnpnp
a
a obcb
a
ε
ε
ε
+
+
++ +
≤
≤ ++−Φ
≤<
Where
oo
j jo
n pnj+≥ >
. Therefore,
0
j
np
a ε
+<
holds for all integers
1k≥
, i.e.,
0
lin lin0
j
n nk
nn
aa
+
→∞ →∞
= =
The Proof is completed.

Y. X. WEN ET AL.
OPEN ACCESS JAMP
24
Theorem 2. Let
:TX X→
be an → a uniformly continuous Φ-pseudo-contractive operator with bounded
range. Suppose that the iterative sequence
{ }
n
x
is defined by Equation (4) satisfying
,lin0 and
nn n
n
. γα α
→∞
<∞ ==∞
∑∑
(14)
then
{ }
n
x
converges strongly to unique fixed point of
T
.
Proof. From Theorem 1, we know that there exists a unique fixed point
q
of
T
.
Putting
{ } { }
00
: :0
n
Msup TxxXsupunx=∈ +≥+
for any given
0
xX∈
,
then
( )
100 000
1-xxTx M αα ≤ +≤
.
Using induction, we have
00
xM≤
for all
0n≥
. Let
0
MM q= +
. Since
( )
1
20
n nnnnn
xxaxTxMasn α
+
−=− ≤→→∞
, therefore,
( )
1
:0
nnn
eTxTxasn
+
− →→∞
by the uniformly continuity of
T
.
From Equation (14) there exists an integer
0
0N>
such that
0
1
06
n
nN α ≤≤ ∀≥
. (15)
By Equations (4), (8) and (15) we have
()() ()
( )( ) ( )
( ) ( )
( )
( ) ( )
( )
( )
( )
( )
2
2
1
22
1
1
22
11
2
1
222
21
21
222
1
1+2 ,
2,
1
2,
2, 2
1 2+2
2 22
23 2
nn nnnnn
n nnnn
nn n
nn
nn nn
nn nn
n nnnn
nnnn n
nn nn
x qxqaTxqu
xqu jxq
Txq j xq
xq
TxTxjxq
Txq j xqM
xqxq
Me Mxq
x qMMe
αγ
αγ
α
α
α
αγ
αα α
α γα
αα
+
+
+
++
+
+
+
−= −−+−+
≤− −−
+− −
≤− −
+− −
+ −−+
≤− +−−
++−Φ −
≤−+ ++
( )
( ) ( )
21
2
1
Φ
Φ
nnn
nnnnn
M xq
xq ocxq
γα
αα
+
+
−−
= −++−−
(16)
for all
nN≥
where
( )
22
23
nn nn
oMMe α αα = +
and
2
2
nn
cM γ =
. It follows from the Lemma 2 that
lim
n
xq
→∞
−
. The Proof is completed.
Last, as an application, the existence, the uniqueness and the approximate method of solution of nonlinear
equation with Φ-accretive operator is obtained. That is
Theorem 3. Suppose that
:AX X→
is an → a uniformly continuous Φ-accretive operator and the range of
either
A
or
IA−
be bounded. For any given
fX∈
, the equation
Ax f=
has unique solution in
X
, and
if
{ }
n
x
is defined by
( )()()
1
10
n nn
xxfIAxu n αα γ
+
=− ++−+≥ 

satisfying the conditions of equation (14) then it converges strongly to the solution.
Proof. We define
:SX X→
by
Sxfx Ax= +−
for all
xX∈
. Clearly,
S
is Φ-pseudo-contractive and
continuous if
A
is Φ-accretive and continuous, and the range of
S
is bounded if
( )
IA−
is. It is easy to see
that
x
∗
is a solution of the equation
Ax f=
if and only if that
x
∗
is a fixed point of
S
. It follows from the
Theorem 2 that
Ax f=
has an unique solution
xX
∗∗
∈
and the iterative sequence
{ }
n
x
is defined by Equa-
tion (4) converges strongly to
x
∗
. i.e., the sequence
{ }
n
x
is an iterative solution of the equation
Ax f=
.
The proof is completed.

Y. X. WEN ET AL.
OPEN ACCESS JAMP
25
Remark. Theorem 2 improves a number of results (for example, Theorem 4.2 of [2], Theorem 2 of [1] and
Corollary 3.3 and 3.4 of [3]) in the following senses.
1) The existence and the convergence of the fixed point for Φ-pseudo-contractive operator are studied simul-
taneously.
2) The operators may not be strongly pseudo-contractive or
φ
-strongly pseudo-contractive.
3) The continuity of operator may not be Lipschitzian.
4) The errors come from the iterative process have been considered appropriately.
REFERENCES
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Strongly Pseudo-Contractive Mappings in Banach Spaces,” Journal of Mathematical Analysis and Applications, Vol. 224, 1998,
pp. 149-165. http://dx.doi.org/10.1006/jmaa.1998.5993
[3] Z. Q. Liu, M. Bounias and S. M. Kang, “Iterative Approximations of Solutions to Nonlinear Equations of
φ
-Strongly Accre-
tive in Banach Spaces,” Rocky Mountain Journal of Mathematics, Vol. 32, 2002, pp. 981-997.
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http://dx.doi.org/10.1090/S0002-9904-1967-11678-1
[5] Y. G. Xu, “Ishikawa and Mann Iterative Processes with Errors for Nonlinear Strongly Accretive Operator Equations,” Journal
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[7] K. Deimling, “Nonlinear Functional Analysis,” Springer-Verlag, Berlin, 1985. http://dx.doi.org/10.1007/978-3-662-00547-7