Applied Mathematics, 2011, 2, 1369-1371
doi:10.4236/am.2011.211192 Published Online November 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
The Equivalence between the Mann and Ishikawa
Iterations for Generalized Contraction Mappings in a Cone
L. Jones Tarcius Doss, T. Esakkiappan
Department of Mat hematics, Anna University, Chennai, India
E-mail: tesakki@gmail.com
Received July 22, 201 1; revised Septe mber 13, 2011; accepted September 20, 2011
Abstract
In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping
in cone subset of a real Banach space is discussed.
Keywords: Mann Iteration, Ishikawa Iterations, Generalized Contraction, Cone
1. Introduction
Generally, the iteration techniques of W.R.Mann [1] and
Shiro Ishikwa [2] are used to find the approximation of
fixed point of a contraction mapping. These iterations are
quite useful even for the cases of where Picard iteration
fails. In this paper, we see the equivalence between these
Mann and Ishikawa iterations for a generalized contra-
ction mapping in a cone. First, we recall the definition of
a cone (refer Huang Long-guang and Zhang Xian [3])
and some of its properties.
Definition 1.1: Let be a real Banach space and a
subset of is said to be a cone if satisfies the
following:
E
P E
1) , P is closed and ;
P
ax by {0}P
,2) for every
P
x
yP and ; ,0ab
3) .
()=PP{0}
The partial ordering with respect to the cone P is
defines by
x
y if and only if 
y
xP. We shall
write <
x
y to indicate that
x
y
but
x
y. Further
x
y will stand for 
y
xintP, where in de-
notes interior of . We now define the generalized
contraction mapping. Let be a real Banach space,
a nonempty convex cone subset of E. Let a self
map of with the property that
tP
PE
P T
P
(, )TxTyMx y

(1.1)
where
and (, )
M
xy satisfy the following:
1) :[0,)[0,)isareal-valued,nondecreasing,
rightcontinuous function;
  (1.2)
2) ()<foreach>0;
ttt (1.3)
3) isnondecreasingon(0,);
(1.4)
4) ():=(())isnonincreasingon(0,);
gtt tt (1.5)

5) (,):=
max, ,,,
Mxy
x
yxTxyTyxTyyTx
(1.6)
T satisfying above conditions is said to be a Generalized
contraction. Below, we see the definition of the two
iteration schemes due to Mann [1] and Ishikawa [2]. Further,
these two iterations are applied to a class of generalized
contraction mapping which is mentioned just above.
Let 00
x
uP
. The Mann iteration is defined by
1=(1)
nnnn
uu
n
Tu
. (1.7)
The Ishikawa iteration is defined by
1=(1),
=(1),
nnnn
nnnn
n
n
x
xTy
yxT



x
(1.8)
where
(0,1), 0,1
nn

. Clearly, the sequences
n
x,
n
u and
n
y
are in because 00
P=
x
uP
and
(0,1)
n
and
[0
n
,1) and from the defi-
nition of cone.
Let
n
w
li
be a sequence in P which is a subset of a
real Banach space. We say that converges to

n
ww
and write if
m =
n
nww
 lim= 0
n
nww where .

is the norm associated with .
E
The main aim of this paper is to show that the con-
vergence of Mann iteration is equivalent to the con-
vergence of Ishikawa iteration in the cone .
P
Below, we sate two results without proof which are
very much useful for our analysis. for proof, one may
refer [4] and [5] respectively.
Lemma 1 [4]
Let
n
a be a nonnegative sequence which satisfies
L. J. T. DOSS ET AL.
1370
n
the following inequality:
1(1 ),
nnn
aa
 (1.9)
where (0,1)
n
=( )
n
for all , and
01
,=
n
n
nn

n
o
. Then . lim n
na
 = 0
Lemma 2 [5] Let be a nonempty closed convex
subset of a Banach space , and T a self-map of
satisfying (1.1). Let
P
{
n
E
}P
satisfy the conditions >0
n
for all and . Then the sequences
0n

,,
nn
1n

,
n
=
n
n
 
,
n
x
yuTxTy and
are bounded.
n
Tu
Clearly, the sequences and

,
nn
ux
n
y
are in
because 00
P=
x
uP and
(0,1)
n and
and from the definition of cone. Here,
is a closed and convex subset of E which also follows
from the definition of cone. Therefore, the above lemma
can be verified for .

[0,1)
n
P
P
2. Main Result
In this section, we discuss the main result which gives
the equivalence of Mann and Ishikawa iterations in the
cone. The analysis is similar to the work of Rhoades and
Soltuz [6].
THEOREM 2.1
Let be a cone subset of a Banach space , and
a self-map of satisfying (1.1)-(1.6). Let {}
P E
T Pn
satisfy the conditions 0
n
for all and 0n
1=
n
n
. Denote by
x
the unique fixed point of T.
Then for , the following are equivalent:
00
uxP
1) the Mann iteration (1.7) converges to
x
;
2) the Ishikawa iteration (1.8) converges to
x
.
Proof: By Lemma 2, both Mann and Ishikawa itera-
tions are bounded. we have to prove the equivalence
between (1.7) and (1. 8). We need to prove that
lim= 0
nn
nxu
 . (2.1)
Set

= maxsup:sup:
sup: sup:
nnj nj
nj nj
rxTyjnuTu
xTu jnuTy jn
 
 
jn
(2.2)
We then have the following




11 11
11 11
111 1
1
1(,
1();
njnnjnn j
nnnn j
nnn n




111 1
11 11
111 1
1
1(,
1();
njnnjnn
nnnn j
nnn n
u TuuTuTuTu
rMuu
rr



 
 
 
 
 
 
)
j




11 11
11 11
111 1
1
1(
1();
njnnjnn
nnnn j
nnn n
,)
j
x
Tux TuTy Tu
rMyu
rr



 
 
 
 
 
 




111 1
11 11
111 1
1
1(,
1().
njnnjnn j
nnnn j
nnn n
u TyuTyTuTy
rMuy
rr



 
 
 
 
 
 
)
From the definition of and all above inequalities
imply that, n
r

111 1
1111
1(
0()
nnnnn
nnnn n
rrr
rrrr


 

)
.


(2.3)
Therefore,
n
r is monotone non-increasing in
and positive, i.e., bounded below. Hence, there exists n
lim
 n
nr, denoted by . We wish to show that .
0r=0r
Suppose not that, . From (2.3), we get the
following, >0r
11 1
1
()
nn nn
n
rrr
gr



1
11
1
() ()
n
nnnn
n
gr gr
rr rr
rr
 

1
n
.
In general, we have that

1
() .
kkk
gr rr
r

Therefore, on summing we obtain,

10
=0 =0
() ()
=.
nn
kkk
kk
gr gr
rr rr
rr

 
 1
n
The right-hand side is bounded and the left-hand side
is unbounded, which leads to a contradiction. Thus
=.ro
Therefore, we have
=0 =0
lim lim
nn nn
nn
xTu uTy
 
 (2.4)
=0 =0
lim lim
nn nn
nn
xTy uTu
 
 (2.5)
)
x
Tyx TyTy Ty
rMyy
rr



 
 
 
 
 
 
We now show that both the iteration schemes are
equivalent. Suppose the Mann iteration converges,then
we have


11
1
1.
nn nnnnnn
nnn n nnnn
xux uTyTu
xuTyxxTu



 
   
Using (2.4), (2.5) , Lemma 1 and above eq uations with
Copyright © 2011 SciRes. AM
L. J. T. DOSS ET AL.
Copyright © 2011 SciRes. AM
1371
the following
:
:
:, ,
()


  
nnnnnnnnn
nn
x uTyxxTu
o
we have lim= 0
n
n
 , that is (2.1) holds.
Then, the relation
0.
nnnn
xxxuxu


This implies that Ishikawa iteration also converges.
Suppose the Ishikawa iteration converges, then we have
11
(1 )
(1 ).
nn nnnnnn
nn nnn nnn
x
uxuTy
x uTy uu Tu



 
 
Tu
Using (2.4), (2.5) , Lemma 1 and above eq uations with
the following
:=, :=,
=( ),
nnnnnnnnn
nn
xuTyu uTu
o



we have lim= 0
n
n
 , that is (2.1) holds.
Then, the relation
0.
nnnn
uxxu xx


This implies that Mann iteration converges. Hence the
theorem.
3. Acknowledgements
The second author Mr. T. Esakkiappan would like to
thank the referee for his valid suggestions. Further, the
same author would like to thank his research supervisor
Prof.P.Vijayaraju for his valuable guidance and support.
4. References
[1] W. R. Mann, “Mean Value Methods in Iteration,” Pro-
ceedings of the American Mathematical Society, Vol. 4,
No. 3, 1953, pp. 506-510.
doi:10.1090/S0002-9939-1953-0054846-3
[2] S. Ishikawa, “Fixed Points by a New Iteration Method,”
Proceedings of the American Mathematical Society, Vol.
44, No. 1, 1974, pp. 147-150.
doi:10.1090/S0002-9939-1974-0336469-5
[3] L.-G. Huang and X. Zhang, “Cone Metric Spaces and
Fixed Point Theorems of Contractive Mappings,” Journal
of Mathematical Analysis and Applications, Vol. 332, No.
2, 2007, pp.1468-1476. doi:10.1016/j.jmaa.2005.03.087
[4] X. Weng, “Fixed Point Iteration for Local Strictly Pseudo-
Contractive Mapping,” Proceedings of the American Ma-
thematical Society, Vol. 113, No. 3, 1991, pp. 727-731.
doi:10.1090/S0002-9939-1991-1086345-8
[5] B. E. Rhoades, “Convergence of an Isikawa-Type Itera-
tion Scheme for a Generalized Contraction,” Journal of
Mathematical Analysis and Applications, Vol. 185, No. 2,
1994, pp. 350-355. doi:10.1006/jmaa.1994.1253
[6] B. E. Rhoades and S. M. Soltuz, “The Eqivalence be-
tween Mann and Ishikawa Iterations Dealing with Gener-
alized Contractions,” International Journal of Mathemat-
ics and Mathematical Sciences, Vol. 2006, 2006, pp. 1-5.