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
yP and ; ,0ab
3) .
()=PP{0}
The partial ordering with respect to the cone P is
defines by
y if and only if
xP. We shall
write <
y to indicate that
y
but
y. Further
y will stand for
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 (, )
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
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
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
xTy
yxT
x
(1.8)
where
(0,1), 0,1
nn
. Clearly, the sequences
n
x,
n
u and
n
are in because 00
P=
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