Some Fixed Point Theorems with G-iteration in Normed Linear
Spaces
Deepak Singh Kaushal
Abstract
In this paper a new type of one step iteration for self mappings is introduced under certain conditions in normed linear space and studied with a contractive conditions of Rafiq [4].
Key words: Common fixed point , Contractive condition, G-iteration, Mann iteration
Mathematics Subject Classification: 47H10, 54H25.
1. INTRODUCTION
Let
X
be a nonempty closed convex subset of a normed linear spaceE
andT X
:
→
X
be a self mapping and{ }
x
n be the sequence then for arbitraryx
0∈
X
Mann[3] iteration process is defined as(
)
1
1
0
n n n n n
x
+= −
λ
x
+
λ
Tx
for n
≥
Similarly Ishikawa[2] iteration process for
{ }
x
n is given by(
)
(
)
1 ' '1
and
1
, for
0
n n n n n n n n n nx
x
Ty
y
x
Tx
n
λ
λ
λ
λ
+= −
+
= −
+
≥
Where
x
0∈
X
is arbitrary and{ }
λ
n,
{ }
λ
n' are sequences of real numbers such that0
≤
λ
n≤
1 , 0
≤
λ
n'≤
1
. By using the concept of Mann iteration process Sahu[5] introduced a new G-iteration process:-Let
T
be a self mapping of Banach space then G-iteration process associated by T is defined in the following manner, Letx x
0,
1∈
X
and(
)
(
)
2 1 11
n n n n n n n nx
+=
µ
−
λ
x
++
λ
Tx
++ −
µ
Tx
forn
≥
0
Where{ }
µ
nand
{ }
λ
n satisfy(i)
λ
0=
µ
0=
1
(ii)
0
<
λ
n<
1,
n
>
0 and
µ
n≥
λ
nfor
n
≥
0
(iii)lim
n0
n→∞
λ
= >
h
(iv)
lim
n1
n→∞
µ
=
Das and Debata [1] generalized the Ishikawa iteration processes from the case of one self mapping to the case of two self mappings
S
and
T
of
X
given by(
)
(
)
1 ' '1
and
1
, for
0
n n n n n n n n n nx
x
Sy
y
x
Tx
n
λ
λ
λ
λ
+= −
+
= −
+
≥
By using above iteration Das and Debata[1] established the common fixed points of quasi-nonexpansive mappings in a uniformly convex Banach space.Several other researchers such as Takahashi and Tamura[6] investigated iteration in a strictly convex Banach space,for the case of two nonexpansive mappings under different assumptions and contractive conditions.
Our aim in this paper is to establish some fixed point theorems by using a more general contractive condition than those of Rafiq [4].We shall use a new type of one step iteration for self mappings and employ the following contractive definition:
Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
78
Where
0
≤ <
δ
1 and
e
xdenotes the exponential function ofx
∈
X
.After this we extend the above contractive definition for a pair of mapping in following manner:
Let
X
be a closed convex subset of normed linear spaceN
Suppose thatS
and
P
are two self mappings ofX
satisfying the following contractive condition(
2
)
,
,
0
... 1.2
( )
L x Sx
Sx
−
Py
≤
e
−δ
x
−
Sx
+
δ
x
−
Py
∀
x y
∈
X L
≥
Where
0
≤ <
δ
1 and
e
x denotes the exponential function ofx
∈
X
.2. MAIN RESULTS
Theorem 2.1: Let
X
be a closed convex subset of normed linear spaceN
and letT
be a self mapping ofX
satisfying the contractive condition of (1.1) and{ }
x
n be the sequence of G-iterates associated withT
then G-iteration process is defined in the following manner:0 1 2 1 1
Let
,
and
(
)
(1
)
(
)
for
0
n n n n n n n n n n n n nx x
X
x
+µ
λ
x
+λ
Tx
+µ
λ
k Tx
λ
k x
n
∈
=
−
+
+ −
−
+
+
−
≥
{ }
{ }
where{
µ
n},
λ
nand
k
nsatisfying
( )
1
if
0
( ) 0
1 , 0
1
for
0.
( )
,
for
0.
( ) lim
lim
where
0.
( ) lim
1.
If{ }converges in
then it converges to a fixed point of .
n n n n n n n n n n n n n n n n
i
k
n
ii
k
n
iii
k
n
iv
h
k
h
v
x
X
T
µ
λ
λ
µ
λ µ
λ
µ
→∞ →∞ →∞=
=
=
=
<
<
<
<
>
≥
≥
≥
= =
>
=
Proof:- If { }
x
n converges onz
∈
X
thenlim
nn→∞
x
=
z
.Now we shall show that
z
is the fixed point ofT
. Consider(
)
(
)
(
)
2 2 2 1 11
n n n n n n n n n n n n n n nz Tz
z
x
x
Tz
z
x
µ
λ
x
λ
Tx
µ
λ
k Tx
λ
k
x
Tz
+ + + + +−
≤
−
+
−
≤
−
+
−
+
+ −
−
+
+
−
−
(
)
(
)
(
)
2 1 11
n n n n n n n n n n n n nz
x
x
Tz
Tx
Tz
k
Tx
Tz
k
x
Tz
µ
λ
λ
µ
λ
λ
+ + +≤ −
+
−
−
+
−
+ −
−
+
−
+
−
−
(
)
(
)
{
}
(
)
(
)
1 1 2 1 1 1 12
1
n n n n n n L x Tx n n n n n n n n n n nz
x
x
Tz
e
x
Tx
x
z
k
Tx
Tz
k
x
Tz
µ
λ
λ
δ
δ
µ
λ
λ
+ + + + − + + +≤
−
+
−
−
+
−
+
−
+ −
−
+
−
+
−
−
…(2.1.1) We observe by the definition of G-iteration that(
)
1 1 1 2 1(1
)
1
n n n n n n n n n n n n n nk
x
Tx
x
x
µ
x
Tx
λ
x
Tx
λ
λ
λ
+ + + + +−
−
−
≤
−
+
−
+
−
(
)
( )(
)
(
)
(
)
1 2 1 2 1 (1 ) 1 1 2 1 1(1
)
1
2
1
n n n n n n n n n n n n n n n n k L x x x Tx x Tx n n n n n n n n n n n n n n n n n n n n nz Tz
z
x
x
Tz
e
k
x
x
x
Tx
x
Tx
x
z
k
Tx
Tz
k
x
Tz
λ µ λ λ λµ
λ
λ
λ
µ
δ
δ
λ
λ
λ
µ
λ
λ
+ + + + + − − − + − + − + + + +−
≤
−
+
−
−
+
−
−
−
+
−
+
−
+
−
+ −
−
+
−
+
−
−
Letting
n
→ ∞
then we have(1
)
z Tz
−
≤ −
h z Tz
−
which is a contradiction.
Hence
z
=
Tz
is a fixed point ofT
.Remark: When
{ } { }
µ
n=
1
and{ } { }
λ
n=
k
n then above G-iterative process reduces to Mann iteration.Theorem 2.2: Let
X
be a closed convex subset of normed linear spaceN
and letS
andP
be a pair of self mappings ofX
satisfying the contractive condition of(1.2) and{ }
x
n be the sequence of G-iterates associated withS
andP
then G-iteration process is defined in the following manner:{ }
{ }
0 1 2 2 2 1 2 1 2 2 2 3 2 2 2 2 2 1 2 1Let
,
and
(
)
(1
) P
(
)
and
(
)
(1
) S
(
)
, for
0
where{
},
and
satisfying
( )
1
if
0.
( ) 0
1 , 0
n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n n nx x
X
x
x
Sx
k
x
k x
x
x
Px
k
x
k x
n
k
i
k
n
ii
k
µ
λ
λ
µ
λ
λ
µ
λ
λ
µ
λ
λ
µ
λ
µ
λ
λ
+ + + + + + + +∈
=
−
+
+ −
−
+
+
−
=
−
+
+ −
−
+
+
−
≥
=
=
=
=
<
<
<
<
1
for
0.
(
)
,
for
0.
( ) lim
lim
where
0.
( ) lim
1.
If{ }converges to in
then is the common fixed point
and .
n n n n n n n n n n n
n
iii
k
n
iv
h
k
h
v
x
z
X
z
S
P
µ
λ µ
λ
µ
→∞ →∞ →∞>
≥
≥
≥
= =
>
=
Proof:-If{ }
x
n converges onz
∈
X
thenlim
nn→∞
x
=
z
.Now we shall show that
z
is the common fixed point ofS
andP
. Consider 2n 3 2n 3z
−
Pz
≤ −
z
x
++
x
+−
Pz
(
)
(
)
(
)
2n 3 n n 2n 2 n 2 2n 21
n n n 1 2n 1 n n 2n 1 1z
x
+µ
λ
x
+λ
T x
+µ
λ
k T x
+λ
k
x
+T z
≤
−
+
−
+
+ −
−
+
+
−
−
(
)
(
)
(
)
2 3 2 2 2 2 2 1 2 11
n n n n n n n n n n n n nz
x
x
Pz
Sx
Pz
k
Px
Pz
k
x
Pz
µ
λ
λ
µ
λ
λ
+ + + + +≤
−
+
−
−
+
−
+ −
−
+
−
+
−
−
(
)
{
(
)
}
(
)
(
)
2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 1 2 12
1
n n L x Sx n n n n n n n n n n n n n n nz
x
x
Pz
e
x
Sx
x
Pz
k
Px
Pz
k
x
Pz
µ
λ
λ
δ
δ
µ
λ
λ
+ − + + + + + + + +≤
−
+
−
−
+
−
+
−
+ −
−
+
−
+
−
−
...(2.2.1)Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
80
(
)
2 2 2 2 2 2 2 3 2 2 2 1 2 1 2 1(1
)
1
n n n n n n n n n n n n n nk
x
Sx
x
x
µ
x
Px
λ
x
Px
λ
λ
λ
+ + + + + + + +−
−
−
≤
−
+
−
+
−
Now putting above values in (2.2.1) then we have(
)
( )(
)
(
)
2 2 2 3 2 2 2 1 2 1 2 1 2 3 2 2 (1 ) 1 2 2 2 3 2 2 2 1 2 1 2 1 2 2 2 1(1
)
1
2
1
n n n n n n n n n n n n n n n n k L x x x Px x Px n n n n n n n n n n n n n n n n n n nz
Pz
z
x
x
Pz
e
k
x
x
x
Px
x
Px
x
Pz
k
Px
Pz
λ µ λ λ λµ
λ
λ
λ
µ
δ
δ
λ
λ
λ
µ
λ
λ
+ + + + + + + + − − − + − + − + + + + + + + +−
≤
−
+
−
−
+
−
−
−
+
−
+
−
+
−
+ −
−
+
−
+
(
−
k
n)
x
2n+1−
Pz
Lettingn
→ ∞
then we have(1
)
z
−
Pz
≤ − +
h h
δ
z
−
Pz
Which is a contradiction.
Hence
z
=
Pz
i.e.z
is a fixed point ofP
.
Similarly we can show that(1
)
z
−
Sz
≤ − +
h
h
δ
z
−
Sz
Hence
z
=
Sz
i.e.z
is a fixed point ofS
.Finally we can say that
z
is a common fixed point ofS
& .
P
This completes the proof of theorem.
REFERENCES
(1)Das, G.and Debata, J.P., “Fixed points of quasi-nonexpansive mappings” Indian Journal of Pure and Applied Mathematics 17,(1986),1263-1269.
(2)Ishikawa,S.,“Fixed points by a new iteration method” , Proceeding of the American Mathematical Society ,44,(1974),147-150.
(3)Mann,W.R., “ Mean value methods in iterations” , Proceeding of the American Mathematical Society , Vol.4,(1953),506-510.
(4)Rafiq,A.,“Common fixed points of quasi contractive operators” General Mathematics,Vol. 16,No.2(2008)49-58.
(5)Sahu ,N.K., “ Some fixed point theorem on quasi contractive mappings” Pure and Applied Mathematika Sciences , Vol.XXXII,No. 1-2,(1990),53-57.
(6)Takahashi,W.and Tamura ,T.,“Convergence theorem for a pair of nonexpansive mappings” , Journal of Convex Analysis,Vol.5,No.1,(1995),45-58.
Deepak Singh Kaushal
Sagar Institute of Science,Technology & Research Bhopal (M.P.)
Email:[email protected]
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