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Some Fixed Point Theorems for Contractive Type Mapping in nbanach
spaces
Mukti Gangopadhyay^{1*}, Mantu Saha^{2†} AND A. P.
Baisnab^{3}
^{
1}Calcutta
Girls B.T. College, 6/1 Swinhoe Street, Kolkata700019, (WB) INDIA.
^{
2}Department
of Mathematics, the University of Burdwan, Burdwan713104, (WB) INDIA.
^{
3}Lady
Brabourne College, Kolkata, (WB) INDIA.
Corresponding Addresses:
^{
*}[email protected],
^{
†}[email protected]
Abstract:
Some Fixed Point Theorems for a class of mappings with contractive
iterates in a setting of nBanach spaces have been proved.
2000 Mathematics
subject classification: 47H10, 54H25.
Keywords:
nBanach space, fixed point, contraction.
1. Introduction
The concept of
2normed spaces was introduced and studied by German Mathematician
Siegfried Gähler [11], [12], [13], [14] in a series of paper as
appeared in 1960’s. Later on Misiak [1] had also developed the notion
of an nnorm in 1989. The concept on ninner product
spaces is also due to Misiak who had studied the same as early as 1980
, see [1]. Following this one sees systematic development in theory of
linear nnormed spaces as made by S. S. Kim and Y. J. Cho [10],
R. Maleceski [7] and H. Gunawan and Mashadi [5]. H. Dutta and B.
Surendra Reddy in [6], have shown that under certain cases convergence
and completion in a nnormed space is equivalent to those in (n
– r) norm, r = 1, 2, …, n – 1. For related works of
nmetric spaces and ninner product spaces one may see
[1], [2] and [3].
Recently H. Gunawan
and M. Mashadi in [5] have proved some fixed point theorems for
contractive mappings acting over a finite dimensional nBanach
space. In this paper we also present some fixed point theorems for
mappings with contractive iterates supposed to act on any nBanach
space.
2.
We
recall some preliminary Definitions related to our findings as
presented here below.
Definition 2.1:
Given a natural number n, let X be a real vector space
of dimension (d may be infinity). A real valued
function on
satisfying following four properties,
(i)
if and only if
are linearly dependent in X.
(ii)
is invariant under permutation of
,
(iii)
for
every
(iv)
for all y and z in X,
is called an nnorm over X and the pair
is called an nnormed spaces.
Definition 2.2.:
A sequence in an nnormed space
is said to converge to an element
(in the nnorm) whenever
for every
.
Definition 2.3.:
A sequence in an nnormed space
is said to be a Cauchy sequence with respect
to nnorm if for every
.
Definition 2.4.:
If every Cauchy sequence in X converges to an element
, then X is said to be complete (with
respect to the nnorm).
A complete nnormed
space is called an nBanach space.
Definition 2.5.:
Let X be a nBanach space and T be a self mapping
of X. T is said to be continuous at x if for
every sequence in X,
as
implies
as
in X.
Example 2.1:
Take the function space of all square integrable functions over the
closed interval [0,1]. Then
where
’s are polynomials with
and
becomes a linearly independent set in
. For a natural number n define an
nnorm over as
(n factors)
Reals denoted by
and
given by
,
where .
Note that in an
nnormed space we have for instance
and
for
all and for all scalars
. Then it is a routine argument to see that
is an nnormed Banach space (see
section 2.1 of [5]).
3. Theorem 3.1
Let T be a
selfmapping of X such that there exists h where
and for all
(3.1)
then T has a
unique fixed point z in X with
for each
.
Proof:
Let, and define a recursive sequence
by
.
Then by (3.1) we
have,
.
Now
is impossible, because
. So we need examining case (i) and (ii) only
as under.
Case
(i) :
Suppose max
Therefore
(3.2)
Case
(ii) :
Suppose
Therefore
implies
(3.3)
From (3.2) and
(3.3) we have
for
all k and for all .
So,
for all k and for all
.
Proceeding in this
way
, where
.
If
, for
, we have
as
.
That means
is Cauchy sequence in X and let
.
Again for
,
Passing on
we have
.
Therefore
and z is a fixed point of T
where for each
.
Uniqueness of z
is obvious.
Theorem 3.2:
Let X be a nBanach space with
. Let
be a sequence of mappings such that
(i)
for all
and
,
with
.
(3.4)
and (ii)
for each
. Then T has a unique fixed point z
in X such that ,
being the unique fixed point of
Proof:
Taking limit as in (3.4) we obtain
for
all and hence T satisfies (3.4), Hence by
Theorem 3.1, T has a unique fixed point say
.
Now for
,
(3.5)
Again
implies
(3.6)
By (3.5) and (3.6)
we have,
or,
or,
.
So, by routine
calculation we get .
Theorem 3.3:
Let X be a nBanach space and
be a sequence of mappings with fixed point
such that
uniformly over
to satisfy,
(3.7)
for allwhere
then
where z is the fixed point of T
in X.
Proof:
Fix , from uniform convergence of
on
there exists an integer k such that
for all and for all
,
for
all where
.
(3.8)
Now
(3.9)
From (3.7) we get,
_{
}since
.
Now
is impossible because
and
.
Hence above gives
Using (3.9) we have
implies
i.e.,
which equals to
, for
using (3.8). Hence
.
Theorem 3.4:
Let S and T be two self mappings of X satisfying
(3.10)
for all
where
. Then S and T have a unique
common fixed point where
for each
.
Proof:
Let and define
by
and
.
Then by (3.10)
(3.11)
Let
(3.12)
In case
Then
,
gives
that means right
hand side of (3.11) equals to
, contrary to the case as assumed above.
Thus
By a similar
argument we arrive at
Therefore (3.11) is
not tenable.
Further
and therefore (3.11) gives
By exactly the same
argument we produce
for
all and therefore for all k we
have
.
N_{OW }
following standard and usual arguments one reaches to conclude that
is a Cauchy in X, and let
. Now for
,
Passing
on above gives
and we conclude that
.
Similarly, we show
that and hence z is a common fixed point of
S and T.
Uniqueness of z
is obvious by virtue of (3.10). Finally taking
, and following iteration scheme as undertaken
we have . So,
.
The proof is now
complete.
References:
[1]
A. Misiak, ninner product spaces, Math. Nachr. 140 (1989)
299319. MR 91a: 46021. Zbl 673 .46012.
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[3]
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[4]
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Acknowledgement:
The first author acknowledges for financial support from University
Grants Commission, Regional Office, Kolkata sanctioning (vide sanction
letter no.F.PSW032/1112 (ERO), dated 08.08.2011) the Minor Research
Project to the author.
