Some Fixed Point Theorems for Contractive Type Mapping in n-banach
Mukti Gangopadhyay1*, Mantu Saha2† AND A. P.
Girls B.T. College, 6/1 Swinhoe Street, Kolkata-700019, (WB) INDIA.
of Mathematics, the University of Burdwan, Burdwan-713104, (WB) INDIA.
Brabourne College, Kolkata, (WB) INDIA.
Some Fixed Point Theorems for a class of mappings with contractive
iterates in a setting of n-Banach spaces have been proved.
subject classification: 47H10, 54H25.
n-Banach space, fixed point, contraction.
The concept of
2-normed spaces was introduced and studied by German Mathematician
Siegfried Gähler , , ,  in a series of paper as
appeared in 1960’s. Later on Misiak  had also developed the notion
of an n-norm in 1989. The concept on n-inner product
spaces is also due to Misiak who had studied the same as early as 1980
, see . Following this one sees systematic development in theory of
linear n-normed spaces as made by S. S. Kim and Y. J. Cho ,
R. Maleceski  and H. Gunawan and Mashadi . H. Dutta and B.
Surendra Reddy in , have shown that under certain cases convergence
and completion in a n-normed space is equivalent to those in (n
– r) norm, r = 1, 2, …, n – 1. For related works of
n-metric spaces and n-inner product spaces one may see
,  and .
Recently H. Gunawan
and M. Mashadi in  have proved some fixed point theorems for
contractive mappings acting over a finite dimensional n-Banach
space. In this paper we also present some fixed point theorems for
mappings with contractive iterates supposed to act on any n-Banach
recall some preliminary Definitions related to our findings as
presented here below.
Given a natural number n, let X be a real vector space
of dimension (d may be infinity). A real valued
satisfying following four properties,
if and only if
are linearly dependent in X.
is invariant under permutation of
for all y and z in X,
is called an n-norm over X and the pair
is called an n-normed spaces.
A sequence in an n-normed space
is said to converge to an element
(in the n-norm) whenever
A sequence in an n-normed space
is said to be a Cauchy sequence with respect
to n-norm if for every
If every Cauchy sequence in X converges to an element
, then X is said to be complete (with
respect to the n-norm).
A complete n-normed
space is called an n-Banach space.
Let X be a n-Banach space and T be a self mapping
of X. T is said to be continuous at x if for
every sequence in X,
Take the function space of all square integrable functions over the
closed interval [0,1]. Then
’s are polynomials with
becomes a linearly independent set in
. For a natural number n define an
n-norm over as
Reals denoted by
Note that in an
n-normed space we have for instance
all and for all scalars
. Then it is a routine argument to see that
is an n-normed Banach space (see
section 2.1 of ).
3. Theorem 3.1
Let T be a
self-mapping of X such that there exists h where
and for all
then T has a
unique fixed point z in X with
Let, and define a recursive sequence
Then by (3.1) we
is impossible, because
. So we need examining case (i) and (ii) only
From (3.2) and
(3.3) we have
all k and for all .
for all k and for all
Proceeding in this
, we have
is Cauchy sequence in X and let
and z is a fixed point of T
where for each
Uniqueness of z
Let X be a n-Banach space with
be a sequence of mappings such that
. Then T has a unique fixed point z
in X such that ,
being the unique fixed point of
Taking limit as in (3.4) we obtain
all and hence T satisfies (3.4), Hence by
Theorem 3.1, T has a unique fixed point say
By (3.5) and (3.6)
So, by routine
calculation we get .
Let X be a n-Banach space and
be a sequence of mappings with fixed point
where z is the fixed point of T
Fix , from uniform convergence of
there exists an integer k such that
for all and for all
From (3.7) we get,
is impossible because
Hence above gives
Using (3.9) we have
which equals to
using (3.8). Hence
Let S and T be two self mappings of X satisfying
. Then S and T have a unique
common fixed point where
Let and define
Then by (3.10)
that means right
hand side of (3.11) equals to
, contrary to the case as assumed above.
By a similar
argument we arrive at
Therefore (3.11) is
and therefore (3.11) gives
By exactly the same
argument we produce
all and therefore for all k we
following standard and usual arguments one reaches to conclude that
is a Cauchy in X, and let
. Now for
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
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299-319. MR 91a: 46021. Zbl 673 .46012.
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H. Gunawan, On n-inner products, n-norms and the Cauchy
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H. Ganawan and Mashadi, On finite dimensional 2-normed spaces, Soochow
J. Math. 27 (2001), 321-329.
Hendra Gunawan and M. Mashadi, On n-normed spaces, IJMMS 27:10
Hemen Dutta, B. Surender Reddy, On nonstandard n-norm on some
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Reddy B. S., Elementary properties of n-Banach spaces, The
Journal of the Indian Academy of Mathematics.
S. S. Kim and Y. J. Cho, Strict convexity in linear n-normed
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The first author acknowledges for financial support from University
Grants Commission, Regional Office, Kolkata sanctioning (vide sanction
letter no.F.PSW-032/11-12 (ERO), dated 08.08.2011) the Minor Research
Project to the author.