1. Introduction
Banach [1] proved a principle, and this famous Banach contraction principle has many generalizations, see [2] - [7], and in 2008, Suzuki [8] established one of those generalizations, and this generalization is called Suzuki principle.
The aim of this paper is to prove a fixed point result generalized from the above mentioned principle in b2-metric space [9].
2. Preliminaries
Before giving our results, these definitions and results as follows will be needed to present.
Definition 2.1 [9] Let X be a nonempty set,
be a real number and let d:
be a map satisfying the following conditions:
1) For every pair of distinct points
, there exists a point
such that
.
2) If at least two of three points
are the same, then
,
3) The symmetry:
for all
.
1) The rectangle inequality:
, for all
.
Then d is called a b2 metric on X and
is called a b2 metric space with parameter s. Obviously, for
, b2 metric reduces to 2-metric.
Definition 2.2 [9] Let
be a sequence in a b2 metric space
.
1) A sequence
is said to be b2-convergent to
, written as
, if all
.
2)
is Cauchy sequence if and only if
, when
. for all
.
3)
is said to be complete if every b2-Cauchy sequence is a b2-convergent sequence.
Definition 2.3 [9] Let
and
be two b2-metric spaces and let
be a mapping. Then f is said to be b2-continuous, at a point
if for a given
, there exists
such that
and
for all
imply that
. The mapping f is b2-continuous on X if it is b2-continuous at all
.
Definition 2.4 [9] Let
and
be two b2-metric spaces. Then a mapping
is b2-continuous at a point
if and only if it is b2-sequentially continuous at x; that is, whenever
is b2-convergent to x,
is b2-convergent to
.
Lemma 2.5 [9] Let
be a b2-metric space and suppose that
and
are b2-convergent to x and y, respectively. Then we have
, for all a in X. In particular, if
is a constant, then
, for all a in X.
Lemma 2.6 [10] Let
be a b2 metric space with
and let
be a sequence in X such that
, (2.1)
for all
and all
, where
. Then
is a b2-Cauchy sequence in
.
3. Main Results
Theorem 3.1. Let
be a complete b2-metric space. Let
be two self-maps and
be defined as follows
(3.1)
Assume there exists
such that for every
, the following condition is satisfied
(3.2)
Then
have a unique common fixed point
.
Proof in (3.2), we take
for
.(3.3)
therefore,
(3.4)
Now we take
in (3.2)
for all
.(3.5)
therefore,
(3.6)
and
(3.7)
Given an arbitrary point
in X thenby
and
we construct a sequence
, for
.
From (3.4), we get
(3.8)
From (3.7) and (3.8) we get
,
that is,
, since
, by Lemma 2.6, we get
is a Cauchy sequence.
Since X is complete, there exists z in X, such that
, that is
, and
.
Now let us give that
, for every
. For
is convergent to 0, and by Lemma 2.5, we get
, thus we have
, thus from the above relation, there exists a point
in X such that
For such
, (3.2) implies that
therefore by Lemma 3.5,
therefore we get
, for each
. (3.9)
Now we show that for each
,
(3.10)
It is obvious that the above inequality is true for
, assume that the relation holds for some
. We get (3.10) is true when we have
if
, then if
, we get the following relation from (3.9) and induction hypothesis, and that is
then (3.10) is proved.
Now we consider the following two possible cases in order to prove that f has a fixed point z in X, and that is
.
Case 1
, therefore,
. First, we prove the following relation
, for
. (3.11)
When
it is obvious, and it follows from (3.6) when
, from (3.10) and take
we have
, then we get
.
Now suppose that (3.11) holds for some
,
Therefore, we get
, that is
, (3.11.1)
then by taking
in (3.6)
, (3.11.2)
using the above two relations, (3.11.1) and (3.11.2) we have
From (3.2) and (3.10) with
and
, we have
Therefore,
(3.12)
So by induction we prove the relation of (3.11).
Now (3.11) and
show that for every
, thus, (3.9) shows that
Therefore
. Furthermore by using Lemma 2.5, we get
so
In the same way,
, thus we have
, that is
, and by using Lemma 2.5 in (3.12), we get
, which claims that
, and that is a contraction.
Case 2.
, and that is when
. We now prove that we can find a subsequence
of
such that
, for
. (3.13)
The contraries of the above relation are as follows
and
for
. If n is even we have
if n is odd then we get
for
. By (3.8) we have
this is impossible. Therefore, one of the following relations is true for every
,
or
That means there exists a subsequence
of
such that (3.13) is true for every
. Thus (3.2) shows that
or
From Lemma 2.5, we have
or
Therefore
, which is impossible unless
. hence z in X is a fixed point of f. From the process of the above proof, we know
, then by
,
it implies
this proves that
. By (3.2) we can prove the uniqueness of the common fixed point z,
, so (3.2) shows that
which is impossible unless
. □
NOTES
*Corresponding author.