Abstract
An ingenious approach to generalize Banach contraction principle was adopted by Suzuki in his seminal papers (Proc Am Math Soc 136:1861–1869, 2008, Nonlinear Anal Theory Methods Appl 71:5313–5317, 2009). In this paper we prove certain common fixed point results for generalized Suzuki contractions in the set-up of b-metric spaces, where the b-metric function is not necessarily continuous. Finally, some examples are presented to verify the effectiveness and applicability of our main results.
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Introduction
There are a lot of generalizations of Banach fixed point principle in the literature. In 2008 Suzuki introduced an interesting generalization of Banach fixed point principle. This interesting fixed-point result is as follows.
Theorem 1
[26] Let (X, d) be a complete metric space, and let T be a mapping on X. Define a non-increasing function \({\theta }\) from [0, 1) into (1/2, 1] by
Assume that there exists \(r\in [0,1)\), such that
for all \(\ x,y\in X\), then there exists a unique fixed-point z of T. Moreover, \({\rm lim}_{{\it n}\rightarrow \infty }T^{{\it n}}x=z\) for all \(x\in X\).
Suzuki proved also the following version of Nemytckii fixed point theorem.
Theorem 2
Let (X, d) be a compact metric space. Let \( T:X\rightarrow X\) be a selfmap, satisfying for all \(x,y\in X\) \(x\ne y\) the condition
Then T has a unique fixed point in X.
This theorem was also generalized in [6].
In addition to the above results, Kikkawa and Suzuki [11] provided a Kannan-type version of the theorems mentioned above. In [21], a Chatterjea-type version is provided, whereas Popescu [20] obtained a Cirić-type version. Recently, Kikkawa and Suzuki also provided multivalued versions in [12, 13].
Very recently, Hussain et al. in [8] have extended Suzuki’s Theorems 1 and 2, as well as Popescu’s results from [20] to the case of metric-type spaces and cone metric-type spaces.
Czerwik in [5] introduced the concept of b-metric space. Since then, several papers deal with fixed point theory for single-valued and multivalued operators in b-metric spaces (see also [1–5, 7–10, 14–17, 19, 24, 25]). Pacurar [19] proved results on sequences of almost contractions and fixed points in b-metric spaces. Recently, Hussain and Shah [9] obtained results on KKM mappings in cone b-metric spaces. Khamsi ([14, 15]) also showed that each cone metric space has a b-metric structure.
The aim of this paper is to present some common fixed point results for two mappings under generalized contractive condition in b-metric space, where the b-metric function is not necessarily continuous. Because many of the authors in their works have used the b-metric spaces in which the b-metric function is assumed to be continuous. From this point of view the results obtained in this paper generalize and extend several ones obtained earlier concerning b-metric space.
Consistent with [5] and [25, p. 264], the following definition and results will be needed in the sequel.
Definition 1
[5] Let X be a (nonempty) set and \(b\ge 1\) be a given real number. A function \(d:X\times X\rightarrow R^{+}\) is a b-metric spaces iff, for all \(x,y,z\in X\), the following condition are satisfied:
-
(b1)
d\((x,y)=0\) iff \(x=y,\)
-
(b2)
d\((x,y)={\mathrm{d}}(y,x),\)
-
(b3)
d\((x,z)\le b[{\mathrm{d}}(x,y)+{\mathrm{d}}(y,z)].\)
The pair \((X,{\mathrm{d}})\) is called a b-metric space.
It should be noted that, the class of b-metric spaces is effectively larger than that of metric spaces, since a b-metric is a metric only if \(b=1.\)
We present an example which shows that a b-metric on X need not be a metric on X. (see also [25, p. 264]):
Example 1
[22] Let (X, d) be a metric space, and \(\rho (x,y)=(d(x,y))^{p},\) where \(p>1\) is a real number. Then \(\rho \) is a b-metric with \(b=2^{p-1}.\)
However, if (X, d) is a metric space, then \((X,\rho )\) is not necessarily a metric space.
For example, if \(X= {\mathbb {R}} \) is the set of real numbers and \(d(x,y)=\left| x-y\right| \) is the usual Euclidean metric, then \(\rho (x,y)=(x-y)^{2}\) is a b-metric on \( {\mathbb {R}} \) with \(b=2,\) but is not a metric on \( {\mathbb {R}} \).
Before stating and proving our results, we present some definition and proposition in b-metric space. We recall first the notions of convergence and completeness in a b-metric space.
Definition 2
[3] Let (X, d) be a b-metric space. Then a sequence \(\{x_{n}\}\) in X is called:
-
(a)
convergent if and only if there exists \(x\in X\) such that \( d(x_{n},x)\rightarrow 0\) as \(n\rightarrow \infty \). In this case, we write \(\lim _{n\rightarrow \infty }x_{n}=x.\)
-
(b)
Cauchy if and only if \(d(x_{n},x_{m})\rightarrow 0\) as \(n,m\rightarrow \infty .\)
Proposition 1
(see remark 2.1 in [3]) In a b-metric space (X, d) the following assertions hold:
-
(i)
a convergent sequence has a unique limit,
-
(ii)
each convergent sequence is Cauchy,
Definition 3
[3] The b-metric space (X, d) is complete if every Cauchy sequence in X converges.
It should be noted that, in general a b-metric function d(x, y) for \(b>1\) is not jointly continuous in all two of its variables. Now we present an example of a b-metric which is not continuous.
Example 2
(see Example 3 in [8]) Let \(X= {\mathbb {N}} \cup \{\infty \}\) and let \(D:X\times X\rightarrow {\mathbb {R}} ^{+}\) be defined by,
Then it is easy to see that for all \(m,n,p\in X,\) we have
Thus, (X, D) is a b-metric space with \(b=\frac{5}{2}.\) In [8], it is proved that D(x, y) is not a continuous function.
Since in general a b-metric is not continuous, we need the following simple lemma about the b-convergent sequences.
Lemma 1
[22] Let \((X,{\mathrm d})\) be a b-metric space with \(b\ge 1\), and suppose that \(\{x_{n}\}\) and \(\{y_{n}\}\) b-converge to x, y, respectively. Then, we have
In particular, if \(x=y\), then \(\lim _{n\rightarrow \infty }{\mathrm{d}}(x_{n},y_{n})=0\). Moreover, for each \(z\in X\) we have
Main result
We start our work by proving the following crucial Theorem.
Theorem 3
Let (X, d) be a complete b-metric space. Let \(T,S:X \longrightarrow X\) be two self-maps and \(\theta :[0,1)\longrightarrow (\frac{ 1}{2},1]\) be defined by
Suppose there exists \(r\in [0,1)\) such that for each \(x,y\in X\), the following condition is satisfied
Then T, S have a unique common fixed point \(z\in X.\)
Proof
At first we show that if z is a fixed point of S or T, then z is a common fixed point of T and S. Let z be a fixed point of T that is \( Tz=z\) then we show that \(Sz=z\). From
it follows
thus \(Sz=z\). Therefore it is enough to show that T have a fixed point. Putting \(y=Sx\) in (2)
it follows
for every \(x\in X\). Hence,
Now, putting \(y=Tx\) in (2)
it follows
for every \(x\in X\). Hence,
and
Let \(x_{0}\in X\) be arbitrary and form the sequence \(\{x_{n}\}\) by, \( x_{2n+1}=Sx_{2n}\) and \(Tx_{2n+1}=x_{2n+2}\) for \(n\in {\mathbb {N}}\cup \{0\}\). We show that \(\{x_{n}\}\) is a Cauchy sequence.
By (4), we have
By (7), we have
Therefore,
Also, by definition of b-metric spaces for all \(\ m\ge n,\) we have
So, we have
Hence, \(\{x_{n}\}\) is a Cauchy sequence. Since X is complete, we conclude \( \{x_{n}\}\) converges to z for some \(z\in X.\) That is
and
Let us prove now that
holds for each \(x\ne z\). Since \({\mathrm{d}}(x_{2n},Sx_{2n})\longrightarrow 0,\) and by Lemma 1
\(\text { thus} \ \underset{{\it n}\longrightarrow \infty }{\lim \sup \text { }} {\mathrm{d}}(x_{2{\it n}},x)>0\), it follows that there exists a \(x_{2{\it n}_{k}}\in X\) such that
Assumption (2) implies that for such \(x_{2n_{k}}\)
hence by Lemma 1
thus for each \(x\ne z\) we get that
We will prove that
for each \(n\in {\mathbb {N}}\). For \(n=1\) this relation is obvious. Suppose that it holds for some \(m\in {\mathbb {N}}\). If \(T^{m}z=z\) then \(T^{m+1}z=Tz\) it follows that the above inequality is true. If \(T^{m}z\ne z\), we can apply (9) and the induction hypothesis, we get that
and (10) is proved by induction.
In order to prove that \(Tz=z\). We consider two possible cases.
Case I. \(0\le r<\frac{1}{\sqrt{2}}\) (and hence \(\theta (r)\le \frac{1-r}{ r^{2}}\)). We will prove first that
for each \(n\in {\mathbb {N}}\). For \(n=1\) it is obvious. For \(n=2\) it follows from (6). Suppose that (11) holds for some \(n>2\). Since
hence \((1-r){\mathrm{d}}(z,Tz)\le b{\mathrm{d}}(z,T^{n}z)\). It follows [using (6) with \( x=T^{n-1}z\)] that
Assumptions (2) and (10) imply that
Thus
So relation (11) is proved by induction.
Now \(Tz\ne z\) and (11) imply that \(T^{n}z\ne z\) for each \(n\in {\mathbb {N}}\). Hence, (9) implies that
Hence \(\lim _{n\rightarrow \infty }{\mathrm{d}}(z,T^{n+1}z)=0.\) On the other hand using Lemma 1, we have
so
Similarly,
therefore \({\mathrm{d}}({\it z},\text { }\underset{{\it n}\longrightarrow \infty }{\lim \text { }} {\it T}^{{\it n}+1}{\it z})=0.\)
Thus \(T^{n+1}z\longrightarrow z\) and, using Lemma 1 in (12), we have
which implies that \({\mathrm{d}}(z,Tz)=0\), a contradiction.
Case II. \(\frac{1}{\sqrt{2}}\le r<1\) (and so \(\theta (r)=\frac{1}{1+r})\). We will prove that there exists a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that
holds for each \(k\in {\mathbb {N}}\). Suppose the contrary
and
holds for each \(n\in {\mathbb {N}}\). Now if n is odd then
if n is even then
holds for each \(n\in {\mathbb {N}}\). Then from (8) we have
which is impossible. Hence one of the following inequalities is satisfied for each \(n\in {\mathbb {N}}\):
or
In other words, there is a subsequence \(\{x_{n_{k}}\}\) of \(\{x_{n}\}\) such that (13) holds for each \(k\in {\mathbb {N}}\). Hence assumption (2) implies that
or
By Lemma 1, we get
or
hence \({\mathrm{d}}(z,Tz)\le 0\), which is possible only if \(Tz=z\).
Thus, we have proved that z is a fixed point of T. The uniqueness of the common fixed point follows easily from (2). Indeed, if \(z,z^{\prime }\) are two common fixed points of T,
then (2) implies that
which is possible only if \(z=z^{\prime }\). This proves that z is a unique common fixed point of T and S. \(\square \)
According to Theorem 3 we get the following result.
Corollary 1
Let \((X,\mathrm{d})\) be a complete b-metric space, and let T be a mapping on X. Define a non-increasing function \(\theta \) from [0, 1) into (1/2, 1] by (1).
Suppose there exists \(r\in [0,1)\) such that for each \(x,y\in X\), the following condition is satisfied
then there exists a unique fixed-point z of T. Moreover, \( lim_{n\rightarrow \infty }T^{n}x=z\) for all \(x\in X\).
Proof
It is enough set \(S=T\) in the Theorem 3 then the desired result is obtained. \(\square \)
Remark 1
Note that for \(b=1,\) Corollary 1 reduces to Theorem.
Corollary 2
Let (X, d) be a complete b-metric space, and \( f,S,T:X\longrightarrow X\) be three self-maps and \(\theta :[0,1)\longrightarrow (\frac{1}{2},1]\) be defined by (1).
Suppose there exists \(r\in [0,1)\) such that for each \(x,y\in X\), the following condition is satisfied
Also, if f is one to one, \(fS=Sf\) and \(fT=Tf\), then we have f, T and S have a unique common fixed point \(z\in X.\)
Proof
By Theorem 3, fT, fS have a unique common fixed point \(z\in X\). That is \(fSz=fTz=z\), since f is one to one it follows that \(Sz=Tz\). From
it follows that
it follows that \(Tz=Sz=z\), hence \(fz=fTz=z.\) \(\square \)
Corollary 3
Let \((X,\mathrm{d})\) be a complete metric space, and \( f,S,T:X\longrightarrow X\) be three self-maps and \(\theta :[0,1)\longrightarrow (\frac{1}{2},1]\) be defined by (1).
Suppose there exists \(r\in [0,1)\) such that for each \(x,y\in X\), the following condition is satisfied
Also, if f is one to one, \(fS=Sf\) and \(fT=Tf\), then we have f, T and S have a unique common fixed point \(z\in X\).
Proof
It is enough to set \(b=1\) in the Corollary 2 then the desired result is obtained. \(\square \)
Now, in order to support the useability of our results, let us introduce the following examples.
Let \(X=[0,\infty )\). Define \(d:X\times X\rightarrow \mathbb {R^{+}}\) by
for all \(x,y\in X\). Then \((X,\mathrm{d})\) is a complete b-metric space for \(b=2\). Define two maps \(T,S:X\rightarrow X\) by
for \(x\in X\). Then for each \(x,y\in X\) we have
On the other hand, we have
Thus T and S satisfy all the hypotheses of Theorem 3 and hence T and S have a unique common fixed point. Indeed, \(r=\dfrac{1}{2\sqrt{2 }}<\frac{\sqrt{5}-1}{2},\) \(\theta (r)=\frac{1}{2}\) and 0 is the unique common fixed point of T and S.
Inspired by [8, Example 4] and [26, Example 1], we present the following example:
Example 3
Let \(X=\{(0,0),(10,12),(12,10),(40,42),(42,40)\}\subset \mathbb {R}^{2}\). Define \(d:X\times X\rightarrow \mathbb {R}^{+}\) by
for all \(x=(x_{1},y_{1}),y=(x_{2},y_{2})\in X\). Then (X, d) is a complete b-metric space for \(b=2\). Define two maps \(T,S:X\rightarrow X\) by
Then for each \(x,y\in X\), if
this implies that
Because,
-
i)
\(\frac{1}{2}\frac{\sqrt{2}}{\sqrt{2}+1}\min \left\{ \mathrm{d}((0,0),T(0,0)),\mathrm{d}((0,0),S(0,0))\right\} \le \mathrm{d}(0,0),y),\quad \forall y\in X\).
-
ii)
\(\frac{1}{2}\frac{\sqrt{2}}{\sqrt{2}+1}\min \left\{ \mathrm{d}((10,12),T(10,12)),\mathrm{d}((10,12),S(10,12))\right\} \le \mathrm{d}(10,12),y),\quad \forall y=(0,0),(40,42),(42,40)\).
-
iii)
\(\frac{1}{2}\frac{\sqrt{2}}{\sqrt{2}+1}\min \left\{ \mathrm{d}((12,10),T(12,10)),\mathrm{d}((12,10),S(12,10))\right\} \le \mathrm{d}(12,10),y),\quad \forall y=(0,0),(40,42),(42,40)\).
-
iv)
\(\frac{1}{2}\frac{\sqrt{2}}{\sqrt{2}+1}\min \left\{ \mathrm{d}((40,42),T(40,42)),\mathrm{d}((40,42),S(40,42))\right\} \le \mathrm{d}(40,42),y),\quad \forall y=(0,0),(10,12),(12,10)\).
-
v)
\(\frac{1}{2}\frac{\sqrt{2}}{\sqrt{2}+1}\min \left\{ \mathrm{d}((42,40),T(42,40)),\mathrm{d}((42,40),S(42,40))\right\} \le \mathrm{d}(42,40),y),\quad \forall y=(0,0),(10,12),(12,10)\).
On the other hand, in all of the cases we have
Thus T satisfy all the hypotheses of Theorem 3 and hence T has a unique fixed point. Indeed, \(r=\dfrac{1}{\sqrt{2}},\) \(\theta (r)=\frac{ \sqrt{2}}{\sqrt{2}+1}\) and (0, 0) is the unique common fixed point of T and S.
But,
that is \(2^{2}+2^{2}\le \frac{r}{4}(2^{2}+2^{2})\) this implies that \(r\ge 4 \). It is contradiction. This proves that Theorem 1 is not applicable to T.
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Roshan, J.R., Hussain, N., Sedghi, S. et al. Suzuki-type fixed point results in b-metric spaces . Math Sci 9, 153–160 (2015). https://doi.org/10.1007/s40096-015-0162-9
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DOI: https://doi.org/10.1007/s40096-015-0162-9