On multiplicative metric spaces: survey
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Abstract
The purpose of this survey is to prove that the fixed point results for various multiplicative contractions are in fact equivalent to the corresponding fixed point results in (standard) metric spaces. For example, such are recent results established by He et al. (Fixed Point Theory Appl. 2014:48, 2014), Mongkolkeha and Sintunavarat (J. Nonlinear Sci. Appl. 8:11341140, 2015) and Abdou (J. Nonlinear Sci. Appl. 9:23472363, 2016) and all others from the list of references. Our results here generalize, complement, and improve recent ones from existing literature.
Keywords
multiplicative metric space fixed point continuous nondecreasing lower semicontinuous mappingMSC
47H10 54H251 Introduction and preliminaries
In 2008, Bashirov et al., among other things, initiated a new kind of spaces, called multiplicative metric spaces (MMS for short). The main idea was that the usual triangular inequality was replaced by a ‘multiplicative triangle inequality’ as follows:
Definition 1.1
([4])
 (m1^{∗})

\(d^{*} ( x,y ) \geq1\) for all \(x,y\in X\) and \(d^{*} ( x,y ) =1\) if and only if \(x=y\),
 (m2^{∗})

\(d^{*} ( x,y ) =d^{*} ( y,x ) \) for all \(x,y\in X\),
 (m3^{∗})

\(d^{*} ( x,z ) \leq d^{*} ( x,y ) \cdot d^{*} ( y,z ) \) for all \(x,y,z\in X\) (multiplicative triangle inequality).
If the operator \(d^{*}\) satisfies (m1^{∗})(m3^{∗}) then the pair \(( X,d^{*} ) \) is called a multiplicative metric space (MMS).
For more details of this new kind of spaces the reader can refer to [1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. For example, it follows from these papers that an MMS \(( X,d^{*} ) \) is complete (sequentially compact) if and only if (standard) metric space (SMS for short) \(( X,\ln d^{*} ) \) is such. Also, MM \(d^{*}\) and (standard) metric (SM for short) \(\ln d^{*}\) induce the same topology on X (see Theorem 2.1 below).
The next definition for SMS is well known.
Definition 1.2
 (1)
\(d ( x,y ) \geq0\) for all \(x,y\in X\) and \(d ( x,y ) =0 \) if and only if \(x=y\),
 (2)
\(d ( x,y ) =d ( y,x ) \) for all \(x,y\in X\),
 (3)
\(d ( x,z ) \leq d ( x,y ) +d ( y,z ) \) for all \(x,y,z\in X\) (standard triangle inequality).
If the operator d satisfies (1)(3) then the pair \(( X,d ) \) is called an SMS.
Remark 1.3
It is clear that \(d^{*}:X\times X\rightarrow[1,+\infty)\) while \(d:X\times X\rightarrow[0,+\infty)\).
Very recently, the authors in [1] proved the following.
Theorem 1.4
([1], Theorem 3.1)
 (i)
\(SX\subset BX\), \(TX\subset AX\);
 (ii)
A and S are weakly commutative, B and T are also weakly commutative;
 (iii)
one of S, T, A and B is continuous;
 (iv)
\(d^{*}(Sx,Ty)\leq ( \max \{d^{*}(Ax,By),d^{*}(Ax,Sx),d^{*}(By,Ty),d^{*}(Sx,By),d^{*}(Ax,Ty)\} ) ^{ \lambda}\), for some \(\lambda\in ( 0,\frac{1}{2} )\), and all \(x,y\in X\).
Theorem 1.5
([1], Theorem 3.2)
 (i)
\(SX\subset BX\), \(TX\subset AX\);
 (ii)
A and S are weakly commutative, B and T are also weakly commutative;
 (iii)
one of S, T, A and B is continuous;
 (iv)
\(d^{*}(S^{p}x,T^{q}y)\leq ( \max \{d^{*}(Ax,By),d^{*}(Ax,S^{p}x),d^{*}(By,T^{q}y),d^{*}(S^{p}x,By), d^{*}(Ax,T^{q}y)\} ) ^{ \lambda}\), for some \(\lambda\in ( 0,\frac{1}{2} ) \) and \(p,q\in\mathbb{Z}^{+}\), and for all \(x,y\in X\).
Remark 1.6
Also recently Abdou [3] proved the following results.
Theorem 1.7
([3], Theorem 3.1)
 (a)
either A or S is continuous, the pair \((S,A)\) is compatible and the pair \((T,B)\) is weakly compatible;
 (b)
either B or T is continuous, the pair \((T,B)\) is compatible and the pair \((S,A)\) is weakly compatible.
Then S, T, A, and B have a unique CFP in X.
Theorem 1.8
([3], Theorem 3.3)
 (a)
either A or S is continuous, the pair \((S,A)\) is compatible and the pair \((T,B)\) is weakly compatible;
 (b)
either B or T is continuous, the pair \((T,B)\) is compatible and the pair \((S,A)\) is weakly compatible.
Then S, T, A, and B have a unique CFP in X.
Remark 1.9
 1^{∘}

\({d^{*}}^{p}\) is also an MM on X.
 2^{∘}

It is sufficient that \(\varphi :[1,\infty )\rightarrow[1,\infty)\) is an increasing function such that \(\varphi ( 1 ) =1\) and \(\varphi ( t ) < t\) for all \(t>1\), further, for such φ we get \(\ln\varphi ( t ) <\ln t\) whenever \(t>1\).
 3^{∘}
 Also, it is clear enough that the function φ in both previous theorems is superfluous. By removing it, (1) and (2) are, respectively, equivalent withand$$\begin{aligned}& \ln{d^{*}}^{p} ( Sx,Ty ) \\& \quad \leq\lambda\max\bigl\{ \ln{d^{*}}^{p} ( Ax,By ) ,\ln {d^{*}}^{p} ( Ax,Sx ) +\ln{d^{*}}^{p} ( By,Ty )\ln \bigl( 1+{d^{*}}^{p} ( Ax,By ) \bigr), \\& \qquad \ln{d^{*}}^{p} ( Ax,Ty ) +\ln{d^{*}}^{p} ( By,Ax ) \ln \bigl( 1+{d^{*}}^{p} ( Ax,By ) \bigr)\bigr\} \end{aligned}$$(3)$$\begin{aligned}& \ln{d^{*}}^{p} \bigl( S^{m}x,T^{q}y \bigr) \\& \quad \leq\lambda\max\bigl\{ \ln{d^{*}}^{p} ( Ax,By ) ,\ln {d^{*}}^{p} \bigl( Ax,S^{m}x \bigr) +\ln{d^{*}}^{p} \bigl( By,T^{q}y \bigr)\ln \bigl( 1+{d^{*}}^{p} ( Ax,By ) \bigr), \\& \qquad \ln{d^{*}}^{p} \bigl( Ax,T^{q}y \bigr) + \ln{d^{*}}^{p} ( By,Ax ) \ln \bigl( 1+{d^{*}}^{p} ( Ax,By ) \bigr)\bigr\} . \end{aligned}$$(4)
2 Main results
Our first result generalizes Theorem 2.1 from [8] as well as Proposition 2.1 from [14]. The proof is immediate, i.e., by using the properties of functions \(t\mapsto\ln t\), \(t>0\) and \(t\mapsto e^{t}\), therefore it is omitted.
Theorem 2.1
Let \(( X,d^{*} ) \) be an MMS. Then the pair \(( X,d ) \) is an SMS where \(d ( x,y ) =\ln d^{*} ( x,y ) \) for all \(x,y\in X\). Conversely, if \(( X,d ) \) is an SMS then \(( X,d^{*} ) \) is an MMS where \(d^{*} ( x,y ) =e^{d ( x,y ) }\) for all \(x,y\in X\).
Hence, \(( X,d^{*} ) \) is an MMS if and only if \(( X,\ln d^{*} ) \) is an SMS that is, \(( X,d ) \) is an SMS if and only if \(( X,e^{d} ) \) is an MMS.
The Banach contraction principle [17], as one of the most important result in the fixed point theory and nonlinear analysis generally, has been generalized a lot, in metric as well as in different spaces which are generalizations of SMS. Our motive in this section is to prove that various famous results, such as the EdelsteinNemitskii, BoydWong, and MeirKeeler ones, as well as some other wellknown results, are equivalent in SMS and MMS. In order to make the text easier to follow, in each of the cases, we will first expose a classical theorem in SMS, after that we will give the appropriate theorem in MMS. Finally, we will prove the result which shows the equivalence between the two of them. The results will be exposed in historical order.
In 1962, Edelstein attempted to prove a fixed point theorem by keeping the completeness of the SMS and replacing the Banach contractive condition by a slightly modified condition. It turned out that completeness is not a sufficient condition for the existence of fixed point for the new contractive condition. In [12, 18] a fixed point theorem for the proposed contraction conditions was proved, with certain restriction on the space.
Now we announce the wellknown EdelsteinNemytskii theorem in the framework of SMS.
Theorem 2.2
([12])
The appropriate fixed point theorem in the setting of MMS is the following one.
Theorem 2.3
Our contribution in this research is the following result.
Proof
The following wellknown generalization of the Banach contraction principle was presented in 1969 in the paper of Boyd and Wong [19].
Theorem 2.5
([19])
Remark 2.6
It is easy to see that if \(\psi_{1}(t)=\alpha(t)t\), α is increasing function such that \(\alpha(t)<1\) for \(t\geq0\), then the Banach result follows.
The analogous result in MMS is presented below.
Theorem 2.7
The next theorem presents the equivalence between these two results.
Proof
A very interesting theorem was proved in 1969 in the paper of Meir and Keeler [20]. They proved that the conclusion of the Banach contraction holds for a wider class of contraction mappings. Their contribution is the following theorem.
Theorem 2.9
([20])
 for each \(\varepsilon>0\) there exists \(\delta>0\) such that$$ \varepsilon\leq d ( x,y ) < \varepsilon+\delta \quad \Rightarrow\quad d ( fx,fy ) < \varepsilon. $$(10)
The respective theorem in MMS is the next one.
Theorem 2.10
 for each \(\varepsilon>1\) there exists \(\delta>1\) such that$$ \varepsilon\leq d^{*} ( x,y ) < \varepsilon\cdot\delta \quad \Rightarrow\quad d^{*} ( fx,fy ) < \varepsilon. $$(11)
As we have mentioned in this section, our contribution is to show that the above stated theorems are equivalent. So we prove the following theorem.
Proof
The proof is complete. □
In the sequel, we consider the famous Kannan, Chatterje, and Zamfirescu results in both contexts. The Banach contractive condition implies the continuity of mapping f. Naturally, the question arises whether there is a contractive condition (sufficient to guarantee the existence of a fixed point) that does not imply continuity of the mapping f. The answer to the given question was published in 1968 in the Kannan paper [21], where he announced the following fixed point theorem.
Theorem 2.12
The analogous result in the framework of MMS is the following one.
Theorem 2.13
Now, our task is to show the equivalence between these two results. So we prove the following.
Proof
A similar type of contractive condition was studied in 1972 by Chatterjea [23]. He proved the following result.
Theorem 2.15
The adequate theorem in terms of MMS is presented below.
Theorem 2.16
Our contribution is the following theorem.
Proof
The proof follows using the same idea as in the previous one, so we omit it. □
In 1972, Zamfirescu [24] united the theorems of Banach, Kannan, and Chatterjea. We give this theorem in both frameworks.
Theorem 2.18
Theorem 2.19
Reich proved in 1971 [25] the following theorem and thus unified the Banach and Kannan theorems.
Theorem 2.21
Remark 2.22
Notice that one obtains the Banach fixed point theorem for \(a=b=0\), as well as the Kannan theorem for \(a=b\) and \(c=0\).
Now we give the adequate theorem in the framework of MMS.
Theorem 2.23
In the same context, we present the equivalence between these two theorems.
Proof
In 1973, Hardy and Rogers [26] generalized Reich’s [25] result and published the following interesting result. We present the mentioned theorem in both contexts.
Theorem 2.25
([22])
Theorem 2.26
We notice that these two theorems are equivalent, i.e., the following holds.
Proof
Since the idea is the same in all theorems so far, to avoid repeating we give this theorem without proof. □
In 1973, Geraghty [27] generalized the Banach contraction principle by considering an auxiliary function in the following way.
Theorem 2.28
([27])
In the sequel we consider a Geraghty type contraction in MMS. Consider the functions \(\beta :[1,\infty)\rightarrow[0,1)\) such that if \(\beta ( t_{n} ) \rightarrow1\) then \(t_{n}\rightarrow1\). One such function β is \(\beta ( t ) =e^{1t}\), \(t\geq1\). Now we formulate the Geraghty type theorem in the framework of MMS.
Theorem 2.29
Also, for a Geraghty type contraction in both frameworks we have the following.
Proof
Another generalization of the Kannan fixed point theorem was given by Bianchini [28] in 1972, i.e., the following very interesting result was proved.
Theorem 2.31
([28])
The corresponding result in MMS is the following.
Theorem 2.32
We have the next result.
Proof
Omitted. □
One of the most general contraction conditions was given by Ćirić [29] in 1974. He defined and investigated quasicontractions, mappings that unified the Banach, Kannan, Chatterjea, and Bianchini contractions. We announce this theorem in both frameworks.
Theorem 2.34
The expected result within the MMS is presented below.
Theorem 2.35
Proof
Omitted. □
Sehgal in 1969 [30], proved an FP theorem using a condition that involves contractive iterate at each point of the space, where the space is complete and the mapping is continuous. Soon afterwards, Guseman in 1970 [31] generalized the Sehgal results for mappings which are not necessarily continuous.
Theorem 2.37
The corresponding result in the framework of MMS is the following.
Theorem 2.38
Our expected result is the following.
Proof
Omitted. □
In 2001, Rhoades [32] came up with the following idea: the Banach contraction can be interpreted as \(d(fx,fy)\leq d(x,y)\varphi(d(x,y))\). This extension is called the weakly contractive mapping, and in the same paper the following interesting result was proved.
Theorem 2.40
([32])
The analogous result in the framework of MMS is the following.
Theorem 2.41
Finally, we have the next result.
Proof
Omitted. □
For MMS we have the analogous result:
The above result for compact metric spaces is generalized to complete metric spaces by CaristiKirk. It is well known that CaristiKirk’s (or CaristiKirkBrowder’s) theorem is essentially equivalent to the Ekeland variational principle [33]. The original proof of CaristiKirk’s theorem was rather complicated and several new proofs were presented (see, e.g., [34]).
Here we give this famous result.
Theorem 2.43
(Caristi [35], 1976)
Let \(( X,d ) \) be a complete metric space and \(\phi :X\rightarrow[0,+\infty)\) be an l.s.c. function. Then any mapping \(f:X\rightarrow X\) satisfying (18) has an FP in X.
The corresponding result in the framework of MMS is the following.
Theorem 2.44
Now, we have the following result.
Proof
Omitted. □
In 1965, Prešić [36] extended the Banach contraction mapping principle to mappings defined on product spaces and proved the following theorem.
Theorem 2.46
The analogous result in the framework of MMS is the following.
Theorem 2.47
Finally, our result is the following.
Proof
Omitted. □
Remark 2.49
According to [37], new results are possible.
3 Conclusion
4 Further work
Encouraged by this research, the idea for the following definitions emerged.
Definition 4.1
 (1)
\(d_{b}^{*}(x,y)=0\) iff \(x=y\);
 (2)
\(d_{b}^{*}(x,y)=d_{b}^{*}(y,x)\);
 (3)
\(d_{b}^{*}(x,z)\leq[d_{b}^{*}(x,y)\cdot d_{b}^{*}(y,z)]^{s}\).
The pair \((X, d_{b}^{*})\) is called a bMMS.
Definition 4.2
 (1)
\(x=y \Leftrightarrow p^{*}(x, x) =p^{*}(x, y) =p^{*}(y, y)\);
 (2)
\(p^{*}(x, x)\leq p^{*}(x, y)\);
 (3)
\(p^{*}(x, y) =p^{*}(y, x)\);
 (4)
\(p^{*}(x, z)\leq {\frac{p^{*}(x, y) \cdot p^{*}(y, z)}{p^{*}(y, y)}}\).
A partial MMS is a pair \((X, p^{*})\) such that \(X \neq\emptyset\) and \(p^{*}\) is a partial MM on X.
For future work, it would be interesting to research analogous results with wellknown results in bmetric and partial metric spaces, respectively.
Notes
Acknowledgements
The authors are indebted to anonymous referees who helped us to improve this text. The first author is thankful to Ministry of Education, Sciences and Technological Development of Serbia.
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