Remarks on Multiplicative Metric Spaces and Related Fixed Points
aa r X i v : . [ m a t h . GN ] D ec Remarks on Multiplicative Metric Spaces and RelatedFixed Points
K. Abodayeh A. Pitea W. Shatanawi ∗ ,T. Abdeljawad , , Department of Mathematics and General Sciences
Prince Sultan University
Riyadh, Saudi Arabia 11586E-mails: [email protected], [email protected], [email protected] of Applied Sciences Department of Mathematics and Informatics
University Politehnica of Bucharest
Bucharest, RomaniaE-mail:[email protected]
Abstract
In this article we studied the relationship between metric spaces and multiplicativemetric spaces. Also, we pointed out some fixed and common fixed point results undersome contractive conditions in multiplicative metric spaces can be obtained from thecorresponding results in standard metric spaces.
The notion of multiplicative metric space was introduced by Bashirov et al. [4]. In 2012,Ozavsar and Cervikel [6] defined the notion of convergence in multiplicative metric spacesand studied some fixed point results in such space. After that, many researchers considerthis space and many results on fixed point theory were considered.We begin by introducing the definition of multiplicative metric space.
Definition 1. [4] Let X be a nonempty set and p : X × X −→ [1 , + ∞ ) . We say that ( X, p ) is a multiplicative metric space if for all x, y, z ∈ X we have:1. p ( x, y ) ≥ and x = y if and only if p ( x, y ) = 1; ∗ (Permanent address) Hashemite University, Zarqa-Jordan, Email:swasfi@hu.edu.jo . p ( x, y ) = p ( y, x ); p ( x, z ) ≤ p ( x, y ) p ( y, z ) . The definition of a multiplicative Cauchy sequence in a multiplicative metric space isgiven as follows:
Definition 2. [6] A sequence { x n } in a multiplicative metric space ( X, p ) is said to bemultiplicative Cauchy sequence if for all ǫ > , there exists N ∈ N such that p ( x n , x m ) < ǫ for all m, n ≥ N . Also, if every multiplicative Cauchy sequence is convergent, then ( X, p ) iscalled a complete multiplicative metric space. For the definitions of open balls and convergence in multiplicative metric spaces, we referthe reader to [6].Ozavsar and Cervikel [6] introduced the concept of multiplicative contraction and provedthat every multiplicative contraction in a complete multiplicative metric space has a uniquefixed point.
Definition 3. [6] Let ( X, p ) be a multiplicative metric space. A mapping f : X → X is calledmultiplicative contraction if there exists a real number λ ∈ [0 , such that p ( f ( x ) , f ( x )) ≤ p ( x , x ) λ for all x , x ∈ X . Theorem 1. [6] Let ( X, p ) be a complete multiplicative metric space and let f : X → X bea multiplicative contraction. Then f has a unique fixed point. The notion of weakly commuting mappings was introduced by Sessa [3] in 1982. While,Jungck [2] initiated the concept of weakly compatible mappings in 1996 as a generalizationof the notion of weakly commuting mappings.Moreover, many authors studied many fixed point theorems for weakly commuting mappingsin metric spaces. See [2], [10], [11], [12], [13], and [14].
Definition 4. [2] Let A and S be self-mappings on a metric space ( X, d ) . Then, A and S are said to be weakly compatible if they commute at their coincident point; that is, Ax = Sx for some x ∈ X implies ASx = SAx . Definition 5. [3] Let S and T be two self-mappings of a metric space ( X, d ) . Then S and T are said to be weak commutative mappings if d ( ST x, T Sx ) ≤ d ( Sx, T x ) , for all x ∈ X . It is clear that if S and T are weak commutative mappings, then S and T are weaklycompatible.He et al. [5] employed the concept of weakly commutative mappings to introduce andprove the following common fixed point theorem in multiplicative metric spaces. Theorem 2. [5] Let ( X, p ) be a complete multiplicative metric space. Suppose that A, B, S and T are four self-mappings of X satisfying the following conditions: . T ( X ) ⊆ A ( X ) and S ( X ) ⊆ B ( X ) ;2. The pairs ( S, A ) and ( T, B ) are weakly commutative;3. One of A, B, S and T is continuous;4. p ( Sx, T y ) ≤ { max { p ( Ax, By ) , p ( Ax, Sx ) , p ( By, T y ) , p ( Ax, T y ) , p ( By, Sx ) }} λ , λ ∈ (0 ,
12 ) . (1) Then A, B, S and T have a unique common fixed point.
In this paper, we study the relationship between the multiplicative metric space andthe standard metric space. Also, we show that the proof of Theorem 1 and Theorem 2 areobtained from the corresponding results in standard metric spaces.
We start by giving the relationship between the multiplicative metric space and the standardmetric space.If we have a multiplicative metric space (
X, p ), then the corresponding metric space(
X, d p ) is given by the following theorem. Theorem 3.
Let ( X, ρ ) be a multiplicative metric space. Define d p : X × X → [0 , + ∞ ) by d p ( x, y ) = ln( p ( x, y )) . Then ( X, d p ) is a metric space. Proof.
Follows from the properties of logarithms. (cid:3)
Moreover, if we have a metric space (
X, d ), then the corresponding multiplicative metricspace (
X, p d ) is given by the following theorem. Theorem 4.
Let ( X, d ) be a metric space. Define p d : X × X → [0 , + ∞ ) by p d ( x, y ) = e d ( x,y ) . Then ( X, p d ) is a multiplicative metric space. Proof.
The proof follows from the properties of exponential functions. (cid:3)
Now, we can transfer and prove many applications considered on multiplicative metricspace to a standard metric space and use their proof in the metric space case.For instance, considering the definition of contraction on multiplicative metric space, theproof the following result is a straightforward.3 heorem 5.
A sequence { x n } is a multiplicative Cauchy sequence in a multiplicative metricspace ( X, p ) if and only if { x n } is a Cauchy sequence in the corresponding metric space ( X, d p ) . Applying the logarithmic function to the multiplicative contraction inequality that havebeen defined in Definition 3, will give us the inequality d p ( f ( x ) , f ( x )) = ln p ( f ( x ) , f ( x )) ≤ λd p ( x , x ) . Note that if (
X, p ) is a complete multiplicative metric space, then the correspondingmetric space (
X, d p ) is also a complete metric space.It is obvious we can get the regular contraction inequality which was introduce by Ba-nach. Therefore, one can prove the result in Theorem 1 using the new metric space ( X, d p )and the Banach contraction theorem.We have furnished all the necessary backgrounds to present the proof of Theorem 1 from thestandard metric space. Proof of Theorem 1:
Since (
X, p ) is a complete multiplicative metric space, the corresponding metric space(
X, d p ) is a complete metric space. Also, since f is a multiplicative contraction, it is a con-traction in ( X, d p ). Therefore it satisfies the Banach contraction conditions and thus it hasa unique fixed point. (cid:3) Now, we furnish all the necessary backgrounds to prove Theorem 2 from the correspond-ing standard metric space.Recall the following definition.
Definition 6. [9] Let X be a nonempty set and let d : X × X −→ [0 , + ∞ ) be a functionsatisfying the following conditions:1. d ( x, y ) = d ( y, x ) .2. If d ( x, y ) = 0 then we have x = y .3. d ( x, y ) ≤ d ( x, z ) + d ( z, y ) for all x, y, z ∈ X .Then the pair ( X, d ) is called the d-metric space. It also appeared under the name of metric-like space [15]. It is clear that every metric space is a d-metric space.4 efinition 7. [9] A sequence { x n } in a d-metric space ( X, d ) is called a Cauchy sequence iffor given ǫ > , there exists N ∈ N such that d ( x n , x m ) < ǫ for all m, n ≥ N .Also, ( X, d ) is called complete if every Cauchy sequence in it is convergent. Definition 8.
A function φ : [0 , ∞ ) → [0 , ∞ ) is said to be contractive modulus if φ ( t ) < t for t > . Definition 9.
A real valued function φ defined on X ⊂ R is said to be upper semicontinuousif lim n →∞ φ ( t n ) ≤ φ ( t ) , for every sequence { t n } with lim n →∞ t n = t . Panthi et al.[1] introduced and proved the following result.
Theorem 6. [1] Let ( X, d ) be a complete d-metric space. Suppose that A, B, S and T arefour self mappings of X satisfying the following conditions:1. T ( X ) ⊆ A ( X ) and S ( X ) ⊆ B ( X ) ,2. d ( Sx, T y ) ≤ φ ( m ( x, y )) where φ is an upper semicontinuous contractive modulus and m ( x, y ) ≤ max { d ( Ax, By ) , d ( Ax, Sx ) , d ( By, T y ) , d ( Ax, T y ) , d ( By, Sx ) }
3. The pairs ( S, A ) and ( T, B ) are weakly compatible.Then A, B, S and T have a unique common fixed point. Now, we utilize Theorem 6 to introduce and prove the following result.
Corollary 2.1.
Let ( X, d ) be a complete metric space. Suppose that A, B, S and T are fourself mappings of X satisfying the following conditions:1. T ( X ) ⊆ A ( X ) and S ( X ) ⊆ B ( X ) ,2. suppose there exists λ ∈ [0 , such that d ( Sx, T y ) ≤ λ max { d ( Ax, By ) , d ( Ax, Sx ) , d ( By, T y ) , d ( Ax, T y ) , d ( By, Sx ) }
3. The pairs ( S, A ) and ( T, B ) are weakly compatible.Then A, B, S and T have a unique common fixed point. Proof.
Define φ : R −→ R by φ ( t ) = λt, where 0 < λ <
1. Then1. φ is a contractive modulus, and 5. φ is an upper semicontinuous on R .From Theorem 6, we get the result. (cid:3) Now, we are ready to present the proof of Theorem 2 from the corresponding standardmetric space.
Proof of Theorem 2:
Take ln to both side in Inequality 1 in Theorem 2, we get d p ( Sx, T y ) ≤ λ max { d p ( Ax, By ) , d p ( Ax, Sx ) , d p ( By, T y ) , d p ( Ax, T y ) , d p ( By, Sx ) } , where λ ∈ (0 , ) which is a special case of the contractive condition in Corollary 2.1. Sinceevery pair of weakly commutative mappings is weakly compatible. All the hypotheses ofCorollary 2.1 hold. Thus the four mappings A, B, S and T have a unique common fixedpoint. (cid:3) Conclusion:
From our discussions, we note that some fixed and common fixed point theorems in mul-tiplicative metric spaces can be deduced from the corresponding standard metric spaces.Moreover, we can formulate many fixed and common fixed point theorems in multiplicativemetric spaces from the corresponding results in standard metric spaces. So the researchersmust be careful in working in multiplicative metric spaces.
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