OALibJOpen Access Library Journal2333-9705Scientific Research Publishing10.4236/oalib.1103896OALibJ-80060ArticlesBiomedical&Life Sciences Business&Economics Chemistry&Materials Science Computer Science&Communications Earth&Environmental Sciences Engineering Medicine&Healthcare Physics&Mathematics Social Sciences&Humanities Unique Common Fixed Point in <i>b</i><sub>2</sub> Metric Spaces JinxingCui1JinweiZhao1LinanZhong1*Department of Mathematics, Yanbian University, Yanji, China* E-mail:lnzhong@ybu.edu.cn(LZ);1110201704101820, August 201728, October 2017 31, October 2017© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

We establish some common fixed and common coincidence point theorems for expansive type mappings in the setting of b 2 metric space. Our results extend some known results in metric spaces to b 2 metric space. The re search is meaningful and I recommend it to be published when the followings have been improved.

<i>b</i><sub>2</sub> Metric Space Common Fixed Point Coincidence Point
1. Introduction

The author in (see    ) discuss coincidence and fixed point existence problems relating to expansive mappings in cone metric spaces (see   ), and also gives fixed point theories for expanding mappings. The author in (see  ) gets the coincidence and common fixed point theories in 2 metric spaces (see    ), using the method in (see    ). In this paper, a known existence theorems of common fixed points for two mappings satisfying expansive conditions in b 2 metric space (see  ), which is the generalization of both 2 metric space and b metric space (see   ).

2. Preliminaries

Before stating our main results, some necessary definitions might be introduced

as follows.

Definition 2.1.   Let X be a nonempty set and s ≥ 1 be a given real number. A function d : X × X → ℝ + is a b metric on X if for all x , y , z ∈ X , the following conditions hold:

1) d ( x , y ) = 0 if and only if x = y .

2) d ( x , y ) = d ( y , x ) .

3) d ( x , y ) ≤ s [ d ( x , y ) + d ( y , z ) ] .

In this case, the pair ( X , d ) is called a b metric space.

Definition 2.2.    Let X be an nonempty set and let d : X × X × X → ℝ be a map satisfying the following conditions:

1) For every pair of distinct points x , y ∈ X , there exists a point z ∈ X such that d ( x , y , z ) ≠ 0 .

2)If at least two of three points x , y , z are the same, then d ( x , y , z ) = 0 .

3) The symmetry:

d ( x , y , z ) = d ( x , z , y ) = d ( y , x , z ) = d ( y , z , x ) = d ( z , x , y ) = d ( z , y , x )

for all x , y , z ∈ X .

4) The rectangle inequality: d ( x , y , z ) ≤ d ( x , y , a ) + d ( y , z , a ) + d ( z , x , a ) for all x , y , z , a ∈ X .

Then d is called a 2 metric on X and ( X , d ) is called a 2 metric space.

Definition 2.3.  Let X be a nonempty set, s ≥ 1 be a real number and let d : X × X × X → ℝ be a map satisfying the following conditions:

1) For every pair of distinct points x , y ∈ X , there exists a point z ∈ X such that d ( x , y , z ) ≠ 0 .

2) If at least two of three points x , y , z are the same, then d ( x , y , z ) = 0 .

3) The symmetry:

d ( x , y , z ) = d ( x , z , y ) = d ( y , x , z ) = d ( y , z , x ) = d ( z , x , y ) = d ( z , y , x )

for all x , y , z ∈ X .

4) The rectangle inequality: d ( x , y , z ) ≤ s [ d ( x , y , a ) + d ( y , z , a ) + d ( z , x , a ) ] for all x , y , z , a ∈ X .

Then d is called a b 2 metric on X and ( X , d ) is called a b 2 metric space with parameter s. Obviously, for s = 1 , b 2 metric reduces to 2 metric.

Definition 2.4.  Let { x n } be a sequence in a b 2 metric space ( X , d ) .

1) A sequence { x n } is said to be b2-convergent to x ∈ X , written as lim n → ∞ x n = x , if all a ∈ X , lim n → ∞ d ( x n , x , a ) = 0 .

2) { x n } is Cauchy sequence if and only if d ( x n , x m , a ) → 0 , when n , m → ∞ . for all a ∈ X .

3) ( X , d ) is said to be b2-complete if every b2-Cauchy sequence is a b2- convergent sequence.

Definition 2.5.  Let ( X , d ) and ( X ′ , d ′ ) be two b 2 metric spaces and let f : X → X ′ be a mapping. Then f is said to be b2-continuous at a point z ∈ X if for a given ε > 0 , there exists δ > 0 such that x ∈ X and d ( z , x , a ) < δ for all a ∈ X imply that d ′ ( f z , f x , a ) < ε . The mapping f is b2-continuous on X if it is b2-continuous at all z ∈ X .

Definition 2.6.  Let ( X , d ) and ( X ′ , d ′ ) be two b 2 metric spaces. Then a mapping f : X → X ′ is b2-continuous at a point x ∈ X ′ if and only if it is b2-sequentially continuous at x; that is,whenever { x n } is b2-convergent to x , { f x n } is b2-convergent to f ( x ) .

Definition 2.7.  Let f and g be self maps of a set X . If w = f x = g x for some x in X , then x is called a coincidence point of f and g , and w is called a point of coincidence of f and g . f and g be weakly compatible means if x ∈ X and f x = g x , then f g x = g f x .

Proposition 2.8.  Let f and g be weakly compatible self maps of a set X . If f and g have a unique point of coincidence w = f x = g x , then w is the unique common fixed point of f and g .

3. Main Results

Theorem 3.1. Let ( X , d ) be a b 2 metric space. Suppose mappings f , g : X → X are onto and satisfy

d ( f x , f y , a ) ≥ α d ( g x , f x , a ) + β d ( g y , f y , a ) + γ d ( g x , g y , a ) (1)

for all x , y , a ∈ X and x ≠ y , where α , β ∈ ℝ , γ > 0 . Suppose the following hypotheses:

1) f X or g X is complete,

2) α + β + γ > 2 ,

3) g X ⊂ f X .

Then f and g have a coincidence point.

Proof. From 2), we get α + γ > 0 or β + γ > 0 . Indeed, if we suppose α + γ ≤ 0 and β + γ ≤ 0 , we have α + β + 2 γ ≤ 0 . Since γ ≥ 0 , we have α + β + γ ≤ 0 . That is a contradiction.

Let x 0 ∈ X , since g X ⊂ f X , we take x 1 ∈ X such that f x 1 = g x 0 . Again, we can take x 2 ∈ X such that f x 2 = g x 1 . Continuing in the same way, we construct two sequences { x n } and { y n } in X such that y n = f x n + 1 = g x n for all n ∈ ℕ .

If g x m − 1 = g x m for some m ∈ ℕ , then f x m = g x m . Thus x m is a coincidence point of f and g .

Now, assume that x n − 1 ≠ x n for all n ∈ ℕ .

Step 1: It is shown that lim n → ∞ d ( y n + 1 , y n + 2 , a ) = 0 .

Suppose β + γ > 0 , take x = x n + 1 , y = x n + 2 into (1). we have

d ( y n , y n + 1 , a ) ≥ α d ( y n , y n + 1 , a ) + ( β + γ ) d ( y n + 1 , y n + 2 , a ) (2)

Then

( 1 + α ) d ( y n , y n + 1 , a ) ≥ ( β + γ ) d ( y n + 1 , y n + 2 , a ) ,     for   all   n ∈ ℕ (3)

Since β + γ > 0 , ( 1 − α ) d ( y n , y n + 1 , a ) ≥ 0 . If 1 − α < 0 , then d ( y n , y n + 1 , a ) = 0 . If 1 − α = 0 , then d ( y n + 1 , y n + 2 , a ) = 0 . Therefore { y n } is constant sequence when 1 ≤ α . Suppose 1 − α > 0 , then 0 < 1 − α β + γ < 1 and

d ( y n + 1 , y n + 2 , a ) ≤ 1 − α β + γ d ( y n , y n + 1 , a ) (4)

Suppose α + γ > 0 , take x = x n + 2 , y = x n + 1 into (1). We have

d ( y n , y n + 1 , a ) ≥ ( α + γ ) d ( y n + 1 , y n + 2 , a ) + β d ( y n , y n + 1 , a ) (5)

Then

( 1 − β ) d ( y n , y n + 1 , a ) ≥ ( α + γ ) d ( y n + 1 , y n + 2 , a ) ,     for   all   n ∈ ℕ (6)

Similarly, since α + γ > 0 , suppose 1 − β > 0 , then 0 < 1 − β α + γ < 1 and

d ( y n + 1 , y n + 2 , a ) ≤ 1 − β α + γ d ( y n , y n + 1 , a ) (7)

Let h = max { 1 − α β + γ , 1 − β α + γ } , we know 0 < h < 1 , applying (4) and (7), we get

d ( y n + 1 , y n + 2 , a ) ≤ h d ( y n , y n + 1 , a ) ≤ ⋯ ≤ h n + 1 d ( y 0 , y 1 , a ) ,   n = 0,1,2, ⋯ (8)

then lim n → ∞ d ( y n + 1 , y n + 2 , a ) = 0 .

Step 2: As { d ( y n , y n + 1 , a ) } is decreasing, if d ( y n − 1 , y n , a ) = 0 , then d ( y n , y n + 1 , a ) = 0 . Since part 2 of Definition 2.3, d ( y 0 , y 1 , y 0 ) = 0 , we have d ( y n , y n + 1 , y 0 ) = 0 for all n ∈ ℕ .

Since d ( y m − 1 , y m , y m ) = 0 , we have

d ( y n , y n + 1 , y m ) = 0 (9)

for all n ≥ m − 1 . For 0 ≤ n < m − 1 , we have m − 1 ≥ n + 1 , and from(9) we have

d ( y m − 1 , y m , y n + 1 ) = d ( y m − 1 , y m , y n ) = 0 (10)

It implies that

d ( y n , y n + 1 , y m ) ≤ s d ( y n , y n + 1 , y m − 1 ) + s d ( y n + 1 , y m , y m − 1 ) + s d ( y m , y n , y m − 1 ) = s d ( y n , y n + 1 , y m − 1 ) (11)

Since d ( y n , y n + 1 , y n + 1 ) = 0 , from the above inequality, we have

d ( y n , y n + 1 , y m ) ≤ s m − n − 1 d ( y n , y n + 1 , y n + 1 ) = 0 (12)

for all 0 ≤ n ≤ m − 1 . From (9) and (12), we have

d ( y n , y n + 1 , y m ) = 0 (13)

for all n , m ∈ ℕ . Now, for all i , j , k ∈ ℕ with i < j , we have

d ( y j − 1 , y j , y i ) = d ( y j − 1 , y j , y k ) = 0 (14)

From (14) and triangular inequality, Therefore

d ( y i , y j , y k ) ≤ s [ d ( y i , y j , y j − 1 ) + d ( y j , y k , y j − 1 ) + d ( y k , y i , y j − 1 ) ] = s d ( y i , y j − 1 , y k ) ≤ ⋯ ≤ s j − i d ( y i , y i , y k ) = 0 (15)

This proves that for all i , j , k ∈ ℕ ,

d ( y i , y j , y k ) = 0 (16)

Step 3: It is proved that the sequence { y n } is a b2-Cauchy sequence. Let m , n ∈ ℕ with m > n . We claim that, there exists n 0 ∈ ℕ , such that

d ( y n , y m , a ) < ε (17)

for all m > n ≥ n 0 , a ∈ X . This is done by induction on m.

Let n ≥ n 0 and m = n + 1 . Then we get

d ( y n , y m , a ) = d ( y n , y n + 1 , a ) < d ( y n − 1 , y n , a ) < ε (18)

Then (17) holds for m = n + 1 .

Assume now that (17) holds for some m ≥ n + 1 . We will show that (17) holds for m + 1 . Take x = x n , y = x m + 1

d ( y n − 1 , y m , a ) = d ( f x n , f x m + 1 , a ) ≥ α d ( g x n , f x n , a ) + β d ( g x m + 1 , f x m + 1 , a ) + γ d ( g x n , g x m + 1 , a ) ≥ α d ( y n , y n − 1 , a ) + β d ( y m + 1 , y m , a ) + γ d ( y n , y m + 1 , a ) (19)

Then

s [ d ( y n − 1 , y n , y m ) + d ( y n − 1 , y n , a ) + d ( y n , y m , a ) ] ≥ α d ( y n , y n − 1 , a ) + β d ( y m + 1 , y m , a ) + γ d ( y n , y m + 1 , a ) (20)

We get

γ d ( y n , y m + 1 , a ) ≤ s [ d ( y n − 1 , y n , y m ) + d ( y n − 1 , y n , a ) + d ( y n , y m , a ) ]     − α d ( y n , y n − 1 , a ) − β d ( y m + 1 , y m , a ) ≤ 2 s ε − ( α + β ) ε (21)

Then

d ( y n , y m + 1 , a ) ≤ 2 s ε − ( α + β ) ε γ = 2 s − α − β γ ε < ε (22)

Thus we have proved that (17) holds for m + 1 . From (17), we know { y n } is a Cauchy sequence in ( X , d ) .

If f X is complete, there exists u ∈ f X and p ∈ X such that y n = g x n = f x n + 1 → u = f p .

If β + γ > 0 , let x = x n + 1 , y = p into (1), We have

d ( y n , u , a ) ≥ α d ( y n , y n + 1 , a ) + β d ( g p , u , a ) + γ d ( y n + 1 , g p , a ) ≥ α d ( y n , y n + 1 , a ) + β d ( g p , u , a )     + γ [ d ( u , g p , a ) s − d ( y n + 1 , u , a ) − d ( y n + 1 , u , g p ) ] (23)

Therefore

( β + γ s ) d ( u , g p , a ) ≤ d ( y n , u , a ) + | α | d ( y n , y n + 1 , a )     + γ d ( y n + 1 , u , a ) + γ d ( y n + 1 , u , g p ) (24)

We take a natural number n 1 such that

d ( y n , u , a ) ≤ β + γ s 4 ε , | α | d ( y n , y n + 1 , a ) ≤ β + γ s 4 ε ,

d ( y n + 1 , u , a ) ≤ β + γ s 4 ε , d ( y n + 1 , u , g p ) ≤ β + γ s 4 ε

for n ≥ n 1 . Thus, we obtain d ( u , g p , a ) ≤ ε . Therefore f p = u = g p .

If α + γ > 0 , let x = p , y = x n + 1 into (1), We get

d ( y n , u , a ) ≥ α d ( g p , u , a ) + β d ( y n , y n + 1 , a ) + γ d ( y n + 1 , g p , a ) (25)

Therefore

( α + γ s ) d ( u , g p , a ) ≤ d ( y n , u , a ) + | β | d ( y n , y n + 1 , a )     + γ d ( y n + 1 , u , a ) + γ d ( y n + 1 , u , g p ) (26)

We take a natural number n 2 such that

d ( y n , u , a ) ≤ α + γ s 4 ε , | β | d ( y n , y n + 1 , a ) ≤ α + γ s 4 ε ,

d ( y n + 1 , u , a ) ≤ α + γ s 4 ε , d ( y n + 1 , u , g p ) ≤ α + γ s 4 ε

for n ≥ n 2 . Thus, we obtain d ( u , g p , a ) ≤ ε . Therefore f p = u = g p .

In short, no matter what the situation is, u is always the point of coincidence of f and g, p is the coincidence point of f and g.

If g X is complete, there exists u ∈ g X ⊂ f X and p , q ∈ X , such that y n = g x n → u = g q = f p . The rest proof is the same as that f X is complete.

Theorem 3.2. Let ( X , d ) be a b 2 metric space. Let f , g be mappings satisfying f X ⊃ g X and (1), for all α , β ∈ ℝ , γ > 1 . If 1). f X or g X is complete, 2). α + β + γ > 2 , 3). f and g is weakly compatible. Then f and g have a common fixed point.

Proof. According to Theorem 3.1, there exists u , p ∈ X such that u = f p = g p . Suppose there also exists v , z ∈ X such that v = f z = g z , choose x = p , y = z into (1), we get

d ( u , v , a ) = d ( f p , f z , a ) ≥ γ d ( g p , g z , a ) = γ d ( u , v , a ) (27)

Therefore, there exists d ( u , v , a ) = 0 , then u = v . f and g have the point of coincidence u . According to Proposition 2.8, u is the unique common fixed point of f and g .

Corollary 3.3. Let ( X , d ) be a complete b 2 metric space. Let f be surjective mapping satisfying d ( f x , f y , a ) ≥ α d ( x , f x , a ) + β d ( y , f y , a ) + γ d ( x , y , a ) , for all x , y , a ∈ X , x ≠ y , where with α , β ∈ ℝ , γ > 0 and α + β + γ > 2 , then f has a fixed point, if γ > 0 , then f has a unique fixed point.

Proof. Follows from Theorem 3.1, by taking g = 1 x , identify map, then we get the result.

4. Conclusion

In this paper, a known existence theorems of common fixed points for two mappings satisfying expansive conditions in b 2 metric space were generalized and improved. Based on the research, a new method to discuss the existence problems of common fixed points for mappings with this type expansive condition was taken out. And the results show that the proposed method is better than the former ones.

Fund

This project is supported by NSFC (grant No. 11761072, 11261062) and Research Fund for the Doctoral Program of Higher Education of China (grant No. 20114407120011).

Cite this paper

Cui, J.X., Zhao, J.W. and Zhong, L.N. (2017) Unique Com- mon Fixed Point in b2 Metric Spaces. Open Access Library Journal, 4: e3896. https://doi.org/10.4236/oalib.1103896

NOTESReferencesSahin, I. and Telci, M. (2010) A Theorem on Common Fixed Points of Expansion Type Mappings in Cone Metric Spaces. Analele Stiintifice ale Universitatii Ovidius Constanta, 18, 329-336.Kadelburg, Z., Murthy, P.P. and Radenovic, S. (2011) Common Fixed Points for Expansive Mappings in Cone Metric Spaces. International Journal of Mathematical Analysis, 5, 1309-1319.Shatanawi, W. and Awawde, F. (2012) Some Fixed and Coincidence Point Theorems for Expansive Maps in Cone Metric Spaces. Fixed Point Theory and Applications.Huang, L.G. and Zhang, X. (2007) Cone Metric Spaces and Fixed Point Theorems of Contractive Mappings. Journal of Mathematical Analysis and Applications, 332, 1468-1476. https://doi.org/10.1016/j.jmaa.2005.03.087Zhang, X. (2010) Common Fixed Point Theorem of Lipschitz Type Mappings on Cone Metric Space. Acta Mathematica Sinica (Chinese Series), 53A, 1139-1148.Piao Y.J. ,et al. (2013)Common Fixed Points for Two Mappings Satisfying Some Expansive Conditions on 2-Metric Spaces Journal of Systems Science and Mathematical 33, 1370-1379.Piao, Y.J. (2008) Unique Common Fixed Point for a Family of Self-Maps with Same Type Contractive Condition in 2-Metric Spaces. Analysis in Theory and Applications, 24, 316-320. https://doi.org/10.1007/s10496-008-0316-9Piao, Y.J. (2012) Uniqueness of Common Fixed Point for a Family of Mappings with ?-Contractive Condition in 2-Metricspaces. Applied Mathematics, 3, 73-77. https://doi.org/10.4236/am.2012.31012 Piao, Y.J. and Jin, Y.F. (2012) New Unique Common Fixed Results for Four Mappings with-Contractive Type in 2-Metric Spaces. Applied Mathematics, 3, 734-737. https://doi.org/10.4236/am.2012.37108Mustafa, Z., Parvaech, V., Roshan, J.R. and Kadelburg, Z. (2014) b2-Metric Spaces and Some Fixed Point Theorems. Fixed Point Theory and Applications, 144. https://doi.org/10.1186/1687-1812-2014-144Czerwik S. ,et al. (1993)Contraction Mappings in b-Metric Spaces Acta Mathematica et Informatica Universitatis Ostraviensis 1, 5-11.Czerwik S. ,et al. (1998)Nonlinear Set-Valued Contraction Mappings in b-Metric Spaces. Atti del Seminario Matematico e Fisico (Dell’Univ di Modena) 46, 263-276.Jungck, G. and Rhoades, B.E. (1998) Fixed Points for Set Valued Functions without Continuity. Indian Journal of Pure and Applied Mathematics, 29, 771-779.