Open Access Library Journal
Vol.04 No.10(2017), Article ID:80060,8 pages
10.4236/oalib.1103896
Unique Common Fixed Point in b2 Metric Spaces
Jinxing Cui*, Jinwei Zhao*, Linan Zhong#
Department of Mathematics, Yanbian University, Yanji, China
Email: #zhonglinan2000@126.com
Copyright © 2017 by authors and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: August 20, 2017; Accepted: October 28, 2017; Published: October 31, 2017
ABSTRACT
We establish some common fixed and common coincidence point theorems for expansive type mappings in the setting of b2 metric space. Our results extend some known results in metric spaces to b2 metric space. The research is meaningful and I recommend it to be published when the followings have been improved.
Subject Areas:
Mathematical Analysis
Keywords:
b2 Metric Space, Common Fixed Point, Coincidence Point
1. Introduction
The author in (see [1] [2] [3] ) discuss coincidence and fixed point existence problems relating to expansive mappings in cone metric spaces (see [4] [5] ), and also gives fixed point theories for expanding mappings. The author in (see [6] ) gets the coincidence and common fixed point theories in 2 metric spaces (see [7] [8] [9] ), using the method in (see [1] [2] [3] ). In this paper, a known existence theorems of common fixed points for two mappings satisfying expansive conditions in metric space (see [10] ), which is the generalization of both 2 metric space and b metric space (see [11] [12] ).
2. Preliminaries
Before stating our main results, some necessary definitions might be introduced
as follows.
Definition 2.1. [11] [12] Let be a nonempty set and be a given real number. A function is a metric on if for all , the following conditions hold:
1) if and only if .
2) .
3) .
In this case, the pair is called a b metric space.
Definition 2.2. [7] [8] [9] Let be an nonempty set and let 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 .
4) The rectangle inequality: for all .
Then d is called a 2 metric on X and is called a 2 metric space.
Definition 2.3. [10] Let be a nonempty set, be a real number and let : 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 .
4) The rectangle inequality: for all .
Then d is called a metric on X and is called a metric space with parameter s. Obviously, for , metric reduces to 2 metric.
Definition 2.4. [10] Let be a sequence in a 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 b2-complete if every b2-Cauchy sequence is a b2- convergent sequence.
Definition 2.5. [10] Let and be two metric spaces and let be a mapping. Then is said to be b2-continuous at a point if for a given , there exists such that and for all imply that . The mapping is b2-continuous on if it is b2-continuous at all .
Definition 2.6. [10] Let and be two 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 , is b2-convergent to .
Definition 2.7. [13] Let and be self maps of a set . If for some in , then is called a coincidence point of and , and is called a point of coincidence of and . and be weakly compatible means if and , then .
Proposition 2.8. [13] Let and be weakly compatible self maps of a set . If and have a unique point of coincidence , then is the unique common fixed point of and .
3. Main Results
Theorem 3.1. Let be a metric space. Suppose mappings are onto and satisfy
(1)
for all and , where . Suppose the following hypotheses:
1) or is complete,
2) ,
3) .
Then and have a coincidence point.
Proof. From 2), we get or . Indeed, if we suppose and , we have . Since , we have . That is a contradiction.
Let , since , we take such that . Again, we can take such that . Continuing in the same way, we construct two sequences and in such that for all .
If for some , then . Thus is a coincidence point of and .
Now, assume that for all .
Step 1: It is shown that .
Suppose , take , into (1). we have
(2)
Then
(3)
Since , . If , then . If , then . Therefore is constant sequence when . Suppose , then and
(4)
Suppose , take into (1). We have
(5)
Then
(6)
Similarly, since , suppose , then and
(7)
Let , we know , applying (4) and (7), we get
(8)
then .
Step 2: As is decreasing, if , then . Since part 2 of Definition 2.3, , we have for all .
Since , we have
(9)
for all . For , we have , and from(9) we have
(10)
It implies that
(11)
Since , from the above inequality, we have
(12)
for all . From (9) and (12), we have
(13)
for all . Now, for all with , we have
(14)
From (14) and triangular inequality, Therefore
(15)
This proves that for all ,
(16)
Step 3: It is proved that the sequence is a b2-Cauchy sequence. Let with . We claim that, there exists , such that
(17)
for all , . This is done by induction on m.
Let and . Then we get
(18)
Then (17) holds for .
Assume now that (17) holds for some . We will show that (17) holds for . Take ,
(19)
Then
(20)
We get
(21)
Then
(22)
Thus we have proved that (17) holds for . From (17), we know is a Cauchy sequence in .
If is complete, there exists and such that .
If , let , into (1), We have
(23)
Therefore
(24)
We take a natural number such that
, ,
,
for . Thus, we obtain . Therefore .
If , let , into (1), We get
(25)
Therefore
(26)
We take a natural number such that
, ,
,
for . Thus, we obtain . Therefore .
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 is complete, there exists and , such that . The rest proof is the same as that is complete.
Theorem 3.2. Let be a metric space. Let be mappings satisfying and (1), for all . If 1). or is complete, 2). , 3). and is weakly compatible. Then and have a common fixed point.
Proof. According to Theorem 3.1, there exists such that . Suppose there also exists such that , choose , into (1), we get
(27)
Therefore, there exists , then . and have the point of coincidence . According to Proposition 2.8, is the unique common fixed point of and .
Corollary 3.3. Let be a complete metric space. Let be surjective mapping satisfying , for all , , where with , and , then has a fixed point, if , then f has a unique fixed point.
Proof. Follows from Theorem 3.1, by taking , 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 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
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NOTES
*The authors contributed equally.
#Corresponding author.