### Common Fixed Point Theorems for Expansive Type Mappings in b-Metric Spaces

#### Abstract

*Abstract-* In this paper, we prove three common fixed point results for expansive type mappings in b-metric spaces. First part of our main result contains two results on weakly compatible mappings under a given expansive condition. First theorem is a generalizations of main result of Deheria et.al.[7] and a slight variation of Mohanta [13]. Second theorem is a generalization of another fixed point result of Mohanta [13]. For the second section of our main result, we have introduced f-extension condition on b-metric spaces. Suppose a mapping g:X→X doesn’t satisfy an extension condition in itself but it have a fixed point. If another mapping f:X→X is introduced such that f and g satisfy an f-extension condition and provide a common fixed point then this is called f-extension. Below, we have given examples of f-extension as well as illustrative example of our third main result.

#### Full Text:

PDF#### References

M. Abbas and G. Jungck, Common fixed point results for non-commuting mappings without continuity in cone metric spaces, Jour. Math. Anal. Appl., 341(2008), 416-420

S. Banach, Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales., Fund. Math. 3(1922), 133–181

P. Chouhan and N. Malviya, Fixed points of expansive type mappings in 2-banach spaces, International Journal of Analysis and Applications, 3(1)(2013), 60-67, http://www.etamaths.com

S. Cze’rwik, Contraction mapping in b-metric spaces, Acta Math. Inform. Univ. Ostr. 1(1993), 5-11

P.Z. Daffer, H. Kaneko, On expansive mappings, Math. Japonica, 37(1992), 733-735.

R. D. Daheriya, R. Jain and M. Ughade, Some fixed point theorem for expansive type mapping in dislocated metric space, ISRN Mathematical Analysis, Volume 2012, Article ID 376832, 5 pages, doi:10.5402/2012/376832.

R.D. Daheriya, M. R. Likhitker and M. Ughade, Some theorems on fixed points & common fixed points of expansive type mappings in b-metric spaces, Open Journal of Applied & Theor. Math. (OJATM) 2(2)(2016), 65-78, ISSN: 2455-7102

M. Demma and Pasquale Vetro, Picard sequence and fixed point results on b metric spaces, Journal of Function Spaces, Volume-2015, Article ID:189861, 6 pages http://dx.doi.org/ 10.1155/ 2015/ 189861. C3-8

R. Jain, R. D. Daheriya and M. Ughade, Fixed point, coincidence point and common fixed point theorems under various expansive conditions in b-metric spaces, International Journal of Scientific and Innovative Math.Research 3(9)(2015), . 26-34.

G. Jungck, Compatible mappings and common fixed points, Int. Jour. Math. Math. Sci. 9(4)(1986), 771-779

G. Jungck, Common fixed points for non-continuous non-self mappings on non-metric spaces, Far East J. Math. Sci. 4(2)(1996), 199–212

S. Kumar, Common fixed point theorems for expansion mappings in various spaces, Acta. Math. Hunger, 118(1-2)(2008), 9-28

S.K. Mohanta, Coincidence points and common fixed points for expansive type mappings in b-metric spaces, Iranian Journal of Mathematical Sciences & Informat. 11(1)(2016), 101-113, DOI: 10.7508/ijmsi.2016.01.009

S. K. Mohanta and R.Maitra, Coincidence points and common fixed points for expansive type mappings in cone b-metric spaces, Applied Mathematics E-Notes 14(2014), 200-208.

A. Muraliraj and R.J. Hussain, Coincidence and fixed point theorems for expansive maps in d−metric spaces, Internat. Journal of Mathematical Analysis 7(44)(2013), 2171 – 2179.

Z. Mustafa, F. Awawdeh and W. Shatanawi, Fixed point theorem for expansive mappings in G-metric spaces, International Journal of Contemporary Mathematical Sciences 5(50)(2010), 2463-2472.

P. Nagpal, Shin Min Kang, Sudhir Kumar Garg and Sanjay Kumar, Several fixed point theorems for expansive mappings in multiplicative metric spaces, International Journal of Pure and Applied Mathematics 107(2)(2016), 357-369, url: http://www.ijpam.eu, doi: 10.12732/ijpam.v107i2.6

M. Öztürk, N. Kaplan, Common fixed points of f-contraction mappings in complex valued metric spaces, Math. Sci. 8(2014)129, doi: 10.1007/s40096-014-0129-2.

H. Piri and H. Afshari, Some fixed point theorems in complete partial b-metric spaces, Adv. Fixed Point Theory, 4(3)(2014), 444-461, http://scik.org

B.E. Rhoades, An expansion mapping theorem, Jnanabha 23(1993), 151- 152.

Rashmi Jain, R. D. Daheriya and Manoj Ughade, Fixed point and common fixed point results for expansive mappings in Gb - cone metric spaces, Journal of Global Research in Mathematical Archives, 2(11)(2014), 1-11, http://www.jgrma.info

A. S. Saluja, A. K., Dhakde, and D. Magarde, Some fixed point theorems for expansive type mapping in dislocated metric space, Mathematical Theorey and Modeling, 3(2013).

I. Sahin and Mustafa Telcia, Theorem on common fixed points of expansion type mappings in cone metric spaces, An. S¸T. Univ. Ovidius Constanta, 18(1)(2010), 329–336.

S. Sessa, On a weak commutativity conditions of mappings in fixed point consideration, Publ. Inst. Math. Beograd, 32(46)(1982), 146-153.

W. Shatanawi and F. Awawdeh, Some fixed and coincidence point theorems for expansive maps in cone metric spaces, Fixed Point Theory and Applications 2012, 2012:19.

S.Z. Wang, B.Y. Li, Z.M. Gao, K. Iseki, Some fixed point theorems for expansion mappings, Math. Japonica, 29(1984), 631-636.

Xianjiu Huang, Chuanxi Zhu, Xi Wen, Fixed point theorems for expanding mappings in partial metric spaces, An. St. Univ. Ovidius Constant 20(1)(2012), 213-224.

Yan Han and Shaoyuan Xu, Some new theorems of expanding mappings without continuity in cone metric spaces, Fixed Point Theory and Applications, 2013, 2013:3

R.Jain, R.D. Daheriya and M. Ughade, Fixed point, coincidence point and common fixed point theorems under various expansive conditions in parametric metric spaces and parametric b-metric spaces, Gazi University Journal of Science, 29(1)(2016), 95-107.

### Refbacks

- There are currently no refbacks.

**MAYFEB Journal of Mathematics**

**Toronto, Ontario, Canada**

**MAYFEB TECHNOLOGY DEVELOPMENT**

**ISSN**

**2371-6193**