Some Common Fixed Point Theorems in Fuzzy Metric Spaces Using the CLRg Property

0*, Shailesh Dhar Diwan**
* Senior Assistant Professor, Department of Applied Mathematics, SSIPMT, Raipur, India.
** Associate Professor, Department of Mathematics, Government P. G. College, Dhamtari, India.
Periodicity:April - June'2016
DOI : https://doi.org/10.26634/jmat.5.2.6007

Abstract

In this paper, the authors have established some common fixed point theorems in fuzzy metric spaces using the Common Limit in the Range (i.e, (CLRg )) property. Since, CLRg property does not require condition of closeness of range and so the results extend, generalize and improve several known results of metric spaces and fuzzy metric space in several ways. The obtained results show that the completeness of space and continuity of mappings are not required. In the case of CLRg property, Containment of ranges of involved mappings and the closeness of subspace are not required. As an application to the main result, the authors present some fixed point theorems for self mappings in fuzzy metric space by using the notion of (EA) property. The (EA) property replaces the completeness requirement of the space with a more natural condition of closeness of the range. The (EA) property also relaxes the continuity of one or more mappings and containment of the range of one mapping into the range of another, which can be used to construct the sequences of some iterates. Some examples are furnished in the paper to support the validity of the results.

Keywords

Fuzzy Metric Space, t-norm, Weakly Compatible Mappings, (CLRg) Property, (EA) Property.

How to Cite this Article?

Sharma,P., and Diwan,S.D. (2016). Some Common Fixed Point Theorems in Fuzzy Metric Spaces Using the CLRg Property. i-manager’s Journal on Mathematics, 5(2), 45-53. https://doi.org/10.26634/jmat.5.2.6007

References

[1]. Aamri M. and Moutawakil D. E., (2002).“Some New Common Fixed Point Theorems Under Strict Contractive Conditions”. Journal of Mathematical Analysis and Applications, Vol.270, No.1, pp.181-188.
[2]. Alamgir K.M. and Sumitra, (2012).“CLRg Property for Coupled Fixed Point Theorems in Fuzzy Metric Spaces”. International Journal of Applied Physics and Mathematics, Vol.2, No.5, pp.355-358.
[3]. Ali J., Imdad M. and Bahuguna D, (2010).“Common Fixed Point Theorems in Menger Spaces with Common Property (EA)”.Computer and Mathematics with Application, Vol.60, No.12, pp.3152-3159.
[4]. Asati A., Singh A. and Asati M., (2014).“Common Fixed Point Theorems in Fuzzy Metric Spaces Using (JCLR) Property”. Journal of Computational and Applied Mathematical Science, Vol.5, No.3, pp.288-293.
[5]. Badshah V.H, Chauhan M.S and Sharma D., “Common Fixed Point Theorems for Compatible Mappings”.International Journal of Theoretical and Applied Sciences, Vol.1, No.2, pp.79-82.
[6]. Chauhan S., Bhatnagar S. and Radenovi C., (2013).“Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces”. Le Matematiche, Vol.Lxviii , No.1, pp.87-98.
[7]. Chauhan S., Khan M.A., Kumar S., (2013). “Unified Fixed Point Theorems in Fuzzy Metric Spaces via Common Limit Range Property”.Journal of Inequalities and Applications, pp.182.
[8]. Chugh, Renu and Sanjay kumar, (2001).“Common Fixed Points for Weakly Compatible Maps”.Proceedings of the Indian Academy of Sciences, Vol.111, No.2, pp.241-247.
[9]. Fang J.X., (1992). “On Fixed Point Theorems in Fuzzy Metric Spaces”.Fuzzy Sets and Systems, Vol.46, pp.107-113.
[10]. Francisco A., RoldanL.Sintunavarat W.,(2016). “Common Fixed Point Theorems in Fuzzy Metric Spaces using the CLRg Property”.Fuzzy Sets and Systems, Vol.282, pp.131-142.
[11]. George A. and Veeramani P., (1994).“On Some Results in Fuzzy Metric Spaces”.Fuzzy Sets and System, Vol.64, pp.395- 399.
[12]. Gong X. and Luo T., (2015). “Common Fixed Point Theorems for Nonlinear Contractive Mappings with CLRg Property inFuzzy Metric Spaces”. Applied Mathematical Sciences, Vol.9, No.33, pp.1625-1632.
[13]. Grabiec M., (1989). “Fixed Points in Fuzzy Metric Spaces”.Fuzzy Sets and Systems, Vol.27, pp.385-389.
[14]. Gregori V. and Sapena A., (2002).“On Fixed Point Theorem in Fuzzy Metric Spaces”.Fuzzy Sets and Systems, Vol.125, pp.245-252.
[15]. Imdad M., Pant B. D. and Chauhan S., (2012).“Fixed Point Theorems in Menger Spaces using the CLRST Property and Applications”.Journal on Nonlinear Analysis and Optimization, Vol.3, No.2, pp.223-225.
[16]. Jain M., Tas K., Kumar S. and Gupta N., (2012). “Coupled Fixed Point Theorems for a Pair of Weakly Compatible Maps along with CLRg Property in Fuzzy Metric Spaces”. Journal on Applied Mathematics, pp.13.
[17]. Jungck G. (1986). “Compatible Mappings and Common Fixed Points”.International Journal of Mathematics and Mathematical Sciences, Vol.9, pp.771-779.
[18]. Jungck G. and Rhoades B.E., (2006).“Fixed Point Theorems for Occasionally Weakly Compatible Mappings”.Fixed Point Theory, Vol.7, pp.286-296.
[19]. Jungck, G., (1976). “Commuting Mappings and Fixed Points”. American Journal on Mathematics, Vol.83, pp.261-263.
[20]. Kramosil O. and Michalek J., (1975).“Fuzzy Metric and Statistical Metric Spaces”.Kybernetika, Vol.11, pp.326-334.
[21]. Kumar S. and Rani A., (2015).“ Common Fixed Point Results using Weakly Compatible Maps”. Journal on Analytics and Numerical Theory, Vol.3, No.2, pp.109-116.
[22]. Manro S. and Vetro C., (2014).“Common Fixed Point Theorems in Fuzzy Metric Spaces Employing CLRS and JCLRST Properties”.FactaUniversitatis-Series Mathematics and Informatics, Vol.29, pp.77-90.
[23]. Manro S., (2013). “A Common Fixed Point Theorem in Intuitionistic Fuzzy Metric Spaces Using Occasionally Converse Commuting Maps and Implicit Relation”.Advances in Fixed Point Theory, Vol.3 , No.1, pp.1-8.
[24]. Mihet D, (2010). “Fixed Point Theorems in Fuzzy Metric Spaces Using Property (EA)”.Nonlinear Analysis. Vol.73, pp.2184- 2188.
[25]. Mishra S.N., “Common Fixed Points of Compatible Mappings in PM Spaces”. Math.Japon. Vol.36, pp.283-289.
[26]. Pant R.P, (1994). “Common Fixed Points of Non-commuting Mappings”.Journal on Mathematical Analysis and Application, Vol.188, pp.436-440.
[27]. Pathak H.K. and Khan M.S., (1995).“Compatible Mappings of Type (B) and Common Fixed Point Theorems of Gregus Type”.Czechoslovak Mathematics Journal, Vol.45, No.120, pp.685-698.
[28]. Singh B. and Chauhan M. S., (2000).“Common Fixed Points of Compatible Maps in Fuzzy Metric Spaces”.Fuzzy Sets and Systems, Vol.115, pp.471-475.
[29]. Schweizer B. and Sclar A, (1960).“Statistical Metric Space”.Pacific Journal on Mathematics, pp.314-334.
[30]. Sessa S., (1986). “On a Weak Commutativity Condition in a Fixed Point Considation”.Publication of Inst. Mathematics, Vol.32, No.46, pp.149-153.
[31]. Shukla D.P., Tiwari S.K., and Shukla S.K., (2013).“Unique Common Fixed Point Theorems for Compatible Mappings in Complete Metric Space”.General Mathematical Notes, Vol.18, No.1, pp.13-23.
[32]. Sintunavarat W., and Kuman P., (2011).“Common Fixed Point Theorems for a Pair of Weakly Compatible Mappings in Fuzzy Metric Spaces”.Journal of Applied Mathematics, Vol.2011, pp.14.
[33]. Sintunavarat W., and Francisco A., (2016).“Common Fixed Point Theorems in Fuzzy Metric Spaces using the CLRg Property”.Fuzzy Sets and Systems, Vol.282, pp.131-142.
[34]. Sumitra and Ibtisam. (2013). “Coupled Fixed Point Theorems with CLRg Property in Fuzzy Metric Spaces”. International Journal of Research and Reviews in Applied Sciences, Vol.15, No.3, pp.322-329.
[35]. Vasuki R.,(1999). “Common Fixed Points for R-weakly Commuting Maps in Fuzzy Metric Spaces”.Indian Journal on Pure Applied Mathematics, Vol.30, pp.419-423.
[36]. Wang S.,(2016). “ Common Fixed Point Theorems for Weakly Compatible Mappings in Fuzzy Metric Spaces using the CLRgProperty ” . Journal on Nonlinear Science and Applications, Vol.9, pp.1043-1051.
[37]. Zadeh L.A., (1965). “Fuzzy Sets”.Inform and Control, Vol.8 , pp.338-353.
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