- Research Article
- Open Access
Common Coupled Fixed Point Theorems for Contractive Mappings in Fuzzy Metric Spaces
© Xin-Qi Hu. 2011
Received: 23 November 2010
Accepted: 27 January 2011
Published: 23 February 2011
Since Zadeh  introduced the concept of fuzzy sets, many authors have extensively developed the theory of fuzzy sets and applications. George and Veeramani [2, 3] gave the concept of fuzzy metric space and defined a Hausdorff topology on this fuzzy metric space which have very important applications in quantum particle physics particularly in connection with both string and -infinity theory.
Bhaskar and Lakshmikantham , Lakshmikantham and Ćirić  discussed the mixed monotone mappings and gave some coupled fixed point theorems which can be used to discuss the existence and uniqueness of solution for a periodic boundary value problem. Sedghi et al.  gave a coupled fixed point theorem for contractions in fuzzy metric spaces, and Fang  gave some common fixed point theorems under -contractions for compatible and weakly compatible mappings in Menger probabilistic metric spaces. Many authors [8–23] have proved fixed point theorems in (intuitionistic) fuzzy metric spaces or probabilistic metric spaces.
In this paper, using similar proof as in , we give a new common fixed point theorem under weaker conditions than in  and give an example which shows that the result is a genuine generalization of the corresponding result in .
First we give some definitions.
Definition 1 (see ).
Definition 2 (see ).
The -norm is an example of -norm of H-type, but there are some other -norms of H-type .
Definition 3 (see ).
A subset is called open if, for each , there exist and such that . Let denote the family of all open subsets of . Then is called the topology on induced by the fuzzy metric . This topology is Hausdorff and first countable.
Definition 4 (see ).
Definition 5 (see ).
Lemma 2 (see ).
Definition 6 (see ).
Definition 7 (see ).
Definition 8 (see ).
Definition 9 (see ).
Definition 10 (see ).
Definition 11 (see ).
3. Main Results
The proof is divided into 4 steps.
This completes the proof of the Theorem 1.
Corollary 2 (see ).
Next we give an example to demonstrate Theorem 1.
We consider the following cases.
it is easy to verified.
The author is grateful to the referees for their valuable comments and suggestions.
- Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.MATHMathSciNetGoogle Scholar
- George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets and Systems 1994,64(3):395–399. 10.1016/0165-0114(94)90162-7MATHMathSciNetView ArticleGoogle Scholar
- George A, Veeramani P: On some results of analysis for fuzzy metric spaces. Fuzzy Sets and Systems 1997,90(3):365–368. 10.1016/S0165-0114(96)00207-2MATHMathSciNetView ArticleGoogle Scholar
- Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis. Theory, Methods & Applications 2006,65(7):1379–1393. 10.1016/j.na.2005.10.017MATHMathSciNetView ArticleGoogle Scholar
- Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,70(12):4341–4349. 10.1016/j.na.2008.09.020MATHMathSciNetView ArticleGoogle Scholar
- Sedghi S, Altun I, Shobe N: Coupled fixed point theorems for contractions in fuzzy metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2010,72(3–4):1298–1304. 10.1016/j.na.2009.08.018MATHMathSciNetView ArticleGoogle Scholar
- Fang J-X: Common fixed point theorems of compatible and weakly compatible maps in Menger spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(5–6):1833–1843. 10.1016/j.na.2009.01.018MATHView ArticleMathSciNetGoogle Scholar
- Ćirić LB, Miheţ D, Saadati R: Monotone generalized contractions in partially ordered probabilistic metric spaces. Topology and its Applications 2009,156(17):2838–2844. 10.1016/j.topol.2009.08.029MATHMathSciNetView ArticleGoogle Scholar
- O'Regan D, Saadati R: Nonlinear contraction theorems in probabilistic spaces. Applied Mathematics and Computation 2008,195(1):86–93. 10.1016/j.amc.2007.04.070MATHMathSciNetView ArticleGoogle Scholar
- Jain S, Jain S, Bahadur Jain L: Compatibility of type (P) in modified intuitionistic fuzzy metric space. Journal of Nonlinear Science and its Applications 2010,3(2):96–109.MATHMathSciNetGoogle Scholar
- Ćirić LB, Ješić SN, Ume JS: The existence theorems for fixed and periodic points of nonexpansive mappings in intuitionistic fuzzy metric spaces. Chaos, Solitons and Fractals 2008,37(3):781–791. 10.1016/j.chaos.2006.09.093MATHMathSciNetView ArticleGoogle Scholar
- \'Cirić L, Lakshmikantham V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stochastic Analysis and Applications 2009,27(6):1246–1259. 10.1080/07362990903259967MATHMathSciNetView ArticleGoogle Scholar
- Ćirić L, Cakić N, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory and Applications 2008, 2008:-11.Google Scholar
- Aliouche A, Merghadi F, Djoudi A: A related fixed point theorem in two fuzzy metric spaces. Journal of Nonlinear Science and its Applications 2009,2(1):19–24.MATHMathSciNetGoogle Scholar
- Ćirić L: Common fixed point theorems for a family of non-self mappings in convex metric spaces. Nonlinear Analysis. Theory, Methods & Applications 2009,71(5–6):1662–1669. 10.1016/j.na.2009.01.002MATHView ArticleMathSciNetGoogle Scholar
- Rao KPR, Aliouche A, Babu GR: Related fixed point theorems in fuzzy metric spaces. Journal of Nonlinear Science and its Applications 2008,1(3):194–202.MATHMathSciNetGoogle Scholar
- Ćirić L, Cakić N: On common fixed point theorems for non-self hybrid mappings in convex metric spaces. Applied Mathematics and Computation 2009,208(1):90–97. 10.1016/j.amc.2008.11.012MATHMathSciNetView ArticleGoogle Scholar
- Ćirić L: Some new results for Banach contractions and Edelstein contractive mappings on fuzzy metric spaces. Chaos, Solitons and Fractals 2009,42(1):146–154. 10.1016/j.chaos.2008.11.010MATHMathSciNetView ArticleGoogle Scholar
- Shakeri S, Ćirić LJB, Saadati R: Common fixed point theorem in partially ordered -fuzzy metric spaces. Fixed Point Theory and Applications 2010, 2010:-13.Google Scholar
- Ćirić L, Samet B, Vetro C: Common fixed point theorems for families of occasionally weakly compatible mappings. Mathematical and Computer Modelling 2011,53(5–6):631–636. 10.1016/j.mcm.2010.09.015MATHMathSciNetView ArticleGoogle Scholar
- Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Applied Mathematics and Computation 2011,217(12):5784–5789. 10.1016/j.amc.2010.12.060MATHMathSciNetView ArticleGoogle Scholar
- Ćirić L, Abbas M, Damjanović B, Saadati R: Common fuzzy fixed point theorems in ordered metric spaces. Mathematical and Computer Modelling 2011,53(9–10):1737–1741. 10.1016/j.mcm.2010.12.050MATHMathSciNetView ArticleGoogle Scholar
- Kamran T, Cakić N: Hybrid tangential property and coincidence point theorems. Fixed Point Theory 2008,9(2):487–496.MATHMathSciNetGoogle Scholar
- Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and its Applications. Volume 536. Kluwer Academic, Dordrecht, The Netherlands; 2001:x+273.Google Scholar
- Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets and Systems 1988,27(3):385–389. 10.1016/0165-0114(88)90064-4MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.