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.
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