© S. Shakeri et al. 2010
Received: 29 October 2009
Accepted: 27 January 2010
Published: 23 February 2010
The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions [1–43]. Recently Nieto and Rodríguez-López [27–29] and Ran and Reurings  presented some new results for contractions in partially ordered metric spaces. The main idea in [27–33] involves combining the ideas of iterative technique in the contraction mapping principle with those in the monotone technique.
Recall that if is a partially ordered set and is such that for implies , then a mapping is said to be nondecreasing. The main result of Nieto and Rodríguez-López [27–33] and Ran and Reurings  is the following fixed point theorem.
The works of Nieto and Rodríguez-López [27, 28] and Ran and Reurings  have motivated Agarwal et al. , Bhaskar and Lakshmikantham , and Lakshmikantham and Ćirić  to undertake further investigation of fixed points in the area of ordered metric spaces. We prove the existence and approximation results for a wide class of contractive mappings in intuitionistic metric space. Our results are an extension and improvement of the results of Nieto and Rodríguez-López [27, 28] and Ran and Reurings  to more general class of contractive type mappings and include several recent developments.
The notion of fuzzy sets was introduced by Zadeh . Various concepts of fuzzy metric spaces were considered in [15, 16, 22, 45]. Many authors have studied fixed point theory in fuzzy metric spaces; see, for example, [7, 8, 25, 26, 39, 46–48]. In the sequel, we will adopt the usual terminology, notation, and conventions of -fuzzy metric spaces introduced by Saadati et al.  which are a generalization of fuzzy metric sapces  and intuitionistic fuzzy metric spaces [32, 37].
Definition (see ).
Classically, a triangular norm on is defined as an increasing, commutative, associative mapping satisfying , for all . These definitions can be straightforwardly extended to any lattice . Define first and .
is a trivial example of a -norm of Hadžić type, but there exist -norms of Hadžić type weaker than  where
Example (see ).
Lemma (see ).
The sequence is said to be convergent to in the -fuzzy metric space (denoted by ) if whenever for every . A -fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.
The proof is the same as that for fuzzy spaces (see [35, Proposition ]).
Lemma (see ).
3. Main Results
Now we present the main result in this paper.
Also suppose that
This research is supported by Young research Club, Islamic Azad University-Ayatollah Amoli Branch, Amol, Iran. The authors would like to thank Professor J. J. Nieto for giving useful suggestions for the improvement of this paper.
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