- Research Article
- Open Access
© S. Shakeri et al. 2010
- Received: 29 October 2009
- Accepted: 27 January 2010
- Published: 23 February 2010
- Open Subset
- Approximation Result
- Wide Class
- Fixed Point Theorem
- Point Theory
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 ).
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.
- Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Applicable Analysis 2008,87(1):109–116. 10.1080/00036810701556151MathSciNetView ArticleMATHGoogle Scholar
- ltun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory and Applications 2010, 2010:-17.Google 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.2005.10.017MathSciNetView ArticleMATHGoogle Scholar
- Gnana Bhaskar T, Lakshmikantham V, Vasundhara Devi J: Monotone iterative technique for functional differential equations with retardation and anticipation. Nonlinear Analysis: Theory, Methods & Applications 2007,66(10):2237–2242. 10.1016/j.na.2006.03.013MathSciNetView ArticleMATHGoogle Scholar
- Björner A: Order-reversing maps and unique fixed points in complete lattices. Algebra Universalis 1981,12(3):402–403.MathSciNetView ArticleMATHGoogle Scholar
- Burgić Dž, Kalabušić S, Kulenović MRS: Global attractivity results for mixed-monotone mappings in partially ordered complete metric spaces. Fixed Point Theory and Applications 2009, 2009:-17.Google Scholar
- Chang SS, Cho YJ, Lee BS, Jung JS, Kang SM: Coincidence point theorems and minimization theorems in fuzzy metric spaces. Fuzzy Sets and Systems 1997,88(1):119–127. 10.1016/S0165-0114(96)00060-7MathSciNetView ArticleMATHGoogle Scholar
- Cho YJ, Pathak HK, Kang SM, Jung JS: Common fixed points of compatible maps of type on fuzzy metric spaces. Fuzzy Sets and Systems 1998,93(1):99–111. 10.1016/S0165-0114(96)00200-XMathSciNetView ArticleMATHGoogle 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 & Fractals 2008,37(3):781–791. 10.1016/j.chaos.2006.09.093MathSciNetView ArticleMATHGoogle Scholar
- Ćirić LB: A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society 1974, 45: 267–273.MathSciNetMATHGoogle Scholar
- Ćirić LB: Coincidence and fixed points for maps on topological spaces. Topology and Its Applications 2007,154(17):3100–3106. 10.1016/j.topol.2007.08.004MathSciNetView ArticleMATHGoogle Scholar
- Ćirić LB, Jesić SN, Milovanović MM, Ume JS: On the steepest descent approximation method for the zeros of generalized accretive operators. Nonlinear Analysis: Theory, Methods & Applications 2008,69(2):763–769. 10.1016/j.na.2007.06.021MathSciNetView ArticleMATHGoogle Scholar
- Deschrijver G, Cornelis C, Kerre EE: On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Transactions on Fuzzy Systems 2004,12(1):45–61. 10.1109/TFUZZ.2003.822678MathSciNetView ArticleMATHGoogle Scholar
- Deschrijver G, Kerre EE: On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems 2003,133(2):227–235. 10.1016/S0165-0114(02)00127-6MathSciNetView ArticleMATHGoogle Scholar
- Deng Z: Fuzzy pseudometric spaces. Journal of Mathematical Analysis and Applications 1982,86(1):74–95. 10.1016/0022-247X(82)90255-4MathSciNetView ArticleGoogle Scholar
- Erceg MA: Metric spaces in fuzzy set theory. Journal of Mathematical Analysis and Applications 1979,69(1):205–230. 10.1016/0022-247X(79)90189-6MathSciNetView ArticleMATHGoogle Scholar
- Qiu D, Shu L, Guan J: Common fixed point theorems for fuzzy mappings under -contraction condition. Chaos, Solitons & Fractals 2009,41(1):360–367. 10.1016/j.chaos.2008.01.003MathSciNetView ArticleMATHGoogle Scholar
- Farnoosh R, Aghajani A, Azhdari P: Contraction theorems in fuzzy metric space. Chaos, Solitons & Fractals 2009,41(2):854–858. 10.1016/j.chaos.2008.04.009MathSciNetView ArticleMATHGoogle Scholar
- Ghaemi MB, Lafuerza-Guillen B, Razani A: A common fixed point for operators in probabilistic normed spaces. Chaos, Solitons & Fractals 2009,40(3):1361–1366. 10.1016/j.chaos.2007.09.016MathSciNetView ArticleMATHGoogle Scholar
- Hussain N: Common fixed points in best approximation for Banach operator pairs with Ćirić type -contractions. Journal of Mathematical Analysis and Applications 2008,338(2):1351–1363. 10.1016/j.jmaa.2007.06.008MathSciNetView ArticleMATHGoogle Scholar
- Ješić SN, Babačev NA: Common fixed point theorems in intuitionistic fuzzy metric spaces and -fuzzy metric spaces with nonlinear contractive condition. Chaos, Solitons & Fractals 2008,37(3):675–687. 10.1016/j.chaos.2006.09.048MathSciNetView ArticleMATHGoogle Scholar
- Kamran T: Common fixed points theorems for fuzzy mappings. Chaos, Solitons & Fractals 2008,38(5):1378–1382. 10.1016/j.chaos.2008.04.031MathSciNetView ArticleMATHGoogle 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.2008.09.020MathSciNetView ArticleMATHGoogle Scholar
- Liu Z, Guo Z, Kang SM, Lee SK: On Ćirić type mappings with nonunique fixed and periodic points. International Journal of Pure and Applied Mathematics 2006,26(3):399–408.MathSciNetMATHGoogle Scholar
- Miheţ D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets and Systems 2004,144(3):431–439. 10.1016/S0165-0114(03)00305-1MathSciNetView ArticleMATHGoogle Scholar
- Pap E, Hadžić O, Mesiar R: A fixed point theorem in probabilistic metric spaces and an application. Journal of Mathematical Analysis and Applications 1996,202(2):433–449. 10.1006/jmaa.1996.0325MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005,22(3):223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Mathematica Sinica 2007,23(12):2205–2212. 10.1007/s10114-005-0769-0MathSciNetView ArticleMATHGoogle Scholar
- Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proceedings of the American Mathematical Society 2007,135(8):2505–2517. 10.1090/S0002-9939-07-08729-1MathSciNetView ArticleMATHGoogle 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.2007.04.070MathSciNetView ArticleMATHGoogle Scholar
- O'Regan D, Petruşel A: Fixed point theorems for generalized contractions in ordered metric spaces. Journal of Mathematical Analysis and Applications 2008,341(2):1241–1252. 10.1016/j.jmaa.2007.11.026MathSciNetView ArticleMATHGoogle Scholar
- Park JH: Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2004,22(5):1039–1046. 10.1016/j.chaos.2004.02.051MathSciNetView ArticleMATHGoogle Scholar
- Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society 2004,132(5):1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleMATHGoogle Scholar
- Rezaiyan R, Cho YJ, Saadati R: A common fixed point theorem in Menger probabilistic quasi-metric spaces. Chaos, Solitons & Fractals 2008,37(4):1153–1157. 10.1016/j.chaos.2006.10.007MathSciNetView ArticleMATHGoogle Scholar
- Rodríguez-López J, Romaguera S: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets and Systems 2004,147(2):273–283. 10.1016/j.fss.2003.09.007MathSciNetView ArticleMATHGoogle Scholar
- Saadati R, Razani A, Adibi H: A common fixed point theorem in -fuzzy metric spaces. Chaos, Solitons & Fractals 2007,33(2):358–363. 10.1016/j.chaos.2006.01.023MathSciNetView ArticleGoogle Scholar
- Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos, Solitons & Fractals 2006,27(2):331–344. 10.1016/j.chaos.2005.03.019MathSciNetView ArticleMATHGoogle Scholar
- Saadati R, Park JH: Intuitionistic fuzzy Euclidean normed spaces. Communications in Mathematical Analysis 2006,1(2):85–90.MathSciNetMATHGoogle Scholar
- Saadati R, Sedghi S, Zhou H: A common fixed point theorem for -weakly commuting maps in -fuzzy metric spaces. Iranian Journal of Fuzzy Systems 2008,5(1):47–53.MathSciNetMATHGoogle Scholar
- Sehgal VM, Bharucha-Reid AT: Fixed points of contraction mappings on probabilistic metric spaces. Mathematical Systems Theory 1972, 6: 97–102. 10.1007/BF01706080MathSciNetView ArticleMATHGoogle Scholar
- Sharma S, Deshpande B: Common fixed point theorems for finite number of mappings without continuity and compatibility on intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2009,40(5):2242–2256. 10.1016/j.chaos.2007.10.011MathSciNetView ArticleMATHGoogle Scholar
- Singh SL, Mishra SN: On a Ljubomir Ćirić fixed point theorem for nonexpansive type maps with applications. Indian Journal of Pure and Applied Mathematics 2002,33(4):531–542.MathSciNetMATHGoogle Scholar
- Vasuki R: A common fixed point theorem in a fuzzy metric space. Fuzzy Sets and Systems 1998,97(3):395–397. 10.1016/S0165-0114(96)00342-9MathSciNetView ArticleMATHGoogle Scholar
- Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.MathSciNetMATHGoogle Scholar
- Kramosil I, Michálek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975,11(5):336–344.MathSciNetMATHGoogle Scholar
- Goguen JA: -fuzzy sets. Journal of Mathematical Analysis and Applications 1967, 18: 145–174. 10.1016/0022-247X(67)90189-8MathSciNetView ArticleMATHGoogle Scholar
- Gregori V, Sapena A: On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets and Systems 2002,125(2):245–252. 10.1016/S0165-0114(00)00088-9MathSciNetView ArticleMATHGoogle Scholar
- Hosseini SB, O'Regan D, Saadati R: Some results on intuitionistic fuzzy spaces. Iranian Journal of Fuzzy Systems 2007,4(1):53–64.MathSciNetMATHGoogle 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)90162-7MathSciNetView ArticleMATHGoogle Scholar
- Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications. Volume 536. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+273.Google Scholar
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