Open Access

Common Fixed Point Theorem in Partially Ordered -Fuzzy Metric Spaces

Fixed Point Theory and Applications20102010:125082

DOI: 10.1155/2010/125082

Received: 29 October 2009

Accepted: 27 January 2010

Published: 23 February 2010

Abstract

We introduce partially ordered -fuzzy metric spaces and prove a common fixed point theorem in these spaces.

1. Introduction

The Banach fixed point theorem for contraction mappings has been generalized and extended in many directions [143]. Recently Nieto and Rodríguez-López [2729] and Ran and Reurings [33] presented some new results for contractions in partially ordered metric spaces. The main idea in [2733] 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 [2733] and Ran and Reurings [33] is the following fixed point theorem.

Theorem 1.1.

Let be a partially ordered set and suppose that there is a metric on such that is a complete metric space. Suppose that is a nondecreasing mapping with
(1.1)

for all where Also suppose the following.

(a) is continuous.

(b)If is a nondecreasing sequence with in

then for all hold.

If there exists an with , then has a fixed point.

The works of Nieto and Rodríguez-López [27, 28] and Ran and Reurings [33] have motivated Agarwal et al. [1], Bhaskar and Lakshmikantham [3], and Lakshmikantham and Ćirić [23] 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 [33] to more general class of contractive type mappings and include several recent developments.

2. Preliminaries

The notion of fuzzy sets was introduced by Zadeh [44]. 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, 4648]. In the sequel, we will adopt the usual terminology, notation, and conventions of -fuzzy metric spaces introduced by Saadati et al. [36] which are a generalization of fuzzy metric sapces [49] and intuitionistic fuzzy metric spaces [32, 37].

Definition (see [46]).

Let be a complete lattice, and a nonempty set called a universe. An -fuzzy set on is defined as a mapping . For each in , represents the degree (in ) to which satisfies .

Lemma (see [13, 14]).

Consider the set and the operation defined by
(2.1)

, and , for every . Then is a complete lattice.

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 .

Definition.

A negation on is any strictly decreasing mapping satisfying and . If , for all , then is called an involutive negation.

In this paper the negation is fixed.

Definition.

A triangular norm ( -norm) on is a mapping satisfying the following conditions:

(i) for all (boundary condition);

(ii) for all (commutativity);

(iii) for all (associativity);

(iv) for all and (monotonicity).

A -norm on is said to be continuous if for any and any sequences and which converge to and we have

(2.2)

For example, and are two continuous -norms on . A -norm can also be defined recursively as an -ary operation ( ) by and

(2.3)

for and .

A -norm is said to be of Hadžić type if the family is equicontinuous at , that is,

(2.4)

is a trivial example of a -norm of Hadžić type, but there exist -norms of Hadžić type weaker than [50] where

(2.5)

Definition.

The 3-tuple is said to be an -fuzzy metric space if is an arbitrary (nonempty) set, is a continuous -norm on and is an -fuzzy set on satisfying the following conditions for every in and in :

(a) ;

(b) for all if and only if ;

(c) ;

(d) ;

(e) is continuous.

If the -fuzzy metric space satisfies the condition:
(2.6)

then is said to be Menger -fuzzy metric space or for short a -fuzzy metric space.

Let be an -fuzzy metric space. For , we define the open ball with center and radius , as

(2.7)

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 induced by the -fuzzy metric .

Example (see [38]).

Let be a metric space. Denote for all and in and let and be fuzzy sets on defined as follows:
(2.8)

Then is an intuitionistic fuzzy metric space.

Example.

Let . Define for all and in , and let on be defined as follows:
(2.9)

for all and . Then is an -fuzzy metric space.

Lemma (see [49]).

Let be an -fuzzy metric space. Then, is nondecreasing with respect to , for all in .

Definition.

A sequence in an -fuzzy metric space is called a Cauchy sequence, if for each and , there exists such that for all ,
(2.10)

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.

Definition.

Let be an -fuzzy metric space. is said to be continuous on if
(2.11)

whenever a sequence in converges to a point , that is, and .

Lemma.

Let be an -fuzzy metric space. Then is continuous function on .

Proof.

The proof is the same as that for fuzzy spaces (see [35, Proposition ]).

Lemma.

If an -fuzzy metric space satisfies the following condition:
(2.12)

then one has and .

Proof.

Let for all . Then by of Definition 2.5, we have and by of Definition 2.5, we conclude that .

Lemma (see [50]).

Let be an -fuzzy metric space in which is Hadži type. Suppose
(2.13)

for some and . Then is a Cauchy sequence.

3. Main Results

Definition.

Suppose that is a partially ordered set and are mappings of into itself. We say that is -nondecreasing if for ,
(3.1)

Now we present the main result in this paper.

Theorem.

Let be a partially ordered set and suppose that there is an -fuzzy metric on such that is a complete -fuzzy metric space in which is Hadži type. Let be two self-mappings of such that there exist and such that is a -nondecreasing mapping and
(3.2)

for all for which and all

Also suppose that

(3.3)

Also suppose that is closed. If there exists an with , then and have a coincidence. Further, if and commute at their coincidence points, then and have a common fixed point.

Proof.

Let be such that Since we can choose such that Again from we can choose such that Continuing this process we can choose a sequence in such that
(3.4)
Since and we have Then from (3.1),
(3.5)
that is, by (3.4), Again from (3.1),
(3.6)
that is, Continuing we obtain
(3.7)

Now we will show that a sequence converges to for each . If for some and for each , then it is easily to show that for all . So we suppose that for all We show that for each

(3.8)
Since from (3.4) and (3.7) we have from (3.1) with and
(3.9)
So by (3.4),
(3.10)
Since by (d) of Definition 2.5
(3.11)
we have
(3.12)
As -norm is continuous, letting we get
(3.13)
Consequently,
(3.14)
By repeating the above inequality, we obtain
(3.15)
Since as it follows that
(3.16)
Thus we proved (3.7). By repeating the above inequality (3.7), we get
(3.17)
Since as and , letting in (3.17) we get
(3.18)

Now we will prove that is a Cauchy sequence which means that for every and there exists such that

(3.19)
Let and be arbitrary. For any we have
(3.20)
Since is nondecreasing with respect to , for all in ,
(3.21)
and hence, by (d) of Definition 2.5,
(3.22)
From (3.17) it follows that
(3.23)
From (3.23) with we get
(3.24)
Thus by (3.22),
(3.25)
Hence we get
(3.26)
From (3.26) and (3.17),
(3.27)
Hence we conclude, as as and , that there exists such that
(3.28)

Thus we proved that is a Cauchy sequence.

Since is closed and as , there is some such that

(3.29)

Now we show that is a coincidence of and Since from (3.3) and (3.29) we have for all then from (3.2) and by (d) of Definition 2.5 we have

(3.30)
Letting we get
(3.31)
for all Therefore,
(3.32)
Hence we get
(3.33)

Hence we conclude that for all Then by (b) of Definition 2.5 we have Thus we proved that and have a coincidence.

Suppose now that and commute at . Set Then

(3.34)
Since from (3.3) we have and as and from (3.2) we get
(3.35)
Letting we get
(3.36)
Hence, similarly as above, we get
(3.37)
Hence we conclude that Since we have
(3.38)

Thus we proved that and have a common fixed point.

Remark.

Note that is -nondecreasing and can be replaced by which is -non-increasing in Theorem 3.2 provided that is replaced by in Theorem 3.2.

Corollary 3.4.

Let be a partially ordered set and suppose that there is an -fuzzy metric on such that is a complete -fuzzy metric space in which is Hadži type. Let be a nondecreasing self-mappings of such that there exist and such that
(3.39)

for all for which and all Also suppose the following.

(i)If is a nondecreasing sequence with in , then for all hold.

(ii) is continuous.

If there exists an with , then has a fixed point.

Proof.

Taking ( = the identity mapping) in Theorem 3.2, then (3.3) reduces to the hypothesis

Suppose now that is continuous. Since from (3.4) we have for all and as from (3.29), then

(3.40)

Corollary 3.5.

Let be a partially ordered set and suppose that there is an -fuzzy metric on such that is a complete -fuzzy metric space in which is Hadži type. Let be a nondecreasing self-mappings of such that there exist and such that
(3.41)

for all for which and all Also suppose the following.

(i)If is a nondecreasing sequence with in , then for all hold.

(ii) is continuous.

If there exists an with , then has a fixed point.

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
Young Research Club, Islamic Azad University-Ayatollah Amoli Branch
(2)
Faculty of Mechanical Engineering
(3)
Faculty of Sciences, Islamic Azad University-Ayatollah Amoli Branch

References

  1. 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
  2. ltun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory and Applications 2010, 2010:-17.Google Scholar
  3. 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
  4. 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
  5. Björner A: Order-reversing maps and unique fixed points in complete lattices. Algebra Universalis 1981,12(3):402–403.MathSciNetView ArticleMATHGoogle Scholar
  6. 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
  7. 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
  8. 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
  9. 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
  10. Ćirić LB: A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society 1974, 45: 267–273.MathSciNetMATHGoogle Scholar
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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
  31. 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
  32. Park JH: Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2004,22(5):1039–1046. 10.1016/j.chaos.2004.02.051MathSciNetView ArticleMATHGoogle Scholar
  33. 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
  34. 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
  35. 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
  36. 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
  37. 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
  38. Saadati R, Park JH: Intuitionistic fuzzy Euclidean normed spaces. Communications in Mathematical Analysis 2006,1(2):85–90.MathSciNetMATHGoogle Scholar
  39. 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
  40. 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
  41. 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
  42. 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
  43. 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
  44. Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.MathSciNetMATHGoogle Scholar
  45. Kramosil I, Michálek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975,11(5):336–344.MathSciNetMATHGoogle Scholar
  46. Goguen JA: -fuzzy sets. Journal of Mathematical Analysis and Applications 1967, 18: 145–174. 10.1016/0022-247X(67)90189-8MathSciNetView ArticleMATHGoogle Scholar
  47. 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
  48. 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
  49. 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
  50. 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

Copyright

© S. Shakeri et al. 2010

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.