Skip to main content

Common Fixed Point Theorem in Partially Ordered -Fuzzy Metric Spaces

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.

References

  1. Agarwal RP, El-Gebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Applicable Analysis 2008,87(1):109–116. 10.1080/00036810701556151

    Article  MathSciNet  MATH  Google 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.2005.10.017

    Article  MathSciNet  MATH  Google 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.2006.03.013

    Article  MathSciNet  MATH  Google Scholar 

  5. Björner A: Order-reversing maps and unique fixed points in complete lattices. Algebra Universalis 1981,12(3):402–403.

    Article  MathSciNet  MATH  Google 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)00060-7

    Article  MathSciNet  MATH  Google 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-X

    Article  MathSciNet  MATH  Google 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.2006.09.093

    Article  MathSciNet  MATH  Google Scholar 

  10. Ćirić LB: A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society 1974, 45: 267–273.

    MathSciNet  MATH  Google 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.2007.08.004

    Article  MathSciNet  MATH  Google 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.2007.06.021

    Article  MathSciNet  MATH  Google 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.2003.822678

    Article  MathSciNet  MATH  Google 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)00127-6

    Article  MathSciNet  MATH  Google Scholar 

  15. Deng Z: Fuzzy pseudometric spaces. Journal of Mathematical Analysis and Applications 1982,86(1):74–95. 10.1016/0022-247X(82)90255-4

    Article  MathSciNet  Google 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)90189-6

    Article  MathSciNet  MATH  Google 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.2008.01.003

    Article  MathSciNet  MATH  Google 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.2008.04.009

    Article  MathSciNet  MATH  Google 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.2007.09.016

    Article  MathSciNet  MATH  Google 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.2007.06.008

    Article  MathSciNet  MATH  Google 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.2006.09.048

    Article  MathSciNet  MATH  Google Scholar 

  22. Kamran T: Common fixed points theorems for fuzzy mappings. Chaos, Solitons & Fractals 2008,38(5):1378–1382. 10.1016/j.chaos.2008.04.031

    Article  MathSciNet  MATH  Google 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.2008.09.020

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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)00305-1

    Article  MathSciNet  MATH  Google 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.1996.0325

    Article  MathSciNet  MATH  Google 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/s11083-005-9018-5

    Article  MathSciNet  MATH  Google 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/s10114-005-0769-0

    Article  MathSciNet  MATH  Google 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/S0002-9939-07-08729-1

    Article  MathSciNet  MATH  Google 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.2007.04.070

    Article  MathSciNet  MATH  Google 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.2007.11.026

    Article  MathSciNet  MATH  Google Scholar 

  32. Park JH: Intuitionistic fuzzy metric spaces. Chaos, Solitons & Fractals 2004,22(5):1039–1046. 10.1016/j.chaos.2004.02.051

    Article  MathSciNet  MATH  Google 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/S0002-9939-03-07220-4

    Article  MathSciNet  MATH  Google 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.2006.10.007

    Article  MathSciNet  MATH  Google 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.2003.09.007

    Article  MathSciNet  MATH  Google 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.2006.01.023

    Article  MathSciNet  Google Scholar 

  37. Saadati R, Park JH: On the intuitionistic fuzzy topological spaces. Chaos, Solitons & Fractals 2006,27(2):331–344. 10.1016/j.chaos.2005.03.019

    Article  MathSciNet  MATH  Google Scholar 

  38. Saadati R, Park JH: Intuitionistic fuzzy Euclidean normed spaces. Communications in Mathematical Analysis 2006,1(2):85–90.

    MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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/BF01706080

    Article  MathSciNet  MATH  Google 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.2007.10.011

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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)00342-9

    Article  MathSciNet  MATH  Google Scholar 

  44. Zadeh LA: Fuzzy sets. Information and Computation 1965, 8: 338–353.

    MathSciNet  MATH  Google Scholar 

  45. Kramosil I, Michálek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975,11(5):336–344.

    MathSciNet  MATH  Google Scholar 

  46. Goguen JA: -fuzzy sets. Journal of Mathematical Analysis and Applications 1967, 18: 145–174. 10.1016/0022-247X(67)90189-8

    Article  MathSciNet  MATH  Google 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)00088-9

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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)90162-7

    Article  MathSciNet  MATH  Google 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 

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R Saadati.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Shakeri, S., Ćirić, L. & Saadati, R. Common Fixed Point Theorem in Partially Ordered -Fuzzy Metric Spaces. Fixed Point Theory Appl 2010, 125082 (2010). https://doi.org/10.1155/2010/125082

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2010/125082

Keywords

  • Open Subset
  • Approximation Result
  • Wide Class
  • Fixed Point Theorem
  • Point Theory