Skip to main content

Fixed points of some new contractions on intuitionistic fuzzy metric spaces

Abstract

We introduce some new contractions on intuitionistic fuzzy metric spaces, and give fixed point results for these classes of contractions. A stability result is established.

1 Introduction and preliminaries

The great interest in the study of various fixed point theories for different classes of contractions on some specific spaces is known. We underline studies on quasi-metric spaces [1, 2], quasi-partial metric spaces [3], convex metric spaces [4], cone metric spaces [57], partially ordered metric spaces [817], partial metric spaces [18], Menger spaces [19], G-metric spaces [20, 21], and fuzzy metric spaces [2225].

The concept of fuzzy set was introduced by Zadeh in 1965 [26]. Ten years later, Kramosil and Michalek introduced the notion of fuzzy metric spaces [24] and George and Veeramani modified the concept in 1994 [27]. Also, they defined the notion of Hausdorff topology in fuzzy metric spaces [27].

In 2004, Park introduced the notion of intuitionistic fuzzy metric space. In his elegant article [28], he showed that for each intuitionistic fuzzy metric space (X,M,N,,), the topology generated by the intuitionistic fuzzy metric (M,N) coincides with the topology generated by the fuzzy metric M.

Actually, Park’s notion is useful in modeling some phenomena where it is necessary to study the relationship between two probability functions. Some authors have introduced and discussed several notions of intuitionistic fuzzy metric spaces in different ways (see, for example, [2931]. Grabiec obtained a fuzzy version of the Banach contraction principle in fuzzy metric spaces in Kramosil and Michalek’s sense [22], and since then many authors have proved fixed point theorems in fuzzy metric spaces [3235].

For necessary notions to our results, such as continuous t-norm, intuitionistic fuzzy metric space and the induced topology, which is denoted by τ ( M , N ) , we refer the reader to [28] and [36].

A sequence { x n } in an intuitionistic fuzzy metric space (X,M,N,,) is said to be Cauchy sequence whenever, for each ε>0 and t>0, there exists a natural number n 0 such that M( x n , x m ,t)>1ε and N( x n , x m ,t)<ε for all n,m n 0 .

The space (X,M,N,,) is called complete whenever every Cauchy sequence is convergent with respect to the topology τ ( M , N ) .

Let (X,M,N,,) be an intuitionistic fuzzy metric space. According to [32], the fuzzy metric (M,N) is called triangular whenever

1 M ( x , y , t ) 1 1 M ( x , z , t ) 1+ 1 M ( z , y , t ) 1

and

N(x,y,t)N(x,z,t)+N(z,y,t)

for all x,y,zX and t>0.

We shall use the above background to develop our new results in this article. Our results are stated on complete triangular intuitionistic fuzzy metric spaces. In this framework, we introduce some new classes of contractive conditions and give fixed point results for them.

2 Main results

Now, we are ready to state and prove our main results.

Theorem 2.1 Let (X,M,N,,) be a complete triangular intuitionistic fuzzy metric space, h[0,1) and let T:XX be a continuous mapping satisfying the contractive condition

1 M ( T x , T y , t ) 1hmax { 1 M ( x , T x , t ) 1 , 1 M ( y , T y , t ) 1 }

for all x,yX. Then T has a fixed point.

Proof Let x 0 X. Put x 1 =T x 0 and x n + 1 = T n + 1 x 0 for all n1.

If x n = x n + 1 for some n, then we have nothing to prove.

Assume that x n x n + 1 for all n. Then

1 M ( x n + 1 , x n , t ) 1 = 1 M ( T x n , T x n 1 , t ) 1 h max { 1 M ( x n , T x n , t ) 1 , 1 M ( x n 1 , T x n 1 , t ) 1 }

for all n.

Now, for each n, put t n =max{ 1 M ( x n , T x n , t ) 1, 1 M ( x n 1 , T x n 1 , t ) 1}.

If t n = 1 M ( x n , T x n , t ) 1, then

1 M ( x n + 1 , x n , t ) 1h ( 1 M ( x n , T x n , t ) 1 ) =h ( 1 M ( x n , x n + 1 , t ) 1 ) ,

which is a contradiction. Thus, t n = 1 M ( x n 1 , T x n 1 , t ) 1 for all n, and so

1 M ( x n + 1 , x n , t ) 1h ( 1 M ( x n 1 , T x n 1 , t ) 1 ) .

But

1 M ( x n , x n 1 , t ) 1 = 1 M ( T x n 1 , T x n 2 , t ) 1 h max { 1 M ( x n 1 , T x n 1 , t ) 1 , 1 M ( x n 2 , T x n 2 , t ) 1 }

and 1 M ( x n , x n 1 , t ) 1h( 1 M ( x n 2 , T x n 2 , t ) 1) for all n.

Thus,

1 M ( x n + 1 , x n , t ) 1h ( 1 M ( x n , x n 1 , t ) 1 ) h n ( 1 M ( x 1 , x 0 , t ) 1 ) .

Hence, for each n>m, we obtain

1 M ( x n , x m , t ) 1 1 M ( x n , x n 1 , t ) 1 + + 1 M ( x m + 1 , x m , t ) 1 ( h n 1 + h n 2 + + h m ) ( 1 M ( x 1 , x 0 , t ) 1 ) h m 1 h ( 1 M ( x 1 , x 0 , t ) 1 ) .

Therefore, { x n } is a Cauchy sequence and so there exists x X such that x n x . Since T is continuous, x n + 1 =T x n T x and so x =T x . □

Theorem 2.2 Let (X,M,N,,) be a complete triangular intuitionistic fuzzy metric space and let T:XX be a selfmap which satisfies the contractive condition

1 M ( T x , T y , t ) 1 [ 1 M ( x , T y , t ) 1 + 1 M ( y , T x , t ) 1 1 M ( x , T x , t ) 1 + 1 M ( y , T y , t ) 1 + 1 t ] ( 1 M ( x , y , t ) 1 )

for all x,yX. Then T has a fixed point.

Proof Let x 0 X. Define the sequence { x n } by x n + 1 =T x n for all n. Then

1 M ( x n + 1 , x n , t ) 1 = 1 M ( T x n , T x n 1 , t ) 1 [ 1 M ( x n , x n , t ) 1 + 1 M ( x n 1 , x n + 1 , t ) 1 1 M ( x n , x n + 1 , t ) 1 + 1 M ( x n 1 , x n , t ) 1 + 1 t ] ( 1 M ( x n , x n 1 , t ) 1 ) = [ 1 M ( x n 1 , x n + 1 , t ) 1 1 M ( x n , x n + 1 , t ) 1 + 1 M ( x n 1 , x n , t ) 1 + 1 t ] ( 1 M ( x n , x n 1 , t ) 1 ) [ 1 M ( x n 1 , x n , t ) 1 + 1 M ( x n , x n + 1 , t ) 1 1 M ( x n , x n + 1 , t ) 1 + 1 M ( x n 1 , x n , t ) 1 + 1 t ] ( 1 M ( x n , x n 1 , t ) 1 ) 1 M ( x n , x n 1 , t ) 1

for all n and t>0. Therefore, { 1 M ( x n , x n 1 , t ) 1} is a non-increasing sequence and so it is convergent to some r0.

If r>0, then by putting

β n = [ 1 M ( x n 1 , x n , t ) 1 + 1 M ( x n , x n + 1 , t ) 1 1 M ( x n , x n + 1 , t ) 1 + 1 M ( x n 1 , x n , t ) 1 + 1 t ] ,

we obtain lim n β n = 2 r 2 r + 1 t and so r 2 r 2 r + 1 t r, which is a contradiction. Thus, r=0.

Note that

1 M ( x n + 1 , x n , t ) 1 β n [ 1 M ( x n , x n 1 , t ) 1 ] β n β n 1 [ 1 M ( x n 1 , x n 2 , t ) 1 ] ( β n β n 1 β 1 ) [ 1 M ( x 1 , x 0 , t ) 1 ]

for all n. Thus, for each m>n, we get

1 M ( x m , x n , t ) 1 1 M ( x n , x n + 1 , t ) 1 + 1 M ( x n + 1 , x n + 2 , t ) 1 + + 1 M ( x m 1 , x m , t ) 1 [ ( β n β n 1 β 1 ) + ( β n + 1 β n β 1 ) + + ( β m 1 β m 2 β 1 ) ] ( 1 M ( x 1 , x 0 , t ) 1 ) .

Now, we consider a n = β n 1 β 2 β 1 . Since lim n a n + 1 a n = lim n β n =0, it follows that k = 1 a k <. Hence, { x n } is a Cauchy sequence and so it converges to some x X.

We claim that x is a fixed point of T.

Since

1 M ( x n + 1 , T x , t ) 1 [ 1 M ( x n , T x , t ) 1 + 1 M ( x , T x n , t ) 1 1 M ( x , T x , t ) 1 + 1 M ( x n , T x n , t ) 1 + 1 t ] ( 1 M ( x n , x , t ) 1 )

for all n, we get 1 M ( x , T x , t ) 1=0 and so T x = x . □

The following example shows that there are discontinuous mappings which satisfy the conditions of Theorem 2.2.

Example 2.1 Let X=[0,2 3 ) endowed with the usual distance d(x,y)=|xy|. Consider M(x,y,t)= t t + d ( x , y ) and N(x,y,t)= d ( x , y ) t + d ( x , y ) for all x,yX and t0. Define the selfmap T on X by

Tx={ 0 , x [ 0 , 2 3 ) , 2 3 , x = 2 3 .

It is easy to check that T satisfies the conditions of Theorem 2.2.

In fact, for x=2 3 and 0y<2 3 , we have

( 1 M ( T x , T y , t ) 1 ) [ 1 M ( x , T x , t ) 1 + 1 M ( y , T y , t ) 1 + 1 t ] = ( | T x T y | t ) [ | x T x | t + | y T y | t + 1 t ] = 2 3 t [ y t + 1 t ] 1 t 2 [ ( 2 3 y ) 2 ( 2 3 ) ( 2 3 y ) ] = [ | x T y | t + | y T x | t ] | x y | t = [ 1 M ( x , T y , t ) 1 + 1 M ( y , T x , t ) 1 ] ( 1 M ( x , y , t ) 1 ) ,

and so

1 M ( T x , T y , t ) 1 [ 1 M ( x , T y , t ) 1 + 1 M ( y , T x , t ) 1 1 M ( x , T x , t ) 1 + 1 M ( y , T y , t ) 1 + 1 t ] ( 1 M ( x , y , t ) 1 ) .

Theorem 2.3 Let (X,M,N,,) be a complete triangular intuitionistic fuzzy metric space, α,β[0,1) with α+β<1 and let T:XX be a continuous mapping which satisfies the contractive condition

1 M ( T x , T y , t ) 1α ( 1 M ( x , T x , t ) 1 ) ( 1 M ( y , T y , t ) 1 ) 1 M ( x , y , t ) 1 +β ( 1 M ( x , y , t ) 1 )

for all x,yX. Then T has a unique fixed point in X.

Proof Let x 0 X. Put x 1 =T x 0 and x n + 1 = T n + 1 x 0 for all n1.

If x n = x n + 1 for some n, then we have nothing to prove.

Assume that x n x n + 1 for all n. Then

1 M ( x n + 1 , x n , t ) 1 = 1 M ( T x n , T x n 1 , t ) 1 α ( 1 M ( x n , T x n , t ) 1 ) ( 1 M ( x n 1 , T x n 1 , t ) 1 ) 1 M ( x n , x n 1 , t ) 1 + β ( 1 M ( x n , x n 1 , t ) 1 ) ,

and so

1 M ( x n + 1 , x n , t ) 1 ( β 1 α ) ( 1 M ( x n , x n 1 , t ) 1 ) ( β 1 α ) n ( 1 M ( x 1 , x 0 , t ) 1 )

for all n.

By using the triangular inequality, for each mn, we obtain

1 M ( x n , x m , t ) 1 1 M ( x n , x n + 1 , t ) 1 + 1 M ( x n + 1 , x n + 2 , t ) 1 + + 1 M ( x m 1 , x m , t ) 1 ( k n + k n + 1 + + k m 1 ) ( 1 M ( x 0 , T x 0 , t ) 1 ) k n 1 k ( 1 M ( x 0 , T x 0 , t ) 1 ) ,

where k= β 1 α . Thus, { x n } is a Cauchy sequence, therefore it converges to some x X. Since t is continuous, it follows T x = x , hence x is a fixed point of T.

Now, suppose that T has another fixed point y x . Then we have

1 M ( x , y , t ) 1 = 1 M ( T x , T y , t ) 1 α ( 1 M ( y , T y , t ) 1 ) ( 1 M ( x , T x , t ) 1 ) 1 M ( x , y , t ) 1 + β ( 1 M ( x , y , t ) 1 ) = β ( 1 M ( x , y , t ) 1 ) < ( 1 M ( x , y , t ) 1 ) ,

which is a contradiction. Hence, T has a unique fixed point. □

We would like to prove that the iterative process utilized above is stable [4, 37]. More accurately, we need this definition.

Definition 2.1 On an intuitionistic fuzzy metric space (X,M,N,,), consider T a selfmap on X, with a fixed point p. For x 0 X, consider the Picard iteration, x n + 1 =T x n , which converges to p. Let ( y n ) be an arbitrary sequence in X. If

[ ( M ( y n + 1 , T y n , t ) 1 ) ( N ( y n + 1 , T y n , t ) 0 ) ] y n p,

we say that the Picard iteration is T-stable.

Corollary 2.1 Provided that the conditions of Theorem  2.3 are fulfilled, suppose that p is the unique fixed point of T. Then the Picard iteration is T-stable.

Proof Indeed, using the triangular condition, we get

1 M ( y n + 1 , p , t ) 1 1 M ( y n + 1 , T y n , t ) 1 + 1 M ( T y n , T p , t ) 1 1 M ( y n + 1 , T y n , t ) 1 + α ( 1 M ( y n , T y n , t ) 1 ) ( 1 M ( p , T p , t ) 1 ) 1 M ( y n , p , t ) 1 + β ( 1 M ( y n , p , t ) 1 ) = 1 M ( y n + 1 , T y n , t ) 1 + β ( 1 M ( y n , p , t ) 1 )

and so

1 M ( y n + 1 , p , t ) 1 1 M ( y n + 1 , T y n , t ) 1+β ( 1 M ( y n , p , t ) 1 ) .

Now, we have to interpret this relation in terms of real sequences. For this purpose, we need the following result, [38].

Lemma 2.1 Let us consider δ[0,1) to be a real number and { ε n } to be a sequence of positive numbers such that lim ε n =0. If { u n } is a sequence of positive real numbers such that u n + 1 δ u n + ε n , then lim u n =0.

Using Lemma 2.1 it follows that lim n y n =p, and the corollary is proved.  □

3 Conclusion

In this work, we introduced some classes of contractive conditions on intuitionistic fuzzy metric spaces endowed with triangular metric. With additional condition of completeness, we introduced new fixed point results for these classes of mappings. A stability result is established.

References

  1. Caristi J: Fixed point theorems for mapping satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.

    Article  MathSciNet  Google Scholar 

  2. Hicks TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 1988, 33: 231–236.

    MathSciNet  Google Scholar 

  3. Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153

    Google Scholar 

  4. Olatinwo MO, Postolache M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 2012, 218(12):6727–6732. 10.1016/j.amc.2011.12.038

    Article  MathSciNet  Google Scholar 

  5. Altun I, Durmaz G: Some fixed point results in cone metric spaces. Rend. Circ. Mat. Palermo 2009, 58: 319–325. 10.1007/s12215-009-0026-y

    Article  MathSciNet  Google Scholar 

  6. Shatanawi W: Some coincidence point results in cone metric spaces. Math. Comput. Model. 2012, 55: 2023–2028. 10.1016/j.mcm.2011.11.061

    Article  MathSciNet  Google Scholar 

  7. Shatanawi W: On w -compatible mappings and common coincidence point in cone metric spaces. Appl. Math. Lett. 2012, 25: 925–931. 10.1016/j.aml.2011.10.037

    Article  MathSciNet  Google Scholar 

  8. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151

    Article  MathSciNet  Google Scholar 

  9. Altun I: Some fixed point theorems for single and multivalued mappings on ordered non-Archimedean fuzzy metric spaces. Iranian J. Fuzzy Syst. 2010, 7(1):91–96.

    MathSciNet  Google Scholar 

  10. Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054

    Google Scholar 

  11. Aydi H, Karapınar E, Postolache M: Tripled coincidence point theorems for weak φ -contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 44

    Google Scholar 

  12. Berinde V: Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 7347–7355. 10.1016/j.na.2011.07.053

    Article  MathSciNet  Google Scholar 

  13. Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28

    Google Scholar 

  14. Rezapour Sh, Amiri P: Some fixed point results for multivalued operators in generalized metric spaces. Comput. Math. Appl. 2011, 61: 2661–2666. 10.1016/j.camwa.2011.03.014

    Article  MathSciNet  Google Scholar 

  15. Shatanawi W, Postolache M: Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60

    Google Scholar 

  16. Shatanawi W, Samet B:On (ψ,φ)-weakly contractive condition in partially ordered metric spaces. Comput. Math. Appl. 2011, 62(8):3204–3214. 10.1016/j.camwa.2011.08.033

    Article  MathSciNet  Google Scholar 

  17. Zhilong L: Fixed point theorems in partially ordered complete metric spaces. Math. Comput. Model. 2011, 54: 69–72. 10.1016/j.mcm.2011.01.035

    Article  Google Scholar 

  18. Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 54

    Google Scholar 

  19. Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535

    Article  MathSciNet  Google Scholar 

  20. Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for (ψ,ϕ) -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012, 63(1):298–309. 10.1016/j.camwa.2011.11.022

    Article  MathSciNet  Google Scholar 

  21. Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870

    Google Scholar 

  22. Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4

    Article  MathSciNet  Google Scholar 

  23. Gregori V, Sapena A: On fixed point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 2002, 125: 245–252. 10.1016/S0165-0114(00)00088-9

    Article  MathSciNet  Google Scholar 

  24. Kramosil O, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetica 1975, 11: 326–334.

    MathSciNet  Google Scholar 

  25. Rafi M, Noorani MSM: Fixed point theorem on intuitionistic fuzzy metric spaces. Iranian J. Fuzzy Syst. 2006, 3(1):23–29.

    MathSciNet  Google Scholar 

  26. Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S0019-9958(65)90241-X

    Article  MathSciNet  Google Scholar 

  27. George A, Veeramani P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 1994, 64: 395–399. 10.1016/0165-0114(94)90162-7

    Article  MathSciNet  Google Scholar 

  28. Park JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051

    Article  MathSciNet  Google Scholar 

  29. Alaca C, Turkoghlu D, Yildiz C: Fixed points in intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2006, 29: 1073–1078. 10.1016/j.chaos.2005.08.066

    Article  MathSciNet  Google Scholar 

  30. Atanassov K: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20: 87–96. 10.1016/S0165-0114(86)80034-3

    Article  MathSciNet  Google Scholar 

  31. Coker D: An introduction to intuitionistic fuzzy metric spaces. Fuzzy Sets Syst. 1997, 88: 81–89. 10.1016/S0165-0114(96)00076-0

    Article  MathSciNet  Google Scholar 

  32. Di Bari C, Vetro C: A fixed point theorem for a family of mappings in a fuzzy metric space. Rend. Circ. Mat. Palermo 2003, 52: 315–321. 10.1007/BF02872238

    Article  MathSciNet  Google Scholar 

  33. Karayilan H, Telci M: Common fixed point theorem for contractive type mappings in fuzzy metric spaces. Rend. Circ. Mat. Palermo 2011, 60: 145–152. 10.1007/s12215-011-0037-3

    Article  MathSciNet  Google Scholar 

  34. Miheţ D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 2004, 144: 431–439. 10.1016/S0165-0114(03)00305-1

    Article  Google Scholar 

  35. Park JS, Kwun YC, Park JH: A fixed point theorem in the intuitionistic fuzzy metric spaces. Far East J. Math. Sci. 2005, 16: 137–149.

    MathSciNet  Google Scholar 

  36. Schweizer B, Sklar A: Statistical metric spaces. Pac. J. Math. 1960, 10: 314–334.

    Google Scholar 

  37. Haghi RH, Postolache M, Rezapour S: On T -stability of the Picard iteration for generalized φ -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971

    Google Scholar 

  38. Berinde V: On stability of some fixed point procedures. Bul. Stiint. Univ. Baia Mare Ser. B Fasc. Mat.-Inform. 2002, 18: 7–14.

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors thank the referees for their remarks on the first version of our article.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Cristiana Ionescu.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors completed the paper together. All authors read and approved the final manuscript.

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

Ionescu, C., Rezapour, S. & Samei, M.E. Fixed points of some new contractions on intuitionistic fuzzy metric spaces. Fixed Point Theory Appl 2013, 168 (2013). https://doi.org/10.1186/1687-1812-2013-168

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1687-1812-2013-168

Keywords