Open Access

Fixed points of some new contractions on intuitionistic fuzzy metric spaces

  • Cristiana Ionescu1Email author,
  • Shahram Rezapour2 and
  • Mohamad Esmaeil Samei2
Fixed Point Theory and Applications20132013:168

DOI: 10.1186/1687-1812-2013-168

Received: 21 March 2013

Accepted: 3 June 2013

Published: 26 June 2013

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.

Keywords

contractive mapping fixed point intuitionistic metric space

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 ) > 1 ε 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 , z X 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 : X X be a continuous mapping satisfying the contractive condition
1 M ( T x , T y , t ) 1 h max { 1 M ( x , T x , t ) 1 , 1 M ( y , T y , t ) 1 }

for all x , y X . 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 n 1 .

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

Thus,
1 M ( x n + 1 , x n , t ) 1 h ( 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 : X X 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 , y X . 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 r 0 .

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 ) = | x y | . 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 , y X and t 0 . Define the selfmap T on X by
T x = { 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 0 y < 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 : X X be a continuous mapping which satisfies the contractive condition
1 M ( T x , T y , t ) 1 α ( 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 , y X . 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 n 1 .

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 m n , 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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, Azarbaijan University of Shahid Madani
(2)
Faculty of Applied Sciences, University Politehnica of Bucharest

References

  1. Caristi J: Fixed point theorems for mapping satisfying inwardness conditions. Trans. Am. Math. Soc. 1976, 215: 241–251.MathSciNetView ArticleGoogle Scholar
  2. Hicks TL: Fixed point theorems for quasi-metric spaces. Math. Jpn. 1988, 33: 231–236.MathSciNetGoogle Scholar
  3. Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153Google 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.2011.12.038MathSciNetView ArticleGoogle 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-yMathSciNetView ArticleGoogle Scholar
  6. Shatanawi W: Some coincidence point results in cone metric spaces. Math. Comput. Model. 2012, 55: 2023–2028. 10.1016/j.mcm.2011.11.061MathSciNetView ArticleGoogle 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.2011.10.037MathSciNetView ArticleGoogle Scholar
  8. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151MathSciNetView ArticleGoogle 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.MathSciNetGoogle 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 359054Google 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 44Google 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.2011.07.053MathSciNetView ArticleGoogle Scholar
  13. Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28Google 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.2011.03.014MathSciNetView ArticleGoogle 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 60Google 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.2011.08.033MathSciNetView ArticleGoogle Scholar
  17. Zhilong L: Fixed point theorems in partially ordered complete metric spaces. Math. Comput. Model. 2011, 54: 69–72. 10.1016/j.mcm.2011.01.035View ArticleGoogle 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 54Google Scholar
  19. Menger K: Statistical metrics. Proc. Natl. Acad. Sci. USA 1942, 28: 535–537. 10.1073/pnas.28.12.535MathSciNetView ArticleGoogle 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.2011.11.022MathSciNetView ArticleGoogle 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 815870Google Scholar
  22. Grabiec M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 1988, 27: 385–389. 10.1016/0165-0114(88)90064-4MathSciNetView ArticleGoogle 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)00088-9MathSciNetView ArticleGoogle Scholar
  24. Kramosil O, Michalek J: Fuzzy metric and statistical metric spaces. Kybernetica 1975, 11: 326–334.MathSciNetGoogle Scholar
  25. Rafi M, Noorani MSM: Fixed point theorem on intuitionistic fuzzy metric spaces. Iranian J. Fuzzy Syst. 2006, 3(1):23–29.MathSciNetGoogle Scholar
  26. Zadeh LA: Fuzzy sets. Inf. Control 1965, 8: 338–353. 10.1016/S0019-9958(65)90241-XMathSciNetView ArticleGoogle 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)90162-7MathSciNetView ArticleGoogle Scholar
  28. Park JH: Intuitionistic fuzzy metric spaces. Chaos Solitons Fractals 2004, 22: 1039–1046. 10.1016/j.chaos.2004.02.051MathSciNetView ArticleGoogle 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.2005.08.066MathSciNetView ArticleGoogle Scholar
  30. Atanassov K: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20: 87–96. 10.1016/S0165-0114(86)80034-3MathSciNetView ArticleGoogle Scholar
  31. Coker D: An introduction to intuitionistic fuzzy metric spaces. Fuzzy Sets Syst. 1997, 88: 81–89. 10.1016/S0165-0114(96)00076-0MathSciNetView ArticleGoogle 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/BF02872238MathSciNetView ArticleGoogle 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/s12215-011-0037-3MathSciNetView ArticleGoogle Scholar
  34. Miheţ D: A Banach contraction theorem in fuzzy metric spaces. Fuzzy Sets Syst. 2004, 144: 431–439. 10.1016/S0165-0114(03)00305-1View ArticleGoogle 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.MathSciNetGoogle 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 658971Google 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.MathSciNetGoogle Scholar

Copyright

© Ionescu et al.; licensee Springer. 2013

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