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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 [5–7], partially ordered metric spaces [8–17], partial metric spaces [18], Menger spaces [19], G-metric spaces [20, 21], and fuzzy metric spaces [22–25].

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, [29–31]. 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 [32–35].

For necessary notions to our results, such as continuous t-norm, intuitionistic fuzzy metric space and the induced topology, which is denoted by Ï„ ( 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≤hmax { 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

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 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.

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

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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

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