Fixed point results in fuzzy metric spaces via α and β κ –admissible mappings with application to integral types

1 Introduction and preliminaries

Firstly, we recalled the concept of continuous t-norm with introduced by Schweizer et al. [1] in 1960.

A binary operation ★ : [0, 1] × [0, 1] → [0, 1] is a continuous t-norm if (i) ★ is associative and commutative, (ii) ★ is continuous, (iii) a ★ 1 = a for all a ∈ [0, 1] and (iv) a ★ b ≤ c ★ d whenever a ≤ c and b ≤ d for all a, b, c, d ∈ [0, 1]. A t-norm ★ is called positive, if a ★ b > 0 for all a, b ∈ (0, 1). The examples of continuous t-norm are Lukasievicz t-norm, product t-norm, minimum t-norm, that is, a ★  _L b = max {a + b - 1, 0}, a ★  _P b = ab and a ★  _M b = min {a, b}, respectively.

George and Veeramani [3] modified the concept of a Hausdorff topology in fuzzy metric spaces due to Kramosil and Michálek [2] as follows. Let X be an arbitrary nonempty set, ★ is a continuous t-norm, and M is a fuzzy set on X × X × ( 0 , ∞ ) .

Definition 1.1. [3] ( X , M , ★ ) is called a fuzzy metric space if satisfying the following conditions,

M ( x , y , t ) > 0 for all x , y ∈ X and t > 0,

Remark 1.2. By Definition 1.1, we can see that 0 < M ( x , y , t ) < 1 , for all x , y ∈ X and t > 0 provided x ≠ y (see [23]).

The open ball B ( x , ι , t ) with a center x ∈ X and a radius 0 < ι < 1 for all t > 0 defined by B ( x , ι , t ) = { y ∈ X : M ( x , y , t ) > 1 - ι } . A subset A in X is called open if for each x ∈ A , there exist t > 0 and 0 < ι < 1 such that B ( x , ι , t ) ⊂ A . Let τ denote the family of all open subsets of X . Then τ is a topology on X , called the topology induced by the fuzzy metric M . This topology is metrizable (see in [24]).

Definition 1.3. [3] Let ( X , M , ★ ) be a fuzzy metric space. A sequence {x _n } in X is said to be convergent to a point x ∈ X if lim n → ∞ M ( x n , x , t ) = 1 for all t > 0. A sequence {x _n } in X is called a Cauchy sequence if, for each 0 < δ < 1 and t > 0, there exits n 0 ∈ ℕ such that M ( x n , x m , t ) > 1 - δ for each n, m ≥ n₀. A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

Lemma 1.4. [6] Let ( X , M , ★ ) be a fuzzy metric space. M ( x , y , · ) is non-decreasing function, for all x , y ∈ X .

Definition 1.5 Let ( X , M , ★ ) be a fuzzy metric space. Then the mapping M is said to be continuous on X × X × ( 0 , ∞ ) if lim n → ∞ M ( x n , y n , t n ) = M ( x , y , t ) , when {(x _n , y _n , t _n )} is a sequence in X × X × ( 0 , ∞ ) which converges to a point, i.e., lim n → ∞ M ( x n , x , t ) = lim n → ∞ M ( y n , y , t ) = 1 , lim n → ∞ M ( x , y , t n ) = M ( x , y , t ) .

Afterwards, many mathematicians presented fixed point theorems in complete fuzzy metric spaces [4–17].

The concept of an altering distance function was defined by Khan et al. [18] in 1984 which improved the metric fixed point theory by introducing a control function.

Definition 1.6. A function φ : [0, 1] → [0, 1] is an altering distance if φ is strictly decreasing, left continuous and φ (ς) =0 if and only if ς = 1. Obviously, we obtain that lim ς → 1 - φ ( ς ) = φ ( 1 ) = 0 . We denote Φ : = { φ : [ 0 , 1 ] → [ 0 , 1 ] : φ is altering distance } .

In 2012, Shen et al. [19] introduced the notion of altering distance in fuzzy metric spaces and gave a fixed point results in complete and compact fuzzy metric spaces. Recently, Došenovića et al. [20] introduced and proved fixed point theorems in complete and compact fuzzy metric spaces using altering distance.

The concept of α-admissible for single valued mappings was introduced by Samet et al. [21]. Afterwards, Salimi et al. [22] introduced β _κ -admissible mapping with generalized the concept of α-admissible. The concept of α-admissible and β _κ -admissible mappings as follows:

Definition 1.7. [21] Let F : X → X and α : X × X → [ 0 , + ∞ ) . Then F is an α-admissible mapping if α ( x , y ) ≥ 1 ⇒ α ( F ( x ) , F ( y ) ) ≥ 1 , x , y ∈ X .

Definition 1.8. [22] Let F : X → X , β : X × ( 0 , + ∞ ) → [ 0 , + ∞ ) and κ : (0, + ∞) → (0, 1). Then F is a β _κ -admissible mapping if β ( x , t ) ≤ κ ( t ) ⇒ β ( F ( x ) , t ) ≤ κ ( t ) , for all x ∈ X , t > 0.

In this paper, inspired by Došenovića et al. [20] and Salimi et al. [22] work mentioned above, we study and prove the existence and uniqueness of fixed points for generalized fuzzy contractive mappings in complete fuzzy metric spaces in the sense of George and Veeramani [3] and give an example to illustrate our main results. As an application, we deduce fixed point theorems in generalized fuzzy contractive mappings of integral type in fuzzy metric spaces.

2 Fixed point results in fuzzy metric spaces via α and β _κ –admissible mappings

In this section, we prove the existence and uniqueness of fixed point theorems in fuzzy metric spaces via α-admissible and β _κ –admissible mappings.

Theorem 2.1. Let ( X , M , ★ ) be a complete fuzzy metric space, F are α and β _κ –admissible mappings and φ ∈ Φ such that α ( x , F ( x ) ) α ( y , F ( y ) ) ≥ 1 implies that

φ ( M ( F ( x ) , F ( y ) , t ) ) ≤ β ( x , t ) β ( y , t ) N ( x , y , t ) ,(2.1) where N ( x , y , t ) = max { φ ( M ( x , y , t ) ) , φ ( M ( x , F ( x ) , t ) ) , φ ( M ( y , F ( y ) , t ) ) } .

Suppose that the following conditions hold:

there exists x 0 ∈ X such that α ( x 0 , F ( x 0 ) ) ≥ 1 and β ( x 0 , t ) ≤ κ ( t ) for all t > 0;

Then F has a unique fixed point x^* in X such that α ( x * , F ( x * ) ) ≥ 1 and β (x, t) <1 for all x ∈ X and t > 0.

Proof 1. Let x 0 ∈ X such that α ( x 0 , F ( x 0 ) ) ≥ 1 and β ( x 0 , t ) ≤ κ ( t ) for all t > 0 (from condition (b)). We define the sequence {x _n } in X such that x n = F ( x n - 1 ) for all n ∈ ℕ .

Since F are α and β _κ -admissible mapping, we have α ( x 0 , x 1 ) = α ( x 0 , F ( x 0 ) ) ≥ 1 , we deduce that α ( x 1 , F ( x 1 ) ) = α ( F ( x 0 ) , F ( x 1 ) ) ≥ 1 and β ( x 1 , t ) = β ( F ( x 0 ) , t ) ≤ κ ( t ) .

By continuing this process, we get α ( x n - 1 , F ( x n - 1 ) ) ≥ 1 for all n ∈ ℕ . This implies that α ( x n - 1 , F ( x n - 1 ) ) α ( x n , F ( x n ) ) ≥ 1 for all n ∈ ℕ . Similarly, we deduce that β ( x n , t ) ≤ κ ( t ) for all n ∈ ℕ ∪ { 0 } and all t > 0. From (2.1), we obtain that φ ( M ( x n , x n + 1 , t ) ) = φ ( M ( F ( x n - 1 ) , F ( x n ) , t ) ≤ β ( x n - 1 , t ) β ( x n , t ) N ( x n - 1 , x n , t )

where N ( x n - 1 , x n , t ) = max { φ ( M ( x n - 1 , x n , t ) ) , φ ( M ( x n - 1 , F ( x n - 1 ) , t ) ) , φ ( M ( x n , F ( x n ) , t ) ) } = max { φ ( M ( x n - 1 , x n , t ) ) , φ ( M ( x n - 1 , x n , t ) ) , φ ( M ( x n , x n + 1 , t ) ) } = max { φ ( M ( x n - 1 , x n , t ) ) , φ ( M ( x n , x n + 1 , t ) ) } .

If max { φ ( M ( x n - 1 , x n , t ) ) , φ ( M ( x n , x n + 1 , t ) ) } = φ ( M ( x n , x n + 1 , t ) ) , then φ ( M ( x n , x n + 1 , t ) ) ≤ β ( x n - 1 , t ) β ( x n , t ) φ ( M ( x n , x n + 1 , t ) ) ≤ κ ( t ) κ ( t ) φ ( M ( x n , x n + 1 , t ) ) ≤ κ ( t ) φ ( M ( x n , x n + 1 , t ) ) < φ ( M ( x n , x n + 1 , t ) ) , a contradiction. Therefore max { φ ( M ( x n - 1 , x n , t ) ) , φ ( M ( x n , x n + 1 , t ) ) } = φ ( M ( x n - 1 , x n , t ) ) ) so we have φ ( M ( x n , x n + 1 , t ) ) ≤ κ ( t ) φ ( M ( x n - 1 , x n , t ) ) ) .

Repeating the same process we conclude that φ ( M ( x n , x n + 1 , t ) ) ≤ ( κ ( t ) ) n φ ( M ( x 0 , x 1 , t ) ) , for all n ∈ ℕ . Letting n→ ∞, we have lim n → ∞ φ ( M ( x n , x n + 1 , t ) ) = 0 .

Since the function φ left-continuous and by condition (M2), we get

lim n → ∞ M ( x n + 1 , x n , t ) = 1 .(2.2)

Now, we prove that {x _n } is a Cauchy sequence. Let prove by contradiction, that is {x _n } is not a Cauchy sequence, then there exists ɛ ∈ (0, 1), which we can find two subsequences {x_m(k)} and {x_n(k)} of {x _n } with m (k) > n (k) ≥ k such that M ( x m ( k ) , x n ( k ) , t ) ≤ 1 - ɛ , for all positive integer k and t > 0.

Assume that m (k) is the least integer exceeding n (k) satisfying the above inequality, that is, equivalent, M ( x m ( k ) - 1 , x n ( k ) , t ) > 1 - ɛ , for all positive integer k and t > 0. By (2.2) follows that for ɛ₁ and ɛ₂ there exist n 1 , n 2 ∈ ℕ such that M ( x m ( k ) , x m ( k ) - 1 , t ) ≥ 1 - ɛ 1 , M ( x n ( k ) , x n ( k ) - 1 , t ) ≥ 1 - ɛ 1 for all n > n₁ and t > 0. M ( x n ( k ) , x n ( k ) - 1 , t 2 ) ≥ 1 - ɛ 2 , for all n > n₂ and t > 0. So, we have M ( x m ( k ) - 1 , x n ( k ) - 1 , t ) ≥ M ( x m ( k ) - 1 , x n ( k ) , t 2 ) ★ M ( x n ( k ) , x n ( k ) - 1 , t 2 ) ≥ ( 1 - ɛ ) ★ ( 1 - ɛ 2 ) for all n > max {n₁, n₂}. Letting n→ ∞, we get M ( x m ( k ) - 1 , x n ( k ) - 1 , t ) ≥ ( 1 - ɛ ) ★ 1 = 1 - ɛ .

Applying (2.1), we have that φ ( 1 - ɛ ) < φ ( M ( x m ( k ) , x n ( k ) , t ) ) ≤ β ( x m ( k ) - 1 , t ) β ( x n ( k ) - 1 , t ) N ( x m ( k ) - 1 , x n ( k ) - 1 , t ) ≤ κ ( t ) κ ( t ) max { φ ( M ( x m ( k ) - 1 , x n ( k ) - 1 , t ) ) , φ ( M ( x m ( k ) - 1 , x m ( k ) , t ) ) , φ ( M ( x n ( k ) - 1 , x n ( k ) , t ) ) } ≤ κ ( t ) max { φ ( 1 - ɛ ) , φ ( 1 - ɛ ) , φ ( 1 - ɛ ) } < φ ( 1 - ɛ )

which is a contradiction.

If it is not possible to find sequences {m (k)} and {n (k)} with m (k) > n (k) ≥ k such that M ( x m ( k ) , x n ( k ) , t ) ≤ 1 - ɛ , for all positive integer k and t > 0, then there exists n 0 ∈ ℕ 0 such that M ( x n 0 + r + 2 , x n 0 + r + 1 , t ) ≤ 1 - ɛ , for any r ∈ ℕ . Since, { M ( x n 0 + r + 2 , x n 0 + r + 1 , t ) } is a monotone and bounded sequence, for any t > 0, i.e. lim r → ∞ M ( x n 0 + r + 2 , x n 0 + r + 1 , t ) = z , for some z ∈ (0, 1 - ɛ]. Then,

φ ( M ( x n 0 + r + 2 , x n 0 + r + 1 , t ) ) ≤ β ( x n 0 + r + 2 , t ) β ( x n 0 + r + 1 , t ) N ( x n 0 + r + 2 , x n 0 + r + 1 , t ) ≤ κ ( t ) κ ( t ) max { φ ( M ( x n 0 + r + 2 , x n 0 + r + 1 , t ) ) , φ ( M ( x n 0 + r + 2 , x n 0 + r + 3 , t ) ) , φ ( M ( x n 0 + r + 1 , x n 0 + r , t ) ) } .

Letting r→ ∞, we have φ ( z ) ≤ κ ( t ) max { φ ( z ) , φ ( z ) , φ ( z ) } < φ ( z ) which is a contradiction. Thus, {x _n } is a Cauchy sequence and ( X , M , ★ ) complete then there exists x^* ∈ X such that x _n  → x^* as n→ ∞, that means, M ( x n , x * , t ) = 1 as n→ ∞, for each t > 0. Since, x _n  ≠ x_n+1 for all n ∈ ℕ ∪ { 0 } , we get 0 < M ( x n , x n + 1 , t ) < 1 for all t > 0. Since, α (x_n-1, x _n ) ≥1 for all n ∈ ℕ . By condition (b), we have α ( x * , F ( x * ) ) ≥ 1 . So, we get α ( x n - 1 , F ( x n - 1 ) ) α ( x * , F ( x * ) ) ≥ 1 for all n ∈ ℕ , with x = x_n-1 and y = x^*. Again, by condition (2.1), we have φ ( M ( x n , F ( x * ) , t ) ) ≤ β ( x n - 1 , t ) β ( x * , t ) N ( x n - 1 , x * , t ) ≤ κ ( t ) κ ( t ) N ( x n - 1 , x * , t ) ≤ κ ( t ) N ( x n - 1 , x * , t ) where N ( x n - 1 , x * , t ) = max { φ ( M ( x n - 1 , x * , t ) ) , φ ( M ( x n - 1 , x n ) , t φ ( M ( x * , F ( x * ) , t ) ) } .

Letting n→ ∞ in the above inequality, we get φ ( M ( x * , F ( x * ) , t ) ) ≤ κ ( t ) max { φ ( 1 ) , φ ( 1 ) , φ ( M ( x * , F ( x * ) , t ) ) } = κ ( t ) φ ( M ( x * , F ( x * ) , t ) ) .

So, we have ( 1 - κ ( t ) ) φ ( M ( x * , F ( x * ) , t ) ) ≤ 0 , for all t > 0. Therefore, φ ( M ( x * , F ( x * ) , t ) ) = 0 , it follows that, M ( x * , F ( x * ) , t ) = 1 and so x * = F ( x * ) . That is x^* is a fixed point of F .

Next, we prove the uniqueness of the fixed point of F . Let y^* be another fixed point of F , x^* ≠ y^* and M ( x * , y * , t ) < 1 such that α ( y * , F ( y * ) ) ≥ 1 and β (x, t) <1 for all x ∈ X and t > 0. So, we get α ( x * , F ( x * ) ) α ( y * , F ( y * ) ) ≥ 1 .

It follows that φ ( M ( x * , y * , t ) ) = φ ( M ( F ( x * ) , F ( y * ) , t ) ) ≤ β ( x * , t ) β ( y * , t ) N ( x * , y * , t ) ≤ κ ( t ) κ ( t ) N ( x * , y * , t ) ≤ κ ( t ) N ( x * , y * , t ) where N ( x * , y * , t ) = max { φ ( M ( x * , y * , t ) ) , φ ( M ( x * , F ( x * ) , t ) , φ ( M ( y * , F ( y * ) , t ) ) .

It follows that φ ( M ( x * , y * , t ) ) ≤ κ ( t ) N ( x * , y * , t ) ≤ κ ( t ) φ ( M ( x * , y * , t ) ) < φ ( M ( x * , y * , t ) ) . which is a contradiction hence M ( x * , y * , t ) = 1 for t > 0, that is, x^* = y^*. Therefore, x^* is the unique fixed point of F such that α ( x * , F ( x * ) ) ≥ 1 and β (x, t) <1. This completes the proof.

In view of Theorem 2.1, we have the following Corollary 2.2.

Corollary 2.2. Let ( X , M , ★ ) be a complete fuzzy metric space, F are α and β _κ –admissible mappings and φ ∈ Φ such that α ( x , F ( x ) ) α ( y , F ( y ) ) ≥ 1 implies that ( φ ( M ( F ( x ) , F ( y ) , t ) ) + λ ) α ( x , F ( x ) ) α ( y , F ( y ) ) ≤ β ( x , t ) β ( y , t ) N ( x , y , t ) + λ , where N ( x , y , t ) = max { φ ( M ( x , y , t ) ) , φ ( M ( x , F ( x ) , t ) ) , φ ( M ( y , F ( y ) , t ) ) } for all x, y ∈ X, x ≠ y, t > 0 and λ > 0. Suppose that the following conditions hold:

there exists x 0 ∈ X such that α ( x 0 , F ( x 0 ) ) ≥ 1 and β ( x 0 , t ) ≤ κ ( t ) for all t > 0;

Then F has a unique fixed point x^* in X such that α ( x * , F ( x * ) ) ≥ 1 and β (x, t) <1 for all x ∈ X and t > 0.

Corollary 2.3. Let ( X , M , ★ ) be a complete fuzzy metric space, F are α and β _κ –admissible mappings and φ ∈ Φ such that α ( x , F ( x ) ) α ( y , F ( y ) ) ≥ 1 implies that α ( x , F ( x ) ) α ( y , F ( y ) ) φ ( M ( F ( x ) , F ( y ) , t ) ) ≤ β ( x , t ) β ( y , t ) N ( x , y , t ) , where N ( x , y , t ) = max { φ ( M ( x , y , t ) ) , φ ( M ( x , F ( x ) , t ) ) , φ ( M ( y , F ( y ) , t ) ) } for all x , y ∈ X , x ≠ y, t > 0. Suppose that the following conditions hold:

there exists x 0 ∈ X such that α ( x 0 , F ( x 0 ) ) ≥ 1 and β ( x 0 , t ) ≤ κ ( t ) for all t > 0;

Then F has a unique fixed point x^* in X such that α ( x * , F ( x * ) ) ≥ 1 and β (x, t) <1 for all x ∈ X and t > 0.

On the other hand, by taking α (x, y) =1 and β (x, t) ² = κ (t) in Corollary (2.3), we deduce the following correct version of Theorem 2.1. in [20].

Corollary 2.4. [20] Let ( X , M , ★ ) be a complete fuzzy metric space and F : X → X . Let φ ∈ Φ and κ : (0, + ∞) → (0, 1) such that: φ ( M ( F ( x ) , F ( y ) , t ) ) ≤ κ ( t ) N ( x , y , t ) , where N ( x , y , t ) = max { φ ( M ( x , y , t ) ) , φ ( M ( x , F ( x ) , t ) ) , φ ( M ( y , F ( y ) , t ) ) } for all x , y ∈ X , x ≠ y, t > 0. Then F has a unique fixed point x^* in X for all x ∈ X and t > 0.

Now, we give an example to support Theorem 2.1.

Example 2.5. Let X = [ 0 , ∞ ) and define d (x, y) = |x - y| with d (x, y) < t. Denote a ★ b is Lukasievicz t-norm for any a, b ∈ [0, 1] and M ( x , y , t ) = 1 - d ( x , y ) t , for x , y ∈ X and t > 0. Then it easy to see that ( X , M , ★ ) is a complete fuzzy metric space [7]. Define the mapping F : X → X by F ( x ) = { x 6 , x ∈ [ 0 , 1 ] , x 2 , x ∈ ( 1 , ∞ ) .

Define α : X × X → [ 0 , ∞ ) , β : X × ( 0 , + ∞ ) → [ 0 , + ∞ ) and κ : (0, + ∞) → (0, 1) by α ( x , y ) = { 1 , x ∈ [ 0 , 1 ] , 0 , otherwise . β ( x , t ) = { 1 2 , x ∈ [ 0 , 1 ] , 1 , otherwise .

Also, define φ (s) =1 - s for all s ∈ (0, 1) and κ ( t ) = 2 3 .

Now, we show that F is an α-admissible mapping. Let x , y ∈ X with α ( x , y ) ≥ 1 , then x, y ∈ [0, 1]. On the other hand, for all x, y ∈ [0, 1], we have F ( x ) = x 6 ≤ 1 . It follows that α ( F ( x ) , F ( y ) ) ≥ 1 .

Hence, F is an α-admissible mapping. In reason of the above arguments, α ( 0 , F ( 0 ) ) ≥ 1 . Let {x _n } is a sequence in X such that α (x _n , x_n+1) ≥1 for all n ∈ ℕ ∪ { 0 } and x _n  → x as n→ ∞, then {x _n } ⊂ [0, 1]. This implies that α (x _n , x) ≥1 for all n ∈ ℕ .

Next, we show that F is β _κ –admissible mapping. Let x ∈ X with β ( x , t ) = 1 2 = 1 2 ≤ κ ( t ) then x, y ∈ [0, 1]. On the other hand, for all x, y ∈ [0, 1], F ( x ) = x 6 < 1 , we have β ( F ( x ) , t ) = 1 2 ≤ κ ( t ) . So, F is β _κ –admissible mapping. We will check that the contractive condition of Theorem 2.1 is fulfilled for 0 ≤ x, y ≤ 1 with F ( x ) = x 6 and F ( y ) = y 6 with x ≠ y, we get α ( x , F ( x ) ) α ( y , F ( y ) ) ≥ 1 .

By condition (M3), suppose that x > y. For this, we consider the following cases:

Case 1: y < x 6 , for each 0 ≤ x, y ≤ 1, we have N ( x , y , t ) = φ ( M ( x , y , t ) )

This implies that φ ( M ( F ( x ) , F ( y ) , t ) ) = 1 - M ( F ( x ) , F ( y ) , t ) = d ( F ( x ) , F ( y ) ) t = | F ( x ) - F ( y ) | t = | x 6 - y 6 | t = | x - y | 6 t < 1 2 ( | x - y | t ) = 1 2 ( d ( x , y ) t ) = β ( x , t ) β ( x , t ) φ ( M ( x , y , t ) ) .

Figure 1. Shows the contractive condition of Example 2.5. Case 1, 3D view which the upper plan is right hand side functions (R.H.S.) and the lower plan is left hand side functions (L.H.S.).

OPEN IN VIEWER

Fig.1

Graph of contractive condition of Example Case1, 3D view.

Figure 2. Shows the contractive condition of Example 2.5. Case 1, 2D view which the red line is right hand side functions (R.H.S.) and the blue line is left hand side functions (L.H.S.).

OPEN IN VIEWER

Fig.2

Graph of contractive condition of Example Case1, 2D view.

Case 2: y ≥ x 6 , for each 0 ≤ x, y ≤ 1, we have N ( x , y , t ) = φ ( M ( x , F ( x ) , t ) ) This implies that φ ( M ( F ( x ) , F ( y ) , t ) ) = 1 - M ( F ( x ) , F ( y ) , t ) = d ( F ( x ) , F ( y ) ) t = | F ( x ) - F ( y ) | t = | x 6 - y 6 | t = | x - y | 6 t < 1 2 ( | x - x 6 | t ) = 5 x 12 t = β ( x , t ) β ( x , t ) φ ( M ( x , y , t ) ) .

Figure 3. Shows the contractive condition of Example 2.5. Case 2, 3D view which the upper plan is right hand side functions (R.H.S.) and the lower plan is left hand side functions (L.H.S.).

OPEN IN VIEWER

Fig.3

Graph of contractive condition of Example Case2, 3D view.

Figure 4. Shows the contractive condition of Example 2.5. Case 2, 2D view which the red line is right hand side functions (R.H.S.) and the blue line is left hand side functions (L.H.S.).

OPEN IN VIEWER

Fig.4

Graph of contractive condition of Example Case2, 3D view.

Therefore, all of conditions in Theorem 2.1 are satisfied. In this example, we have 0 ∈ X is a unique fixed point of F . We can see that Corollarys 2.2 and 2.3 can be applicable to this example.

Now, we show that Corollary 2.4 cannot be applied to F . Taking x = 2, y = 4 and t = 3, we can see that F are not α-admissible and β _κ –admissible mappings. φ ( M ( F ( x ) , F ( y ) , t ) ) = 1 - M ( F ( x ) , F ( y ) , t ) = | F ( x ) - F ( y ) | t = | 4 - 16 | 3 = 4 > 4 9 = 2 3 ( d ( x , y ) t ) .

Therefore Corollary 2.4 cannot be applied to F . This complete the prove.

As an application of our results obtained in previous sections, we present some fixed point results for contraction mappings via α and β _κ –admissible mappings with altering distance of an integral type. For this purpose, we first define the following class of functions: Γ = { φ ( s ) : ( 0 , + ∞ ) → ( 0 , + ∞ ) } such that φ is nonnegative, Lebesgue integrable, summable on each compact subset of positive real numbers and satisfies 0 < ∫ 0 ζ φ ( s ) ds < 1 for each 0 < ζ < 1 .

The following theorem is our result on the existence of fixed points or contraction mappings via α and β _κ –admissible mappings of integral type in fuzzy metric spaces.

Theorem 3.1. Let ( X , M , ★ ) be a complete fuzzy metric space, F are α and β _κ –admissible mappings. Let function φ ∈ Φ and φ ∈ Γ with α ( x , F ( x ) ) α ( y , F ( y ) ) ≥ 1 , implies

∫ 0 φ ( M ( F ( x ) , F ( y ) , t ) ) φ ( s ) ds ≤ β ( x , t ) β ( y , t ) ∫ 0 N ( x , y , t ) φ ( s ) ds ,(3.1) where N ( x , y , t ) = max { φ ( M ( x , y , t ) ) , φ ( M ( x , F ( x ) , t ) ) , φ ( M ( y , F ( y ) , t ) ) } holds for all x , y ∈ X , x ≠ y and t > 0. Suppose that the following conditions hold:

there exists x 0 ∈ X such that α ( x 0 , F ( x 0 ) ) ≥ 1 and β ( x 0 , t ) ≤ κ ( t ) for all t > 0;

Then f has a unique fixed point x^* in X such that α ( x * , F ( x * ) ) ≥ 1 and β (x, t) <1 for all x ∈ X and t > 0.

Proof. Following the lines in the proof of Theorem 2.1 and we consider ψ : [0, 1] → [0, 1] define by ψ ( δ ) = ∫ 0 δ φ ( s ) ds for all δ ∈ (0, 1). It is clear that ψ is altering distance functions. Then the results follow immediately from Theorem 2.1. This completes the proof.

Corollary 3.2. Let ( X , M , ★ ) be a complete fuzzy metric space, F are α and β _κ –admissible mappings. Let φ ∈ Φ and φ ∈ Γ with α ( x , F ( x ) ) α ( y , F ( y ) ) ≥ 1 , implies α ( x , F ( x ) ) α ( y , F ( y ) ) ∫ 0 φ ( M ( F ( x ) , F ( y ) , t ) ) φ ( s ) ds ≤ β ( x , t ) β ( y , t ) ∫ 0 N ( x , y , t ) φ ( s ) ds , where N ( x , y , t ) = max { φ ( M ( x , y , t ) ) , φ ( M ( x , F ( x ) , t ) ) , φ ( M ( y , F ( y ) , t ) ) } for all x , y ∈ X , x ≠ y, t > 0. Then F has a unique fixed point x^∗ in X such that α ( x ∗ , F ( x ∗ ) ) ≥ 1 and β (x, t) <1 for all x ∈ X and t > 0.

In the present work we introduced some classes of fuzzy contractive mappings with altering distance via α and β _κ –admissible mappings in the fuzzy metric spaces. Also, we derived the existence and uniqueness of fixed points for this mappings in the complete fuzzy metric spaces and we also give some example and graph which demonstrate the validity of the hypotheses of our main results.

Finally, we showed some fixed point results for mappings satisfying some classes of fuzzy contractive mappings with altering distance via α and β _κ –admissible mappings of an integral type.

Competing interests

The authors declare that they have no competing interests.

Authors contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.