New fuzzy φ -fixed point results employing a new class of fuzzy contractive mappings

1 Introduction

In 1975, Kramosil and Michalek [11] introduced the notion of fuzzy metric space. Later, George and Veeramani [5] modified the concept of fuzzy metric spaces due to Kramosil and Michalek [11] with a view to have a Hausdorff topology. Like other areas in mathematics, fuzzy metric fixed point theory is also flourishing and by now there exists a considerable literature on fuzzy metric fixed point theory (e.g. [2, 22–24]). In the same continuation, Gregori and Sapena [8] introduced the idea of fuzzy contractive mappings and proved certain fixed point results for such mappings. As a marked improvement, Mihet [14] introduced the notion of fuzzy ψ-contractive mappings which generalizes the concept of fuzzy contractive mappings. Recently, Wardowski [25] defined the concept of fuzzy H -contractive mappings and utilized the same to prove some fixed point results in complete fuzzy metric spaces. Thereafter, Gregori and Miñana [7] also showed that the class of fuzzy H -contractive mappings are included in the class of fuzzy ψ-contractive mappings. Very recently, Melliani et al. [12] introduced the concept of F Z -contractive mappings and showed that several concepts discussed earlier namely: fuzzy contractive, fuzzy ψ-contractive and fuzzy H -contractive can be derived using the concept of F Z -contractive mappings.

In 2014, Jleli et al. [10] introduced the notion of φ-fixed point and obtained some φ-fixed point results in metric spaces. Thereafter, Sezen et al. [21] introduced the concept of fuzzy φ-fixed point and also established some existence and uniqueness of fuzzy φ-fixed points for some mappings in fuzzy metric spaces.

In this paper, following this direction of research, we introduce the concept of ‘ ( F Z , F, φ)-contractive mappings’ and utilize the same to improve, generalize and unify several previously known concepts such as: fuzzy contractive [8], fuzzy ψ-contractive [14], fuzzy H -contractive [25] and fuzzy F Z -contractive mappings [12]. We also establish some φ-fixed point theorems for such mappings in complete fuzzy metric spaces (in the sense of George and Veeramani [5]) besides deriving several consequences in fuzzy metric spaces. As an application, we prove an existence and uniqueness result for the solution of first order periodic differential equation.

2 Preliminaries

Throughout this paper ℕ denotes the set of all natural numbers, ℚ denotes the set of all rational numbers, ℕ 0 = ℕ ∪ { 0 } and ℝ stands for the set of all real numbers. To make our paper self contained, we begin with some definitions and basic properties of fuzzy metric spaces.

Definition 2.1. [18] A binary operation * : [0, 1] × [0, 1] → [0, 1] is said to be a continuous triangular norm (or continuous t-norm) if it satisfies the following conditions:

* is associative and commutative,

For basic examples of continuous t-norm, one may consider the following:

* (a, b) = a · b, for all a, b ∈ [0, 1],

The concept of fuzzy metric spaces in the sense of George and Veeramani [5] is defined as follows:

Definition 2.2. [5] An ordered triple (X, M, *) is said to be a fuzzy metric space if X is a non-empty set, * is a continuous t-norm and M : X² × (0 . ∞) → [0, 1] is a fuzzy set satisfying the following conditions:

M (x, y, t) >0,

In Definition 2.2, if the condition (G4) is replaced by the following condition:

* (M (x, z, t) , M (z, y, s)) ≤ M (x, y, max {t, s}), for all x, y, z ∈ X and t > 0,

A fuzzy metric space (X, M, *) is said to be strong fuzzy metric space if the condition (G4) is replaced by the following condition:

* (M (x, z, t) , M (z, y, t)) ≤ M (x, y, t), for all x, y, z ∈ X and t > 0.

Remark 2.1.

In view of (G1) and (G2), we have 0 < M (x, y, t) <1, for all x ≠ y and t > 0 (cf. [13]).

Example 2.1. [5] Let (X, d) be a metric space. Define M : X × X × (0, ∞) → [0, 1] as M ( x , y , t ) = t t + d ( x , y ) , ∀ x , y ∈ X and t > 0 .

Then (X, M, *) is a fuzzy metric space with respect to the t-norm: * (a, b) = a · b (or * (a, b) = min {a, b}), for all a, b ∈ [0, 1].

Definition 2.3. [5] Let (X, M, *) be a fuzzy metric space.

A sequence {x _n } ⊆ X is said to be convergent to x ∈ X if and only if lim n → ∞ M ( x n , x , t ) = 1 , for all t > 0.

A fuzzy metric space in which every Cauchy sequence is convergent is called a complete fuzzy metric space.

Definition 2.4. [8] Let (X, M, *) be a fuzzy metric space. A mapping T : X → X is said to be fuzzy contractive mapping if there exists k ∈ (0, 1) such that: 1 M ( Tx , Ty , t ) - 1 ≤ k ( 1 M ( x , y , t ) - 1 ) , for all x, y ∈ X and t > 0.

Let Ψ be the class of all functions ψ : (0, 1] → (0, 1] such that ψ is continuous, non-decreasing and ψ (t) > t, for all t ∈ (0, 1).

Definition 2.5. [14] Let (X, M, *) be a fuzzy metric space. A mapping T : X → X is said to be fuzzy ψ-contractive if there exists ψ ∈ Ψ such that M ( Tx , Ty , t ) ≥ ψ ( M ( x , y , t ) ) , ∀ x , y ∈ X and t > 0 .

Let H be the class of all functions η : (0, 1] → [0, ∞) which satisfy the following conditions:

η transforms (0, 1] onto [0, ∞),

Definition 2.6. [25] Let (X, M, *) be a fuzzy metric space. A mapping T : X → X is said to be fuzzy H -contractive with respect to η ∈ H if there exists k ∈ (0, 1) such that η ( M ( Tx , Ty , t ) ) ≤ k η ( M ( x , y , t ) ) , for all x, y ∈ X and t > 0 .

Remark 2.2. [7] If η ∈ H , then η is continuous and bijective. Further, the two mappings k · η : (0, 1] → [0, ∞) and η^-1 : [0, ∞) → (0, 1], defined in its obvious sense are bijective, continuous and strictly decreasing mappings.

Melliani et al. [12] proposed the class F Z of auxiliary functions as follows:

Definition 2.7. [12] Let F Z be the set of all functions ζ : ( 0 , 1 ] × ( 0 , 1 ] → ℝ which satisfy the following conditions:

ζ (1, 1) =0,

Example 2.2. [12, 15] The following functions ζ : ( 0 , 1 ] × ( 0 , 1 ] → ℝ belong to F Z :

ζ ( t , s ) = λ ( 1 s - 1 ) - ( 1 t - 1 ) , for all t, s ∈ (0, 1] and λ ∈ (0, 1).

The authors in [12], utilized the class F Z to introduce the concept of F Z -contractive mappings as follows:

Definition 2.8. [12] Let (X, M, *) be a fuzzy metric space. A mapping T : X → X is said to be F Z -contractive with respect to ζ ∈ F Z if the following condition is satisfied: ζ ( M ( Tx , Ty , t ) , M ( x , y , t ) ) ≥ 0 ,(2.1) for all x, y ∈ X and t > 0 .

Also, the authors in [12] proved the following result.

Theorem 2.1. [12] Let (X, M, *) be a complete strong fuzzy metric space and T : X → X. If T is an F Z -contractive mapping with respect to ζ ∈ F Z , then it has a unique fixed point.

Sezen et al. [21], introduced the notion of fuzzy φ-fixed point as under:

Definition 2.9. [21] Let X be a non-empty set, φ : X → (0, 1] a given function and T : X → X. An element z ∈ X is said to be a fuzzy φ-fixed point of the mapping T if and only if it is a fixed point of T and φ (z) =1.

The following lemma is needed in the sequel.

Lemma 2.1. [3] Let (X, M, *) be a fuzzy metric space and {x _n } ⊆ X be such that lim n → ∞ M ( x n , x_n+1, t) =1, for all t > 0. If {x _n } is not a Cauchy sequence, then there exist 0 < ϵ < 1,  t₀ > 0 and two subsequences {x _{n k} } and {x _{m k} } of {x _n } with m k > n k ≥ k , for all k ∈ ℕ such that M ( x n k , x m k , t 0 ) ≤ 1 - ϵ and lim k → ∞ M ( x n k , x m k - 1 , t 0 2 ) = 1 - ϵ .

Firstly, we observe that the condition (ζ^*1) given in Definition 2.7 is unnecessary in the proof of Theorem 2.1 and if one takes x = y in (2.1), then ζ (1, 1) ≥0. Also, the condition (ζ^*3) can be weakened by considering only those sequences which are involved in the condition (ζ2) of Definition 3.1. In view of our observations, we slightly modify the Definition 2.7 in order to enlarge the family of functions ζ as under:

Definition 3.1. Let Z M be the set of all functions ζ : ( 0 , 1 ] × ( 0 , 1 ] → ℝ which satisfy the following conditions:

ζ ( t , s ) < 1 s - 1 t , for all t, s ∈ (0, 1),

Remark 3.1. Observe that F Z ⊂ Z M . The converse inclusion is not true as substantiated by the following examples:

Example 3.1. Let ζ : ( 0 , 1 ] × ( 0 , 1 ] → ℝ be the function defined by ζ ( t , s ) = { 1 , if ( t , s ) = ( 1 , 1 ) , 1 s - 1 t , otherwise .

It is easy to show that ζ satisfies (ζ1) and (ζ2) but it does not satisfy (ζ^*1) as ζ (1, 1) >0.

Example 3.2. Let k ∈ ℝ be such that k < 1 and ζ : ( 0 , 1 ] × ( 0 , 1 ] → ℝ a function defined by ζ ( t , s ) = { 2 ( 1 s - 1 t ) , if t < s , k ( 1 s - 1 ) - ( 1 t - 1 ) , otherwise . Clearly, ζ satisfies (ζ1) and (ζ2) but it does not satisfy (ζ^*3). Indeed, for 0 < t, s < 1, if   0 < t < s < 1, then ζ ( t , s ) = 2 ( 1 s - 1 t ) < ( 1 s - 1 t ) . Otherwise, when   0 < s ≤ t < 1, we have ζ ( t , s ) = k ( 1 s - 1 ) - ( 1 t - 1 ) < ( 1 s - 1 t ) . Let {t _n }, {s _n } be sequences in (0, 1] such that lim n → ∞ t n = lim n → ∞ s n = δ < 1 and s _n  < t _n , for all n ∈ ℕ , then lim sup n → ∞ ζ ( t n , s n ) = lim sup n → ∞ ( k ( 1 s n - 1 ) - ( 1 t n - 1 ) ) = ( k - 1 ) ( 1 δ - 1 ) < 0 . Therefore, ζ satisfies (ζ1) and (ζ2). However, if we take t n = 1 2 + 1 2 n + 2 and s n = 1 2 + 1 n + 2 , for all n ∈ ℕ , then we have lim sup n → ∞ ζ ( t n , s n ) = lim sup n → ∞ 2 ( 2 n + 4 n + 4 - 2 n + 2 n + 2 ) = 0 , so that ζ does not satisfy (ζ^*3).

Example 3.3. Let ζ : ( 0 , 1 ] × ( 0 , 1 ] → ℝ be the function defined by ζ ( t , s ) = { 1 , if ( t , s ) = ( 1 , 1 ) , 2 ( 1 s - 1 t ) , if t < s , k ( 1 s - 1 ) - ( 1 t - 1 ) , otherwise . Then ζ satisfies (ζ1) and (ζ2) but it does not satisfy (ζ^*1) and (ζ^*3).

Now, let F be the set of all functions F : (0, 1] ³ → (0, 1] which satisfy the following conditions:

min {a, b} ≥ F (a, b, c), for all a, b, c ∈ (0, 1],

Example 3.4. Essentially, the following functions F : (0, 1] ³ → (0, 1] belong to F :

F (a, b, c) = a · b · c,

Next, we introduce the notion of ( F Z , F , φ ) - contractive mappings as follows:

Definition 3.2. Let (X, M, *) be a fuzzy metric space, F ∈ F and φ : X → (0, 1] a given function. A mapping T : X → X is said to be an ( F Z , F , φ ) -contractive mapping if there exists a function ζ ∈ Z M such that ζ ( F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) ≥ 0 ,(3.1) for all x, y ∈ X and t > 0.

To substantiate Definition 3.2, we provide the following example:

Example 3.5. Let X = [0, 1] and d (x, y) = |x - y|, for all x, y ∈ X. Let M be a fuzzy set on X² × (0, ∞) given by M ( x , y , t ) = t t + d ( x , y ) , t > 0 and * is a t-norm given by a * b = a · b, for all a, b ∈ [0, 1]. Then (X, M, *) is a fuzzy metric space. Define a mapping T : X → X by T ( x ) = x x + 1 , ∀ x ∈ X . Define two essential functions: F : (0, 1] ³ → (0, 1] and φ : X → (0, 1] by F ( a , b , c ) = a · b · c and φ ( x ) = 1 , for all a, b, c ∈ (0, 1]  and  x ∈ X. It is clear that φ is continuous and F ∈ F . Then for all x, y ∈ X, t > 0, we have 1 F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) - 1 = | Tx - Ty | t = 1 ( x + 1 ) ( y + 1 ) | x - y | t and 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 = | x - y | t . For any λ ∈ [ 1 2 , 1 ) and for all x, y ∈ X, t > 0, we have λ ( 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 ) - ( 1 F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) - 1 ) = ( λ - 1 ( x + 1 ) ( y + 1 ) ) | x - y | t ≥ 0 . Hence, T is an ( F Z , F , φ ) -contractive mapping with respect to ζ ( t , s ) = λ ( 1 s - 1 ) - ( 1 t - 1 ) .

Remark 3.2. Taking F (a, b, c) = a · b · c and φ (x) =1 in Definition 3.2, we deduce the definition of F Z -contractive mappings introduced by Melliani et al. [12].

Remark 3.3. Setting F (a, b, c) = a · b · c, φ (x) =1 and ζ (as in Example 2.2((i) , (iii) , (v) , (vi))) in Definition 3.2, one can deduce the definitions of fuzzy contractive [8], fuzzy φ-weak contractive [1], fuzzy ψ-contractive [14] and fuzzy H -contractive mappings [25] respectively.

In what follows F _T denotes the set of all fixed points of T (F _T  = {x ∈ X : Tx = x}) and U _φ stands for the set of all ones of the function φ (U _φ  = {x ∈ X : φ (x) =1}).

Now, we are equipped to state and prove our first main result which runs as follows:

Theorem 3.1. Let (X, M, *) be a complete fuzzy metric space, F ∈ F and φ : X → (0, 1] a continuous function. If T : X → X is an ( F Z , F , φ ) -contractive mapping with respect to ζ ∈ Z M , then F _T  ⊆ U _φ and T has a unique fuzzy φ-fixed point.

Proof. Firstly, we prove that F _T  ⊆ U _φ . Let x be a fixed point of T. Applying (3.1) with x = y, we get ζ ( F ( 1 , φ ( x ) , φ ( x ) ) , F ( 1 , φ ( x ) , φ ( x ) ) ) ≥ 0 .(3.2) We claim that F (1, φ (x) , φ (x)) =1. For the sake of contradiction, assume that F (1, φ (x) , φ (x)) <1. From (3.2) and (ζ1), we have 0 ≤ ζ ( F ( 1 , φ ( x ) , φ ( x ) ) , F ( 1 , φ ( x ) , φ ( x ) ) ) < 1 F ( 1 , φ ( x ) , φ ( x ) ) - 1 F ( 1 , φ ( x ) , φ ( x ) ) = 0 , which is a contradiction. Hence, F (1, φ (x) , φ (x)) =1. Now, using (F1) and the fact that F (1, φ (x) , φ (x)) = 1, we have φ ( x ) ≥ min { 1 , φ ( x ) } ≥ F ( 1 , φ ( x ) , φ ( x ) ) = 1 , which implies that φ (x) =1 and hence F _T  ⊆ U _φ .

Now, let x₀ ∈ X be an arbitrary point and {x _n } be the Picard sequence (with initial point x₀) defined by x n = T n x 0 , ∀ n ∈ ℕ . If there exists n 0 ∈ ℕ such that x _{n 0}  = x_n0+1, then x _{n 0} is a fixed point of T and hence a fuzzy φ-fixed point of T (as F _T  ⊆ U _φ ). Assume that x _n  ≠ x_n+1, for all n ∈ ℕ 0 . In view of the definition of fuzzy metric, we have M (x _n , x_n+1, t) <1, for all t > 0. If there exists some m 0 ∈ ℕ such that F (M (x _{m 0} , x_m0+1, t) , φ (x _{m 0} ) , φ (x_m0+1)) =1, then due to (F1) we have 1 = F ( M ( x m 0 , x m 0 + 1 , t ) , φ ( x m 0 ) , φ ( x m 0 + 1 ) ) ≤ min { M ( x m 0 , x m 0 + 1 , t ) , φ ( x m 0 ) } ≤ M ( x m 0 , x m 0 + 1 , t ) < 1 which is a contradiction. Hence, F ( M ( x n , x n + 1 , t ) , φ ( x n ) , φ ( x n + 1 ) ) < 1 , ∀ n ∈ ℕ 0 . Setting x = x_n-1 and y = x _n , for all n ∈ ℕ in (3.1) and utilizing (ζ1), we have 0 ≤ ζ ( F ( M ( Tx n - 1 , Tx n , t ) , φ ( Tx n - 1 ) , φ ( Tx n ) ) , F ( M ( x n - 1 , x n , t ) , φ ( x n - 1 ) , φ ( x n ) ) ) = ζ ( F ( M ( x n , x n + 1 , t ) , φ ( x n ) , φ ( x n + 1 ) ) , F ( M ( x n - 1 , x n , t ) , φ ( x n - 1 ) , φ ( x n ) ) ) < 1 F ( M ( x n - 1 , x n , t ) , φ ( x n - 1 ) , φ ( x n ) ) - 1 F ( M ( x n , x n + 1 , t ) , φ ( x n ) , φ ( x n + 1 ) ) , which implies that F ( M ( x n - 1 , x n , t ) , φ ( x n - 1 ) , φ ( x n ) ) < F ( M ( x n , x n + 1 , t ) , φ ( x n ) , φ ( x n + 1 ) ) .(3.3)

Therefore, {F (M (x _n , x_n+1, t) , φ (x _n ) , φ (x_n+1))} is an increasing sequence of positive real numbers in (0, 1]. Hence there is some r (t) ≤1 for all t > 0 such that lim n → ∞ F ( M ( x n , x n + 1 , t ) , φ ( x n ) , φ ( x n + 1 ) ) = r ( t ) . Our claim is r (t) =1 for all t > 0. Assume on contrary that r (t) <1 for some t₀ > 0. Denote t _n  = F (M (x _n , x_n+1, t₀) , φ (x _n ) , φ (x_n+1)) and s _n  = F (M (x_n-1, x _n , t₀) , φ (x_n-1) , φ (x _n )), for all n ∈ ℕ . In view of (3.3), we have s _n  < t _n , for all n ∈ ℕ . Now, applying (ζ2), we get 0 ≤ lim sup n → ∞ ζ ( F ( M ( x n , x n + 1 , t 0 ) , φ ( x n ) , φ ( x n + 1 ) ) , F ( M ( x n - 1 , x n , t 0 ) , φ ( x n - 1 ) , φ ( x n ) ) ) < 0 , which is a contradiction. Hence, we obtain lim n → ∞ F ( M ( x n , x n + 1 , t ) , φ ( x n ) , φ ( x n + 1 ) ) = 1 .(3.4) From (3.4) and (F1), we get lim n → ∞ M ( x n , x n + 1 , t ) = 1 , ∀ t > 0 .(3.5) and lim n → ∞ φ ( x n ) = 1 .(3.6) Next, we show that the sequence {x _n } is Cauchy. On contrary, suppose it is not Cauchy. In view of Lemma 2.1, there exist 0 < ϵ < 1,  t₀ > 0 and two subsequences {x _{n k} } and {x _{m k} } of {x _n } with m _k  > n _k  ≥ k, for all k ∈ ℕ such that M ( x n k , x m k , t 0 ) ≤ 1 - ϵ and lim k → ∞ M ( x n k , x m k - 1 , t 0 2 ) = 1 - ϵ .(3.7) Using (3.1) and (ζ1) with x = x_{n k -1} and y = x_{m k -1}, we get 0 ≤ ζ ( M ( x n k , x m k , t 0 ) , M ( x n k - 1 , x m k - 1 , t 0 ) ) < 1 M ( x n k - 1 , x m k - 1 , t 0 ) - 1 M ( x n k , x m k , t 0 ) , which implies that M ( x n k , x m k , t 0 ) > M ( x n k - 1 , x m k - 1 , t 0 ) .(3.8) Making use of (3.7), (3.8) and (G4), we have 1 - ϵ ≥ M ( x n k , x m k , t 0 ) > M ( x n k - 1 , x m k - 1 , t 0 ) ≥ * ( M ( x n k - 1 , x n k , t 0 2 ) , M ( x n k , x m k - 1 , t 0 2 ) ) . Taking k→ ∞ and making use of (3.5), (3.7) and (T3), we obtain lim k → ∞ M ( x n k , x m k , t 0 ) = 1 - ϵ ,(3.9) and lim k → ∞ M ( x n k - 1 , x m k - 1 , t 0 ) = 1 - ϵ .(3.10) By using (3.6), (3.9), (3.10), (F2) and the continuity of F, we have lim k → ∞ F ( M ( x n k , x m k , t 0 ) , φ ( x n k ) , φ ( x m k ) ) = F ( 1 - ϵ , 1 , 1 ) = 1 - ϵ ,(3.11) and lim k → ∞ F ( M ( x n k - 1 , x m k - 1 , t 0 ) , φ ( x n k - 1 ) , φ ( x m k - 1 ) ) = F ( 1 - ϵ , 1 , 1 ) = 1 - ϵ .(3.12) Set t _k  = F (M (x _{n k} , x _{m k} , t₀) , φ (x _{n k} ) , φ (x _{m k} )) and s _k  = F (M (x_{n k -1}, x_{m k -1}, t₀) , φ (x_{n k -1}) , φ (x_{m k -1})), for all k ∈ ℕ . In view of (3.3), we have s _k  < t _k , for all k ∈ ℕ . Now, applying (ζ2), we get 0 ≤ lim sup n → ∞ ζ ( F ( M ( x n k , x m k , t 0 ) , φ ( x n k ) , φ ( x m k ) ) , F ( M ( x n k - 1 , x m k - 1 , t 0 ) , φ ( x n k - 1 ) , φ ( x m k - 1 ) ) ) < 0 , which is a contradiction. Therefore, {x _n } is a Cauchy sequence in (X, M, *). The completeness of X ensure the existences of z ∈ X such that lim n → ∞ M ( x n , z , t ) = 1 .(3.13) As φ is continuous function, from (3.6) and (3.13), we have φ ( z ) = 1 .(3.14) If there exists a subsequence {x _{n k} } of {x _n } such that x _{n k}  = z or Tx _{n k}  = Tz, for all k ∈ ℕ , then this together with (3.14) show that z is a fuzzy φ-fixed point of T. Otherwise, we can assume that x _n  ≠ z and Tx _n  ≠ Tz, for all n ∈ ℕ 0 . Now, using (F1), we have F ( M ( x n , z , t ) , φ ( x n ) , φ ( z ) ) ≤ M ( x n , z , t ) < 1 and F ( M ( x n + 1 , Tz , t ) , φ ( x n + 1 ) , φ ( Tz ) ) ≤ M ( x n + 1 , Tz , t ) < 1 . Setting x = x _n and y = z, for all n ∈ ℕ in (3.1) and utilizing (ζ1), we obtain 0 ≤ ζ ( F ( M ( x n + 1 , Tz , t ) , φ ( x n + 1 ) , φ ( Tz ) ) , F ( M ( x n , z , t ) , φ ( x n ) , φ ( z ) ) ) < 1 F ( M ( x n , z , t ) , φ ( x n ) , φ ( z ) ) - 1 F ( M ( x n + 1 , Tz , t ) , φ ( x n + 1 ) , φ ( Tz ) ) which implies that F ( M ( x n + 1 , Tz , t ) , φ ( x n + 1 ) , φ ( Tz ) ) > F ( M ( x n , z , t ) , φ ( x n ) , φ ( z ) ) . Letting n → ∞ in the above inequality and using (3.6), (3.13), (3.14), (F2) and the continuity of F, we obtain F ( M ( z , Tz , t ) , 1 , φ ( Tz ) ) ≥ F ( 1 , 1 , 1 ) = 1 , which (in view of the condition (F1)) yields that M ( z , Tz , t ) = 1 .(3.15) Therefore, due to (3.14) and (3.15), we conclude that z is a fuzzy φ-fixed point of T.

Finally, we have to show that the fuzzy φ-fixed point of T is unique. For the sake of contradiction, we assume that z and w are two distinct fuzzy φ-fixed points of T. Applying (3.1) and (ζ1) with x = z and y = w, we have 0 ≤ ζ ( F ( M ( Tz , Tw , t ) , φ ( Tz ) , φ ( Tw ) ) , F ( M ( z , w , t ) , φ ( z ) , φ ( w ) ) ) = ζ ( F ( M ( z , w , t ) , φ ( z ) , φ ( w ) ) , F ( M ( z , w , t ) , φ ( z ) , φ ( w ) ) ) < 1 F ( M ( z , w , t ) , 1 , 1 ) - 1 F ( M ( z , w , t ) , 1 , 1 ) = 0 , a contradiction. Therefore, the fuzzy φ-fixed point of T is unique. This completes the proof. □

The following examples exhibit the utility of Theorem 3.1.

Example 3.6. Let (X = [0, 1] , d) be metric space with d (x, y) = |x - y|, for all x, y ∈ X. Then (X, d) is a complete metric space. Let M be a fuzzy set on X² × (0, ∞) given by M ( x , y , t ) = t t + d ( x , y ) , t > 0 and * is a t-norm given by a * b = a · b, for all a, b ∈ [0, 1]. Then (X, M, *) is a complete fuzzy metric space. Define a mapping T : X → X by T ( x ) = x x + 1 , ∀ x , y ∈ X . Let F and φ be given as in Example 3.5. Then T is an ( F Z , F , φ ) -contractive mapping with respect to ζ ( t , s ) = λ ( 1 s - 1 ) - ( 1 t - 1 ) . Therefore, Theorem 3.1 can be applied to T and the unique φ-fixed point of T is 0.

Example 3.7. Let X = (0, ∞) and M be a fuzzy set on X² × (0, ∞) given by M ( x , y , t ) = min { x , y } max { x , y } , t > 0 . Define t-norm * : [0, 1] × [0, 1] → [0, 1] by a * b = a · b. Then (X, M, *) is a complete non-Archimedean fuzzy metric space. Consider a mapping T : X → X given by T ( x ) = { x , if x ∈ ( 0 , 1 ) , 1 , if x ∈ [ 1 , ∞ ) and define the following essential functions: F : (0, 1] ³ → (0, 1] and φ : X → (0, 1] by F ( a , b , c ) = a · b · c , ∀ a , b , c ∈ ( 0 , 1 ] and φ ( x ) = { x , if x ∈ ( 0 , 1 ) , 1 , if x ∈ [ 1 , ∞ ) . It is clear that φ is continuous and F ∈ F . Now, we choose ζ ( t , s ) = { 1 , if ( t , s ) = ( 1 , 1 ) , 1 s - 1 t , otherwise . It is easy to show that ζ ∈ Z M . In order to show that T is an ( F Z , F , φ ) -contractive mapping two cases arise:

Case1: Let (t, s) = (1, 1), then ζ (t, s) =1 > 0.

Case2: Let t, s ∈ (0, 1) and (t, s) ≠ (1, 1), then we have the following three cases:

Firstly, if (x, y) ∈ (0, 1) × (0, 1), then we get ζ ( t , s ) = ζ ( F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) = 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) = max { x , y } min { x , y } · 1 xy - max { x , y } min { x , y } · 1 xy = 0 .

Secondly, if (x, y) ∈ [1, ∞) × [1, ∞), then we get ζ ( t , s ) = ζ ( F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) = 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) = max { x , y } min { x , y } - 1 ≥ 0 .

Finally, if (x, y) ∈ (0, 1) × [1, ∞), then we have ζ ( t , s ) = ζ ( F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) = 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) = y x - 1 x = ( y - 1 ) 1 x ≥ 0 . Similarly, if (x, y) ∈ [1, ∞) × (0, 1), then we have ζ ( x , y ) = ( x - 1 ) 1 y ≥ 0 . Therefore, all the conditions of Theorem 3.1 are satisfied and hence, T has a unique φ-fixed point. Observe that T (1) =1 and φ (1) =1 and 1 is the unique fuzzy φ-fixed point of T.

Theorem 3.1 can be improved as follows.

Theorem 3.2. Let (X, M, *) be a complete fuzzy metric space, F ∈ F , ζ ∈ Z M and φ : X → [0, ∞) a continuous function. If T : X → X is a mapping satisfying ζ ( F ( M ( T n x , T n y , t ) , φ ( T n x ) , φ ( T n y ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) ≥ 0 , for all x, y ∈ X, t > 0 and for some n ∈ ℕ , then F _T  ⊂ U _φ and T has a unique fuzzy φ-fixed point.

Proof. In view of Theorem 3.1, T ⁿ has a unique fuzzy φ-fixed point (say z, that is, T ⁿ z = z and φ (z) =1) and F _{T ⁿ}  ⊂ U _φ . Now, observe that T ⁿ  (Tz) = T (T ⁿ z) = Tz. Thus, Tz is also a fixed point of T ⁿ . Now, we have Tz ∈ F _{T ⁿ}  ⊂ U _φ and hence, φ (Tz) =1. Therefore, Tz is also a fuzzy φ-fixed point of T ⁿ . Hence, we must have Tz = z (due to the uniqueness of the fuzzy φ-fixed point of T ⁿ ensured by Theorem 3.1). Furthermore, F _T  = F _{T ⁿ}  ⊂ U _φ . This completes the proof. □

The following example shows that Theorem 3.1 is a genuine extension of Theorem 3.1.

Example 3.8. Let X = [0, 1]. Define t-norm * : [0, 1] → [0, 1] by a * b = a · b, for all a, b ∈ [0, 1]. Let M be the standard fuzzy metric, that is, M ( x , y , t ) = t t + d ( x , y ) , for all x , y ∈ X and t > 0 . Then (X, M, *) is a complete fuzzy metric space. Define a mapping T : X → X by T ( x ) = { 1 , if x ∈ [ 0 , 1 ] ∩ ℚ , 0 , if x ∈ [ 0 , 1 ] ∩ ℚ c . We need the following essential functions: F : (0, 1] ³ → (0, 1] and φ : X → (0, 1] which are defined by F ( a , b , c ) = a · b · c and φ ( x ) = 1 for all a, b, c ∈ (0, 1]  and  x ∈ X . It is clear that φ is continuous and F ∈ F .

Now, we show that T is not an ( F Z , F , φ ) -contractive mapping. On contrary, we assume that there exists ζ ∈ Z M such that ζ (F (M (Tx, Ty, t) , φ (Tx) , φ (Ty)) , F (M (x, y, t) , φ (x) , φ (y))) ≥0, for all x, y ∈ X and t > 0. From the definition of fuzzy metric M and (ζ1), we have ζ ( F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) = ζ ( M ( Tx , Ty , t ) , M ( x , y , t ) ) = ζ ( t t + d ( Tx , Ty ) , t t + d ( x , y ) ) < t + d ( x , y ) t - t + d ( Tx , Ty ) t = d ( x , y ) - d ( Tx , Ty ) t . Put x = 1 and y = 1 2 , we obtain ζ ( F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) = d ( 1 , 1 2 ) - d ( T 1 , T 1 2 ) t = - 1 2 t < 0 , a contradiction. Therefore, T is not an ( F Z , F , φ ) -contractive mapping.

Now, we show that T² is an ( F Z , F , φ ) -contractive mapping. Consider the function ζ ∈ Z _M given by ζ ( t , s ) = k ( 1 s - 1 ) - ( 1 t - 1 ) , for all t, s ∈ (0, 1] and k ∈ (0, 1).

For all x, y ∈ X and t > 0, we have ζ ( F ( M ( T 2 x , T 2 y , t ) , φ ( T 2 x ) , φ ( T 2 y ) ) , F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) = ζ ( M ( T 2 x , T 2 y , t ) , M ( x , y , t ) ) = k ( 1 M ( x , y , t ) - 1 ) - ( 1 M ( T 2 x , T 2 y , t ) - 1 ) = k ( t + d ( x , y ) t - 1 ) - ( t + d ( 1 , 1 ) t - 1 ) = λ d ( x , y ) t ≥ 0 . Hence, T² is an ( F Z , F , φ ) -contractive mapping and all the conditions of Theorem 3.1 are satisfied. Thus, T has a unique fuzzy φ-fixed point (namely x = 1).

In this section, as consequences of Theorem 3.1, we deduce several corollaries which can be viewed as generalizations of various results in the existing literature.

Taking F (a, b, c) = a . b . c, for all a, b, c ∈ (0, 1] and φ (x) =1, for all x ∈ X in Theorem 3.1, we deduce the following corollary which is a sharpened version of Theorem 2.1.

Corollary 4.1. [12] Let (X, M, *) be a complete fuzzy metric space and T : X → X. If there exists ζ ∈ Z M such that ζ ( M ( Tx , Ty , t ) , M ( x , y , t ) ) ≥ 0 , for all x, y ∈ X, then T has a unique fuzzy fixed point.

Corollary 4.2. Let (X, M, *) be a complete fuzzy metric space and T : X → X. Assume that 1 F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) - 1 ≤ λ ( 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 ) , for all x, y ∈ X, t > 0 and λ ∈ (0, 1). Then T has a unique fuzzy φ-fixed point.

Proof. The proof follows from Theorem 3.1 and Example 2.2 (i). □

Corollary 4.3. Let (X, M, *) be a complete fuzzy metric space and T : X → X. Assume that 1 F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) - 1 ≤ ( 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 ) - φ ( 1 F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) - 1 ) , for all x, y ∈ X,  t > 0, where φ : [0, ∞) → [0, ∞) with φ (r) >0, for all r > 0 and φ (0) =0. Then T has a unique fuzzy φ-fixed point.

Proof. In view of Theorem 3.1 and Example 2.2 (iii), the result follows. □

Corollary 4.4. Let (X, M, *) be a complete fuzzy metric space, ψ ∈ Ψ and T : X → X. Assume that F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) ≥ ψ ( F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) , for all x, y ∈ X,  t > 0. Then T has a unique fuzzy φ-fixed point.

Proof. The proof follows from Theorem 3.1 and Example 2.2 (v). □

Corollary 4.5. Let (X, M, *) be a complete fuzzy metric space, η ∈ H and T : X → X. Assume that η ( F ( M ( Tx , Ty , t ) , φ ( Tx ) , φ ( Ty ) ) ) ≤ k η ( F ( M ( x , y , t ) , φ ( x ) , φ ( y ) ) ) , for all x, y ∈ X,  t > 0. Then T has a unique fuzzy φ-fixed point.