Proving Fixed-Point Theorems Employing Fuzzy ( σ , Z ) -Contractive-Type Mappings

: In this article, the concept of fuzzy ( σ , Z ) -contractive mappings is introduced in the setting of fuzzy metric spaces. Thereafter, we utilize our newly introduced concept to prove some existence and uniqueness theorems in M -complete fuzzy metric spaces. Our obtained theorems extend and generalize the corresponding results in the existing literature. Moreover, some examples are adopted to exhibit the utility of the newly obtained results.


Introduction and Motivation
In the theory of fuzzy sets and systems, many researchers have attempted to formulate an appropriate definition of fuzzy metric space (e.g., [1][2][3]). The most natural and widely acceptable definition is essentially due to Kramosil and Michálek [4]. Grabiec [5] is one of the earliest mathematicians to study the theory of the fixed point in fuzzy metric spaces. In doing so, he introduced the notions of G-Cauchyness and the G-completeness of fuzzy metric spaces and extended the fixed-point theorems of Banach and Edelstein from metric spaces to fuzzy metric spaces introduced by Kramosil and Michálek. It has been observed that the notions of G-Cauchy sequences and G-completeness are relatively strong. With a view toward having a Hausdorff topology on a fuzzy metric space, George and Veeramani [6] modified the definition of the fuzzy metric space due to Kramosil and Michálek [4] and also established some valuable related results.
On the other hand, the concept of the σ-admissible mappings was introduced by Samet et al. (see [26], Definition 2.2) in metric spaces. In [27], Gopal and Vetro extended this notion to the setting of fuzzy metric spaces (see Definition 8, given later). Employing this notion, they introduced the concept of σ-ψ-fuzzy contractive mappings and proved a theorem that ensures the existence of a fixed point for this types of mappings. Their presented theorem extends, generalizes, and improves the corresponding results given in the literature.
Employing the function ξ that satisfies the above condition, Shukla et al. unified and extended the contractive-type mappings introduced in [7-9,29] by introducing the following interesting class of mappings: Definition 4 ( [28]). Let T be a self-mapping of an FMS (K, M, * ). The mapping T is said to be for each α, β ∈ K with Tα = Tβ and t > 0.
Let K be a nonempty set, x 0 ∈ K and T : K → K. A sequence {x n } ⊆ K is called a Picard sequence of T based at x 0 if x n = Tx n−1 = T n x 0 , ∀ n ∈ N. Definition 5 ([28]). Let T be a self-mapping of an FMS (K, M, * ) and ξ ∈ Z. Assume that {α n } is any Picard sequence for all n ∈ N. The quadruple (K, M, T, ξ) is said to have the property (S) if for each n ∈ N and t > 0 with

Definition 6 ([28]
). Let T be a self-mapping of an FMS (K, M, * ) and ξ ∈ Z. Assume that {α n } is any Picard sequence for all n ∈ N. The quadruple (K, M, T, ξ) is said to have the property (S ) if for each n ∈ N and t > 0 with 0 < lim n→∞ inf m>n M(α n , α m , t) < 1 and Notice that the condition (S ) is weaker than the condition (S) (see [28], Example 3.18). Shukla et al. [28] proved the following theorem as a consequence of their study. Theorem 1. Let (K, M, * ) be an M-complete FMS and T : K → K be a fuzzy Z-contractive mapping with respect to ξ ∈ Z. If the quadruple (K, M, T, ξ) has the property (S), then T admits a unique fixed point.

Definition 8 ([27]
). Let T be a self-mapping of an FMS and σ : For examples of the σ-admissible mapping of fuzzy metric spaces, we refer the reader to [27,32]. Now, we add another two examples of σ-admissible mappings.
Then, T is a σ-admissible mapping.
Then, T admits a fixed point.

Main Results
Throughout this article, (K, M, * ) is a fuzzy metric space in the George and Veeramani sense. First of all, we start by introducing the notion of fuzzy (σ, Z )-contractive mappings, which include many existing and familiar concepts as special cases. Definition 9. Let T be a self-mapping of an FMS (K, M, * ). We say that T is a fuzzy (σ, Z )contractive with respect to ξ ∈ Z if there is σ : for all α, β ∈ K, t > 0 with Tα = Tβ.

Remark 2.
By adopting the functions ξ and σ suitably in Definition 9, we deduce some well-known contractions as demonstrated below (for all α, β ∈ K and t > 0): (a) If σ(α, β, t) = 1, for each α, β ∈ K and t > 0, then Definition 9 reduces to Definition 4. (b) Taking ξ(l, s) = ψ(s), for each l, s ∈ (0, 1] and ψ ∈ Ψ in Definition 9, we deduce Definition 7. It is worth mentioning here that every fuzzy Z-contractive is a fuzzy (σ, Z )-contractive mapping, but the reverse is not in general true, as demonstrated by the following example: T is not a fuzzy Z-contractive mapping. On the contrary, we assume that T is a fuzzy Z-contractive with respect to some ξ ∈ Z. Take x, y ∈ K such that Tx = Ty. Since M(x, y, t) = M(Tx, Ty, t) = t t+|x−y| ∈ (0, 1), using Remark 1, we have for all t > 0, which is a contradiction. Hence, T is not a fuzzy Z-contractive mapping. To show that T is a fuzzy (σ, Z )-contractive mapping, we need to define two essential functions: It is clear that ξ ∈ Z. Then, for all x, y ∈ K, t > 0, we have which shows that T is a fuzzy (σ, Z )-contractive mapping.
Now, we are able to formulate our first main result as follows: Theorem 3. Let (K, M, * ) be an M-complete FMS and σ : K × K × (0, ∞) → (0, ∞). Assume that T : K → K is a fuzzy (σ, Z )-contractive mapping and the following properties hold: (a) T is σ-admissible; Then, T admits a fixed point.
Proof. Pick out an arbitrary point α 0 in K such that σ(α 0 , Tα 0 , t) ≤ 1, for each t > 0, and consider a Picard sequence {α n } in K, that is, In case α n 0 = α n 0 +1 , for some n 0 ∈ N 0 , then the fixed point of the mapping T is nothing but α n 0 . Assume that α n+1 = α n , for each n ∈ N 0 . As T is σ-admissible, we have: The induction on n, gives rise to: Moreover, if for some m > n, α n = α m , then the contractive condition (2) and Equation (3) imply that: Continuing in this way, one can show that Since α n = α m for some m > n, we have α n+1 = α m+1 . This together with the above relation leads to a contradiction. Therefore, α n = α m for each m > n. In view of the condition (d), there exists k 0 ∈ N such that σ(α n , α m , t) ≤ 1, ∀ m, n ∈ N with m > n ≥ k 0 and t > 0.
Applying the contractive condition (2) and making use of the above inequality, we obtain and hence, In the above inequality, taking the infimum over m(> n) and letting a n (t) = inf m>n M(α n , α m , t) we obtain that a n (t) ≤ a n+1 (t), for each t > 0, and hence, {a n (t)} is a nondecreasing and bounded. Therefore, there exists a(t) such that lim n→∞ a n (t) = a(t).
Our claim is to justify that a(t) = 1, for each t > 0. On the contrary, we assume that a(s) > 1, for some s > 0. From the fact that the quadruple (K, M, T, ξ) owns the property (S), we obtain lim Equation (4) gives rise to Taking n → ∞ in the above relation and using Equation (5), we obtain which is a contradiction to the assumption (a(s) > 1 for some s > 0). This contradiction concludes that, for each t > 0, lim for all t > 0. Therefore, Tγ = γ, due to the uniqueness of the limit.
In order to support the above-obtained result, we provide an example. Precisely, we show that Theorem 3 can be used to cover this example while Theorem 1 is not applicable.
Finally, to show that T is a fuzzy (σ, Z )-contractive mapping, we only need to consider the case α = A 2 and β ∈ {A 1 , A 3 , A 4 , A 5 }. In this case, σ(α, β, t) = e which shows that T is a fuzzy (σ, Z )-contractive mapping. Therefore, all the hypotheses of Theorem 3 are satisfied. This ensures that the mapping T admits a fixed point (namely x = A 1 ). However, T is not a fuzzy Z-contractive mapping. On the contrary, we assume T is fuzzy Z-contractive with respect to to some ξ ∈ Z. Take α = A 2 and β = A 4 . As M(α, β, t) = e for all t > 0, which is a contradiction. Hence, T is not a fuzzy Z-contractive mapping.
One of the advantages of σ-admissible mappings is that the continuity of the mapping is no longer required for the existence of a fixed point provided that the space under consideration satisfies a suitable condition (namely (e ) given in the next theorem). Precisely, we state and prove the following theorem: (c) There exists α 0 ∈ K with σ(α 0 , Tα 0 , t) ≤ 1, for each t > 0; (d) For each sequence {α n } of K with the property that σ(α n , α n+1 , t) ≤ 1, for each t > 0, there exists k 0 ∈ N such that σ(α n , α m , t) ≤ 1, for each m, n ∈ N with m > n ≥ k 0 , t > 0; (e ) If {α n } is a sequence in K such that lim n→∞ α n = α ∈ K and σ(α n , α n+1 , t) ≤ 1, for each n ∈ N and t > 0, then σ(α n , α, t) ≤ 1.
Then, T admits a fixed point.
Proof. The frame of the proof is the same as that in the previous theorem (Theorem 3). Therefore, for a Cauchy sequence {α n } in a complete FMS (K, M, * ), there exists γ ∈ K such that lim n→∞ (α n , γ, t) = 1, ∀ t > 0.
Next, we support Theorem 4 by an example in which the mapping T is not continuous. Moreover, we show the applicability of Theorem 4 over Theorems 1 and 3. , for all x, y ∈ K and all t > 0. Then, (K, M, * ) is an M-complete fuzzy metric space. Consider the mapping T : K → K defined by It is obvious that T is not continuous at x = 1, and hence, Theorem 3 cannot be applied to this example. Define two essential functions ξ : (0, 1] × (0, 1] → R and σ : . Let x, y ∈ K such that σ(x, y, t) ≤ 1. Then, either x, y ∈ [0, 1] or x = y ∈ (1, ∞). In case x, y ∈ [0, 1], by the definition of T, we have Tx, Ty ∈ [0, 1], and hence, σ(Tx, Ty) = 1. In the other case, if x = y ∈ (1, ∞), then again, by the definition of T, we have Tx = Ty ∈ (1, ∞), and hence, σ(Tx, Ty) = 1. Therefore, T is a σ-admissible mapping. Furthermore, 1 ∈ K and σ(1, T1, t) = σ(1, 1 2 , t) = 1. Further, let {α n } be a sequence in K such that lim n→∞ α n = x with k 0 = 1 and σ(α n , α n+1 , t) ≤ 1, for all n ∈ N. From the definition of α, it follows that α n ∈ [0, 1], for all n ∈ N, if we assume that x ∈ (1, ∞), then we assume which is a contradiction of the assumption that lim n→∞ α n = x. Thus, we have x ∈ [0, 1]. Therefore, α(α n , x, t) ≤ 1 and α(α n , α m , t) ≤ 1 for all m, n ∈ N and t > 0. Finally, we show that T is a fuzzy (α, Z)-contractive mapping. To do so, for all x, y ∈ x with Tx = Ty, we consider the following four cases. Case I: If x, y ∈ [0, 1], then (as α(x, y, t) = 1 and M(Tx, Ty, t) > M(x, y, t)), and we have Subcase II: If M(Tx, Ty, t) ≥ M(x, y, t), then we have Case III: This case is similar to that in Case II. Case IV: If x, y ∈ (1, ∞), then (as α(x, y, t) = e ( |x−y| t ) and M(Tx, Ty, t) = M(x, y, t)), and we have Hence, in all cases, T is a fuzzy (α, Z)-contractive mapping. Therefore, all the hypotheses of Theorem 4 are satisfied. Hence, T has a fixed point (namely x = 0). However, T is not a fuzzy Z-contractive mapping. To see this, we consider the case that x, y ∈ (1, ∞) and take into account Remark 1; we have M(Tx, Ty, t) = M(x, y, t), and hence, which impossible; hence, T is not a fuzzy Z-contractive mapping. Now, by an example (see also [28], Example 3.10), we show that the assumption (b) of Theorems 3 and 4 is not superfluous. x } for all x, y ∈ K, t > 0.
Then, (K, M, * ) is an M-complete fuzzy metric space where * is the product t-norm. Define a mapping T : K → K by Tx = x + 1 for all x ∈ K. Then, T is a fuzzy (σ, Z )-contractive mapping with respect to the functions ξ : (0, 1] × (0, 1] → R and σ : K × K × (0, ∞) → (0, ∞) by From the definition of σ, it is very clear that the conditions (a), (c), (d), (e), and (e ) of Theorems 3 and 4 are satisfied. Moreover, trivial calculations show that the condition (b) does not hold, that is the quadruple (K, M, T, ξ) does not have the property (S). Notice that T does not have a fixed point.
Next, the following example shows that the assumption (c) of Theorems 3 and 4 is not superfluous. Furthermore, we define two essential functions: ξ : (0, 1] × (0, 1] → R and σ : Then, for all x, y ∈ K such that Tx = Ty and t > 0, we have which shows that T is fuzzy (σ, Z )-contractive. Moreover, it is easy to show that the conditions (a), (b), (d), (e), and (e ) of Theorems 3 and 4 hold. Now, note that there is no x 0 in K such that σ(x 0 , Tx 0 , t) ≤ 1 for t > 0. Thus, the condition (c) of Theorems 3 and 4 does not hold. Observe that the mapping T does not have a fixed point.
In addition, assume that lim n→∞ inf m>n M(T n α, T m α, t) > 0, for all x ∈ K and t > 0. Then, T admits a fixed point.
Proof. Following the same lines of the proof of Theorem 4 and taking into account that the quadruple (K, M, T, ξ) owns the property (S ) instead of the property (S) with the fact that lim n→∞ inf m>n M(T n α, T m α, t) > 0, for all x ∈ K and t > 0, we obtain the required result.
Next, we discuss the uniqueness of the fixed point in Theorems 3-5. In order to ensure the uniqueness of the fixed point, we add one more sufficient condition to the hypothesis of the theorems. Precisely, we take into account the following condition: (h) For each α, β ∈ Fix(T), we have σ(α, β, t) ≤ 1, for all t > 0. Theorem 6. In addition to the hypothesis of Theorems 3-5, assume that the condition (h) holds. Then, the fixed point of T is unique.
In addition, assume that lim n→∞ inf m>n M(α n , α m , t) > 0, for all x ∈ K and t > 0. Then, T admits a fixed point.
This completes the proof.

Conclusions
Motivated by the results of Shukla et al. [28] and Gopal et al. [27], we introduced the notion of fuzzy (σ, Z )-contractive mappings, which enlarge and unify the class of fuzzy Z-contractive mappings introduced in [28] and the family of σ-ψ-fuzzy contractive mappings obtained in [27]. The new class of mappings covers all the concepts introduced in [7-9,27-29]. Our newly introduced notion was utilized to prove some results in Mcomplete fuzzy metric spaces. Finally, some examples were adopted to demonstrate that our newly presented results are a proper extension of Shukla et al.'s results [28].