Fuzzy Θ f -contractive mappings and their fixed points with applications

1 Introduction

Zadeh [1] introduced fruitful notion of fuzzy set which led the fuzzification of almost entire mathematics. Like other mathematical notions, the idea of fuzzy metric was introduced by several mathematicians [2–4]. The most natural definition of fuzzy metric space is due to Kramosil and Michalek [5]. Fuzzy metric space is one of the significant generalizations of metric space because of its interesting applications in engineering sciences, medical sciences, mathematical programming, modeling theory, image processing etc.

Grabiec [6] was the first mathematician who initiated the study of fixed point theory in fuzzy metric spaces. He introduced Banach contraction in fuzzy metric spaces and extended the fixed point theorems of Banach and Eldelstein to fuzzy metric spaces in the sense of Kramosil and Michalek. Following Grabiec’s work, Fang [7] further established some new fixed point theorems for contractive type mappings. George and Veeramani [8] modified the fuzzy metric space due to Kramosil and Michalek [5] with a view to have a Hausdorff topology. Afterwards, many authors followed this direction of research by investigating different types of fuzzy contractive mappings. In this regards, Gregori and Sapena [9] bring in the notion of fuzzy contractive mapping and obtained a fuzzy version of Banach contraction principal for such type of mappings in fuzzy metric spaces in the sense of George and Veeramani as well as Kramosil and Michalek’s sense.

Recently, starting from an auxiliary function satisfying suitable properties, many authors introduced various kind of contractions and obtained interesting fuzzy fixed point results such as: fuzzy ψ-contractive mappings [10], fuzzy H -contractive mappings [11], fuzzy Z -contractive mappings [12], fuzzy F Z -contractive mappings [13]. Several other related results can be found in [14, 25].

In this paper, inspired by the work of Jleli et al. [26, 27] in metric spaces, we proposed a new classes of contractive mappings termed as: fuzzy Θ _f -contractive and fuzzy Θ _f -weak contractive mappings by using an auxiliary function Θ _f  : (0, 1) → (0, 1) satisfying suitable properties. These new classes include the classes of Gregori and Sapena [9], and Tirado [14]. Thereafter, we establish some fixed point theorems in M-complete fuzzy metric spaces. Moreover, we furnish an application to functional equations under suitable conditions.

2 Preliminaries

With a view to have a self-contained presentation, we collect the relevant background material needed in the proofs of our results.

Definition 2.1. [28] A binary operation * : [0, 1] ² → [0, 1] is said to be a continuous t-norm if for all r₁, r₂, r₃ ∈ [0, 1], the following conditions are satisfied:

r₁ * r₂ = r₂ * r₁ and r₁ * (r₂ * r₃) = (r₁ * r₂) * r₃,

The most commonly used t-norms are: r₁ *  _p r₂ = r₁ · r₂, r₁ *  _m r₂ = min {r₁, r₂} and r₁ *  _L r₂ = max {r₁ + r₂ - 1, 0} which known as product, minimum and Lukasiewicz t-norms respectively.

In [5], Kramosil and Michalek introduced the notion of fuzzy metric space as follows:

Definition 2.2. [5] Let Y be a non-empty set, * a continuous t-norm and M : Y 2 × ( 0 , ∞ ) → [ 0 , 1 ] a fuzzy set. An ordered triple ( Y , M , * ) is said to be a fuzzy metric space (in the sense of Kramosil and Michalek) if for all a, b, c ∈ Y and s, t > 0, the following assumptions are fulfilled:

M ( a , b , 0 ) = 0 ,

In view to have a Hausdorff topology on fuzzy metric space, George and Veeramani [8] modified the definition of fuzzy metric space due to Kramosil and Michalek as follows:

Definition 2.3. [8] Let Y be a non-empty set, * a continuous t-norm and M : Y 2 × ( 0 , ∞ ) → [ 0 , 1 ] a fuzzy set. An ordered triple ( Y , M , * ) is said to be a fuzzy metric space (in the sense of George and Veeramani) if for all a, b, c ∈ Y and s, t > 0, the following assumptions are fulfilled:

M ( a , b , s ) > 0 ,

Definition 2.4. [8] Let ( Y , M , * ) be a fuzzy metric space. For s > 0, the open ball B ( a , r , s ) with a center a ∈ Y and radius r ∈ (0, 1) is defined by: B ( a , r , s ) = { b ∈ Y : M ( a , b , s ) > 1 - r } . A subset A ⊂ Y is called open if for each a ∈ A, there exist s > 0 and r ∈ (0, 1) such that B ( a , r , s ) ⊂ A . The family of all open subsets of Y is a topology on Y, called the topology induced by the fuzzy metric M .

Example 2.1. [8] Let (Y, d) be a metric space. Define M : Y × Y × ( 0 , ∞ ) → ( 0 , 1 ] as M ( a , b , s ) = s s + d ( a , b ) , ∀ a , b ∈ Y and s > 0 . Then ( Y , M , * ) is a fuzzy metric space with respect to the product t-norm (or minimum t-norm). M is known as the standard fuzzy metric.

Definition 2.5. [8] Let ( Y , M , * ) be a fuzzy metric space. A sequence { a n } n ∈ ℕ is called convergent and converges to a ∈ Y, if for all s > 0, lim n → ∞ M ( a n , a , s ) = 1 , that is, for each r ∈ (0, 1) and s > 0, there exists n 0 ∈ ℕ such that M ( a n , a , s ) > 1 - r , for all n ≥ n₀.

Definition 2.6. [8] Let ( Y , M , * ) be a fuzzy metric space. A sequence { a n } n ∈ ℕ is called M-Cauchy, if for each ɛ ∈ (0, 1) and s > 0, there exists n 0 ∈ ℕ such that M ( a m , a n , s ) > 1 - ɛ , for all m, n ≥ n₀.

The fuzzy metric space ( Y , M , * ) is called M-complete, if every M-Cauchy sequence of Y converges in Y.

Remark 2.1. [6, 29] Let ( Y , M , * ) be a fuzzy metric space.

The limit of the convergent sequence in Y is unique.

In [9], Gregori and Sapena bring in the notion of fuzzy contractive mapping and fuzzy contractive sequence as follows:

Definition 2.7. [9] Let ( Y , M , * ) be a fuzzy metric space. A mapping S : Y → Y is called fuzzy contractive if 1 M ( Sa , Sb , s ) - 1 ≤ λ ( 1 M ( a , b , s ) - 1 ) ,(2.1) for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1).

Definition 2.8. [9] A sequence { a n } n ∈ ℕ in a fuzzy metric space ( Y , M , * ) is called fuzzy contractive if 1 M ( a n + 1 , a n + 2 , s ) - 1 ≤ λ ( 1 M ( a n , a n + 1 , s ) - 1 ) , for all s > 0 , n ∈ ℕ and for some λ ∈ (0, 1).

Utilizing the above definitions, Gregori and Sapena [9] proved the following fixed point theorem:

Theorem 2.1. [9] Let ( Y , M , * ) be an M-complete fuzzy metric space in which fuzzy contractive sequences are M-Cauchy. If S : Y → Y is a fuzzy contractive mapping, then S admits a unique fixed point.

Definition 2.2. [14] Let ( Y , M , * ) be a fuzzy metric space. A mapping S : Y → Y is said to be Tirado’s contraction if 1 - M ( Sa , Sb , s ) ≤ λ ( 1 - M ( a , b , s ) ) ,(2.2) for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1).

Tirado [14] proved the following theorem as a consequence of his study:

Theorem 2.2. [14] Let ( Y , M , * L ) be an M-complete fuzzy metric space. If S : Y → Y is a Tirado’s contraction mapping, then S admits a unique fixed point.

In this section, we establish some fixed point theorems for self-mappings of the M-complete fuzzy metric space.

Let Ω be the family of all functions Θ _f  : (0, 1) → (0, 1) such that Θ _f is non-decreasing, continuous and satisfying the following condition:

for each sequence {α _n } ⊂ (0, 1) , lim n → ∞ Θ f ( α n ) = 1 ⇔ lim n → ∞ α n = 1 .

In the following lines, we furnish some examples of the function Θ _f  ∈ Ω.

Example 3.1. Let Θ _f  : (0, 1) → (0, 1) be a function defined as: Θ f ( α ) = e 1 - 1 α , ∀ α ∈ ( 0 , 1 ) .

Example 3.2. Let Θ _f  : (0, 1) → (0, 1) be a function defined as: Θ f ( α ) = 1 - e ( α - 10 ) e ( 1 - 1 α ) , ∀ α ∈ ( 0 , 1 ) .

Example 3.3. Let Θ _f  : (0, 1) → (0, 1) be a function defined as: Θ f ( α ) = 1 - cos ( π α 2 ) , ∀ α ∈ ( 0 , 1 ) .

Example 3.4. Let Θ _f  : (0, 1) → (0, 1) be a function defined as: Θ f ( α ) = 1 + sin ( π 2 ( α - 1 ) ) , ∀ α ∈ ( 0 , 1 ) .

Example 3.5. Let Θ _f  : (0, 1) → (0, 1) be a function defined as: Θ f ( α ) = e α - 1 , ∀ α ∈ ( 0 , 1 ) .

Now, we introduce the concept of fuzzy Θ _f -contractive mapping as follows:

Definition 3.1. Let ( Y , M , * ) be a fuzzy metric space. A mapping S : Y → Y is called fuzzy Θ _f -contractive with respect to Θ _f  ∈ Ω if there exists λ ∈ (0, 1) such that M ( Sa , Sb , s ) < 1 ⇒ Θ f ( M ( Sa , Sb , s ) ) ≥ [ Θ f ( M ( a , b , s ) ) ] λ ,(3.1) for all a, b ∈ Y and s > 0.

Example 3.6. Let ( Y , M , * ) be a fuzzy metric space. A mapping S : Y → Y which satisfies condition (2.1) (fuzzy contractive mapping) is a fuzzy Θ _f -contractive with Θ f ( α ) = e 1 - 1 α . It is clear that the condition (2.1) also holds, for all a, b ∈ Y such that Sa = Sb.

Example 3.7. Let ( Y , M , * ) be a fuzzy metric space. A mapping S : Y → Y which satisfies condition (2.2) (Tirado contraction mapping) is a fuzzy Θ _f -contractive with Θ _f  (α) = e^α-1 . Observe that the condition (2.2) also holds, for all a, b ∈ Y such that Sa = Sb.

Example 3.8. Let Y = [0, 1] and M be a fuzzy set on Y² × (0, ∞) given by M ( a , b , s ) = s s + d ( a , b ) , s > 0 , where d is the usual metric. Then, ( Y , M , * ) is a fuzzy metric space with the product t-norm *. Define a mapping S : Y → Y by S ( a ) = a a + 1 , ∀ a ∈ Y . Also define a function Θ _f  : (0, 1) → (0, 1) by Θ f ( α ) = e 1 - 1 α . It is clear that Θ _f  ∈ Ω.

For all a, b ∈ Y with M ( Sa , Sb , s ) < 1 and λ = 1 4 , we have 1 - 1 M ( Sa , Sb , s ) = - d ( Sa , Sb ) s = 1 ( a + 1 ) ( b + 1 ) ( - d ( a , b ) s ) = 1 ( a + 1 ) ( b + 1 ) ( 1 - 1 M ( a , b , s ) ) > 1 4 ( 1 - 1 M ( a , b , s ) ) = λ ( 1 - 1 M ( a , b , s ) ) . Therefore, Θ f ( M ( Sa , Sb , s ) ) = e 1 - 1 M ( Sa , Sb , s ) > [ e 1 - 1 M ( a , b , s ) ] λ = [ Θ f ( M ( a , b , s ) ) ] λ , which shows that, S is a fuzzy Θ _f -contractive mapping.

Remark 3.1. Every fuzzy Θ _f -contractive mapping is continuous.

To accomplish this, let { a n } n ∈ ℕ ⊂ Y be any sequence and a ∈ Y such that lim n → ∞ M ( a n , a , s ) = 1 and M ( Sa n , Sa , s ) < 1 , for all n ∈ ℕ . Using (3.1), we obtain Θ f ( M ( Sa n , Sa , s ) ) ≥ [ Θ f ( M ( a n , a , s ) ) ] λ > Θ f ( M ( a n , a , s ) ) . As Θ _f is non-decreasing, we have M ( Sa n , Sa , s ) > M ( a n , a , s ) , so that lim n → ∞ M ( Sa n , Sa , s ) = 1 . Thus, T is continuous.

Next, we define fuzzy Θ _f -weak contractive as under:

Definition 3.2. Let ( Y , M , * ) be a fuzzy metric space. A mapping S : Y → Y is called a fuzzy Θ _f -weak contractive with respect to Θ _f  ∈ Ω if there exists λ ∈ (0, 1) such that M ( Sa , Sb , s ) < 1 ⇒ Θ f ( M ( Sa , Sb , s ) ) ≥ [ Θ f ( N ( a , b , s ) ) ] λ ,(3.2) for all a, b ∈ Y and s > 0, where N ( a , b , s ) = min { M ( a , b , s ) , M ( a , Sa , s ) , M ( b , Sb , s ) } .

Remark 3.2. One can easily check that every Θ _f -contractive mapping is a Θ _f -weak contractive mapping, but the converse is not true in general as substantiated in Example 3.9.

Example 3.9. Let Y = [0, 1] and M be a fuzzy set on Y² × (0, ∞) given by M ( a , b , s ) = s s + d ( a , b ) , for all s > 0 and d is the usual metric. Then ( Y , M , * ) is a fuzzy metric space, where * is a product t-norm. Define a mapping S : Y → Y as S ( a ) = { 1 2 , a ∈ [ 0 , 1 ) , 1 4 , a = 1 . As S is not continuous, due to Remark 3.1, S is not fuzzy Θ _f -contractive mapping. Observe that for a ∈ [0, 1), b = 1 and s > 0, we have M ( Sa , Sb , s ) = s s + d ( Sa , Sb ) = s s + 1 4 < 1 . Then, the following two cases are distinguished:

Case I: If a ∈ [ 0 , 1 4 ) and b = 1, then min { M ( a , 1 , s ) , M ( a , Sa , s ) , M ( 1 , S 1 , s ) } = M ( a , 1 , s ) = s s + | a - 1 | . Case II: If a ∈ [ 1 4 , 1 ) and b = 1, then min { M ( a , 1 , s ) , M ( a , Sa , s ) , M ( 1 , S 1 , s ) } = M ( 1 , S 1 , s ) = s s + 3 4 . So, on both the cases if we take Θ f ( α ) = e 1 - 1 α , α ∈ ( 0 , 1 ) and λ = 1 3 , we see that S is a fuzzy Θ _f -weak contractive.

Now, we are equipped to prove our first main result as follows:

Theorem 3.1. Let ( Y , M , * ) be an M-complete fuzzy metric space. If S : Y → Y is a fuzzy Θ _f -contractive, then S admits a unique fixed point.

Proof. Let a₀ ∈ Y and define {a _n } by a n = Sa n - 1 , for all n ∈ ℕ . If there exists m 0 ∈ ℕ such that a _{m 0}  = a_m0+1, then a _{m 0} is a fixed point of S and the proof is finished. Suppose that a _n  ≠ a_n+1, for all n ∈ ℕ , that is, M ( Sa n - 1 , Sa n , s ) < 1 , for all n ∈ ℕ , s > 0 . Using (2.1), we have 1 > Θ f ( M ( Sa n - 1 , Sa n , s ) ) ≥ [ Θ f ( M ( a n - 1 , a n , s ) ) ] λ = [ Θ f ( M ( Sa n - 2 , Sa n - 1 , s ) ) ] λ ≥ [ Θ f ( M ( a n - 2 , a n - 1 , s ) ) ] λ 2 ⋮ ≥ [ Θ f ( M ( a 0 , a 1 , s ) ) ] λ n .(3.3) Taking n→ ∞ (on both the sides), we get lim n → ∞ Θ f ( M ( a n , a n + 1 , s ) ) = 1 , which, in view of the condition (H) gives arise lim n → ∞ M ( a n , a n + 1 , s ) = 1 .(3.4) Now, we show the Cauchyness of the sequence {a _n }. On contrary, we assume that the sequence {a _n } is not Cauchy. Then there are ɛ ∈ (0, 1), s₀ > 0 and two subsequences {a _{n k} } , {a _{m k} } of {a _n } such that m _k  > n _k  ≥ k, for all k ∈ ℕ and M ( a m k , a n k , s 0 ) ≤ 1 - ɛ .

In view of Remark 2.1 (b), we get M ( a m k , a n k , s 0 2 ) ≤ 1 - ɛ .(3.5) Suppose that n _k is the least integer exceeding m _k satisfying inequality (3.5). Then, we have M ( a m k - 1 , a n k , s 0 2 ) > 1 - ɛ .(3.6)

Using inequality (2.1) with a = a_{m k -1}, b = a_{n k -1} and s = s₀, we get Θ f ( M ( a m k , a n k , s 0 ) ) ≥ [ Θ f ( M ( a m k - 1 , a n k - 1 , s 0 ) ) ] λ(3.7) > Θ f ( M ( a m k - 1 , a n k - 1 , s 0 ) ) .

As Θ _f is non-decreasing, we have M ( a m k , a n k , s 0 ) > M ( a m k - 1 , a n k - 1 , s 0 ) .(3.8) Now, using (3.5) and (3.6) in (3.8), we get 1 - ɛ ≥ M ( a m k , a n k , s 0 ) > M ( a m k - 1 , a n k - 1 , s 0 ) ≥ M ( a m k - 1 , a n k , s 0 2 ) * M ( a n k , a n k - 1 , s 0 2 ) > ( 1 - ɛ ) * M ( a n k , a n k - 1 , s 0 2 ) , which on letting k→ ∞ and using (T3) yields lim n → ∞ M ( a m k , a n k , s 0 ) = 1 - ɛ .(3.9) and lim n → ∞ M ( a m k - 1 , a n k - 1 , s 0 ) = 1 - ɛ .(3.10) Taking the limit as k→ ∞ in (3.7), using (3.9) and (3.10) and the continuity of Θ _f , we obtain Θ f ( 1 - ɛ ) ≥ [ Θ f ( 1 - ɛ ) ] λ , which is a contradiction. Hence, {a _n } is Cauchy. The M-completeness of ( Y , M , * ) ensures the existence of a ∈ Y such that lim n → ∞ M ( a n , a , s ) = 1 , ∀ s > 0 . As S is continuous, we have lim n → ∞ M ( a n + 1 , Sa , s ) = 1 . By the uniqueness of the limit, we get Sa = a.

To show that a is the unique fixed point of S, we assume that c ∈ Y is also a fixed point of S such that a ≠ c. Then we get Θ f ( M ( a , c , s ) ) ≥ [ Θ f ( M ( a , c , s ) ) ] λ , ∀ s > 0 , a contradiction. Thus, the fixed point a is unique. This completes the proof. □

Example 3.10. Let ( Y , M , * ) be a fuzzy metric space given as in Example 3.8. Then it is M-complete fuzzy metric space. Consider the mapping S and the function Θ _f as given in Example 3.8. Then S is a fuzzy Θ _f -contractive mapping. Hence, by Theorem 3.1, S possesses a unique fixed point (namely a = 0).

Example 3.11. Let Y = { 3 3 n : n ∈ ℕ } ∪ { 3 } and M be a fuzzy set on Y² × (0, ∞) given by M ( a , b , s ) = { a / b , if a ≤ b , b / a , if b ≤ a , for all s > 0. Then ( Y , M , * ) is an M-complete fuzzy metric space, where * is a product t-norm. Now, define S : Y → Y by S ( a ) = { 3 3 n - 1 , if a = 3 3 n , n ∈ ℕ , 3 , if a = 3 . If we take Θ _f  (α) = α,  α ∈ (0, 1], then for all n , m ∈ ℕ with m < n and s > 0, we have Θ f ( M ( S ( 3 3 m ) , S ( 3 3 n ) , s ) ) = M ( S ( 3 3 m ) , S ( 3 3 n ) , s ) = 3 3 m - 1 / 3 3 n - 1 = [ Θ f ( M ( 3 3 m , 3 3 n , s ) ) ] 1 / 3 and Θ f ( M ( S ( 3 ) , S ( 3 3 n ) , s ) ) = M ( S ( 3 ) , S ( 3 3 n ) , s ) = 3 / 3 3 n - 1 > [ 3 / 3 3 n ] 1 / 3 = [ Θ f ( M ( 3 , 3 3 n , s ) ) ] 1 / 3 . Therefore, the mapping S is fuzzy Θ _f -contractive with λ = 1/3. Observe that S possesses a unique fixed point (namely a = 3).

Remark 3.3.

On setting Θ f ( α ) = e 1 - 1 α , Theorem 3.1 reduces to Theorem 2.1 due to Gregori and Sapena.

As in Remark 3.3, on setting Θ _f  (α) = α and Θ f ( α ) = 1 + sin ( π 2 ( α - 1 ) ) in Theorem 3.1, we obtain the following corollaries which are new addition to the existing literature.

Corollary 3.1. Let ( Y , M , * ) be an M-complete fuzzy metric space. Suppose that S : Y → Y is a mapping such that M ( Sa , Sb , s ) ≥ [ M ( a , b , s ) ] λ , for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1). Then S admits a unique fixed point.

Corollary 3.2. Let ( Y , M , * ) be an M-complete fuzzy metric space. Suppose that S : Y → Y is a mapping such that M ( Sa , Sb , s ) < 1 ⇒ 1 + sin ( π 2 ( M ( Sa , Sb , s ) - 1 ) ) ≥ [ 1 + sin ( π 2 ( M ( a , b , s ) - 1 ) ) ] λ , for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1). Then S has a unique fixed point.

Now, we prove the following relatively more general result employing fuzzy Θ _f -weak contractive mapping.

Theorem 3.2. Let ( Y , M , * ) be an M-complete fuzzy metric space. If S : Y → Y is a fuzzy Θ _f -weak contractive mapping, then S admits a unique fixed point.

Proof. Let a₀ ∈ Y and define a sequence {a _n } in Y by a n = Sa n - 1 , ∀ n ∈ ℕ . If there exists m 0 ∈ ℕ such that a _{m 0}  = a_m0+1, then a _{m 0}  = Sa _{m 0} and hence conclusion holds. Suppose that a _n  ≠ a_n+1, i . e ., M ( a n , a n + 1 , s ) < 1 , for all n ∈ ℕ and s > 0, therefore using (3.2), we have Θ f ( M ( Sa n - 1 , Sa n , s ) ) ≥ [ Θ f ( N ( a n - 1 , a n , s ) ) ] λ ,(3.11) where N ( a n - 1 , a n , s ) = min { M ( a n - 1 , a n , t ) , M ( a n - 1 , Sa n - 1 , s ) , M ( a n , Sa n , s ) } = min { M ( a n - 1 , a n , s ) , M ( a n , a n + 1 , s ) } . If for some n ∈ ℕ , N ( a n - 1 , a n , s ) = M ( a n , a n + 1 , s ) , then (3.11) reduces to Θ f ( M ( a n , a n + 1 , s ) ) ≥ [ Θ f ( M ( a n , a n + 1 , s ) ) ] λ > Θ f ( M ( a n , a n + 1 , s ) ) , which is a contradiction. Therefore, we have N ( a n - 1 , a n , s ) = M ( a n - 1 , a n , s ) , for all n ∈ ℕ and henceforth (3.11) yields

Θ f ( M ( a n , a n + 1 , s ) ) ≥ [ Θ f ( M ( a n - 1 , a n , s ) ) ] λ ,(3.12) for all n ∈ ℕ . Again, using (3.12), we obtain

1 > Θ f ( M ( a n , a n + 1 , s ) ) ≥ [ Θ f ( M ( a 0 , a 1 , s ) ) ] λ n ,(3.13) for all n ∈ ℕ . By taking n→ ∞ in (3.13), we obtain lim n → ∞ Θ f ( M ( a n , a n + 1 , s ) ) = 1 , and hence in veiw of condition (H), we get

lim n → ∞ M ( a n , a n + 1 , s ) = 1 .(3.14) The proof of Cauchyness of the sequence {a _n } follows on lines of the proof of Theorem 3.1 and hence it is omitted. Now, due to the M-completeness of ( Y , M , * ) there exists some a ∈ Y such that lim n → ∞ M ( a n , a , s ) = 1 , s > 0 . Next, we show the existence of fixed point of S. Let P = { n ∈ ℕ : a n + 1 = Sa } . We distinguish two cases using P. Firstly, if P is infinite, then there exists {a_{n k +1}} ⊆ {a_n+1} such that lim k → ∞ a n k + 1 = Sa , so that Sa = a. Secondly, if P is finite, then a_n+1 ≠ Sa for infinitely many n ∈ ℕ . This ensures the existence of {a_{n k +1}} ⊆ {a_n+1} such that M ( a n k + 1 , Sa , s ) < 1 , ∀ k ∈ ℕ . As S is a fuzzy Θ _f -weak contractive, we have

Θ f ( M ( a n k + 1 , Sa , s ) ) ≥ [ Θ f ( N ( a n k , a , s ) ) ] λ ,(3.15) where N ( a n k , a , s ) = min { M ( a n k , a , s ) , M ( a n k , a n k + 1 , s ) , M ( a , Sa , s ) } . If M ( a , Sa , s ) < 1 , then we have

lim n → ∞ N ( a n k , a , s ) = min { 1 , 1 , M ( a , Sa , s ) } = M ( a , Sa , s ) .(3.16) Letting k→ ∞ in (3.15), using (3.16) and the continuity of Θ _f , we get Θ f ( M ( a , Sa , s ) ) ≥ [ Θ f ( M ( a , Sa , s ) ) ] λ , a contradiction as λ ∈ (0, 1). Therefore, we must have M ( a , Sa , s ) = 1 , that is, a remains fixed under S. Finally, we prove the uniqueness of the fixed point of S. On contrary, assume that c be other fixed point of S. Applying (3.2), we get

Θ f ( M ( a , c , s ) ) = Θ f ( M ( Sa , Sc , s ) ) ≥ [ Θ f ( N ( a , c , s ) ) ] λ ,(3.17) where N ( a , c , s ) = min { M ( a , c , s ) , M ( a , Sa , s ) , M ( c , Sc , s ) } . Next, if M ( a , c , s ) < 1 , then N ( a , c , s ) = min { M ( a , c , s ) , 1 , 1 } = M ( a , c , s ) .(3.18) Making use of (3.18) in (3.17), we obtain Θ f ( M ( a , c , s ) ) ≥ [ Θ f ( M ( a , c , s ) ) ] λ ,(3.18) a contradiction. Therefore, M ( a , c , s ) = 1 , ∀ s > 0 . This concludes the proof.□

Next, we furnish the following illustrative example which shows that Theorem 3.2 is a genuine extension of Theorems 2.1, 2.2 and 3.1.

Example 3.12. Let Y = [0, 1] and M be a fuzzy set on Y² × (0, ∞) given by M ( a , b , s ) = s s + d ( a , b ) , for all s > 0 and d is the usual metric. Then ( Y , M , * ) is an M-complete fuzzy metric space, where * is a product t-norm. Consider the self-mapping S to be the same as given in Example 3, that is, S : Y → Y defined as S ( a ) = { 1 2 , a ∈ [ 0 , 1 ) , 1 4 , a = 1 . As we have seen in Example 3, S is not fuzzy Θ _f -contractive mapping for any Θ _f , but S is a fuzzy Θ _f -weak contractive mapping with Θ f ( α ) = e 1 - 1 α , for all α ∈ (0, 1). Then Theorem 3.1 is not applicable while Theorem 3.2 can be applied to S. Observe that S possesses a unique fixed point (namely a = 1 2 ). Notice that S is neither fuzzy contractive nor Tirado’s contraction mapping. Indeed, for a = 1 and b = 3 4 , the conditions (2.1) and (2.2) yields 1 4 s = 1 M ( S 1 , S 3 4 , s ) - 1 ≤ k ( 1 M ( 1 , 3 4 , s ) - 1 ) = k 4 s , and 1 4 s + 1 = 1 - M ( S 1 , S 3 4 , s ) ≤ k ( 1 - M ( 1 , 3 4 , s ) ) = k 4 s + 1 , which are impossible as k ∈ (0, 1). Therefore, Theorems 2.1 and 2.2 can not be applied here.

Finally, we end this section by deducing some corollaries which are new addition to the existing literature. On taking Θ f ( α ) = e 1 - 1 α , Θ _f  (α) = e^1-α, Θ _f  (α) = α and Θ f ( α ) = 1 - cos ( π 2 ( α ) ) in Theorem 3, we obtain the following corollaries respectively:

Corollary 3.3. Let ( Y , M , * ) be an M-complete fuzzy metric space. If S : Y → Y is a mapping such that for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1) ( 1 M ( Sa , Sb , s ) - 1 ) ≤ λ ( 1 N ( a , b , s ) - 1 ) , where N ( a , b , t ) = min { M ( a , b , s ) , M ( a , Sa , s ) , M ( b , Sb , s ) } , then S admits a unique fixed point.

Corollary 3.4. Let ( Y , M , * ) be an M-complete fuzzy metric space. If S : Y → Y is a mapping such that for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1) 1 - M ( Sa , Sb , s ) ≤ λ ( 1 - N ( a , b , s ) ) , where N ( a , b , t ) = min { M ( a , b , s ) , M ( a , Sa , s ) , M ( b , Sb , s ) } , then S admits a unique fixed point.

Corollary 3.5. Let ( Y , M , * ) be an M-complete fuzzy metric space. If S : Y → Y is a mapping such that M ( Sa , Sb , s ) ≥ [ N ( a , b , s ) ] λ , for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1), where N ( a , b , s ) = min { M ( a , b , s ) , M ( a , Sa , s ) , M ( b , Sb , s ) } , then S admits a unique fixed point.

Corollary 3.6. Let ( Y , M , * ) be an M-complete fuzzy metric space. If S : Y → Y is a mapping such that, for all a, b ∈ Y, s > 0 and for some λ ∈ (0, 1), M ( Sa , Sb , s ) < 1 ⇒ 1 - cos ( π 2 M ( Sa , Sb , s ) ) ≥ [ 1 - cos ( π 2 N ( a , b , s ) ) ] λ , where N ( a , b , s ) = min { M ( a , b , s ) , M ( a , Sa , s ) , M ( b , Sb , s ) } , then S admits a unique fixed point.

In this section, we give an application to dynamic programming showing the utility of Theorem 3.1. In fact, we establish a result on the solution of functional equation (4.1) occurring in dynamic programming. Precisely, we show that equation (4.1) admits a unique solution.

Let W and D are two Banach spaces and consider the next functional equation:

q ( z ) = sup w ∈ D { f ( z , w ) + F ( z , w , q ( μ ( z , w ) ) ) } , z ∈ W(4.1) where F : W × D × ℝ → ℝ and f : W × D → ℝ are bounded, and μ : W × D → W. Let B (W) be the class of all bounded real valued functions on W. Define a fuzzy norm N ( u , s ) = e - ∥ u ∥ s induced by the norm ∥ u ∥ = sup z ∈ W ∥ u ( z ) ∥ , for any arbitrary a ∈ B (W) and s > 0. Therefore, (B (W) , N, *) is a fuzzy Banach space endowed with the fuzzy metric M defined as M ( a , b , s ) = e - ∥ a - b ∥ s , where * is the product t-norm. To show the existence of a solution of the functional equation (4.1), we define the operator S : B (W) → B (W) by

( Sa ) ( z ) = sup w ∈ D { f ( z , w ) + F ( z , w , a ( μ ( z , w ) ) ) } ,(4.2) for all z ∈ W and a ∈ B (W). It is clear that S is well defined since f and F are bounded.

Now, we prove the following theorem to ensure the existence of solution of functional equation (4.1)

Theorem 4.1. Assume that there exists λ ∈ (0, 1) such that | F ( z , w , a ( μ ( z , w ) ) ) - F ( z , w , b ( μ ( z , w ) ) ) | ≤ λ ∥ a - b ∥ , for all (z, w) ∈ W × D and a, b ∈ B (W). Then functional equation (4.1) has a unique bounded solution.

Proof. Consider the self-operator S given in (4.2). The existence of a unique solution of Equation (4.1) is equivalent to the existence of fixed point of S.

Now, let r > 0 be an arbitrary real number, z ∈ W and a, b ∈ B (W) with Sa ≠ Sb. Then there exist w₁, w₂ ∈ D such that

( Sa ) ( z ) ≥ f ( z , w 1 ) + F ( z , w 1 , a ( μ ( z , w 1 ) ) ) ,(4.3)

( Sb ) ( z ) ≥ f ( z , w 2 ) + F ( z , w 2 , v ( μ ( z , w 2 ) ) ) .(4.4) Also, we have ( Sa ) ( z ) < f ( z , w 2 ) + F ( z , w 2 , a ( μ ( z , w 2 ) ) ) + r ,(4.5) ( Sb ) ( z ) < f ( z , w 1 ) + F ( z , w 1 , b ( μ ( z , w 1 ) ) ) + r .(4.6) Then using (4.3), (4.6) and condition (ii), it follows that

( Sb ) ( z ) - ( Sa ) ( z ) < F ( z , w 1 , b ( μ ( z , w 1 ) ) ) - F ( z , w 1 , a ( μ ( z , w 1 ) ) ) + r ≤ | F ( z , w 1 , b ( μ ( z , w 1 ) ) ) - F ( z , w 1 , a ( μ ( z , w 1 ) ) ) | + r ≤ λ ∥ b - a ∥ + r .(4.7) Similarly using (4.4), (4.5) and condition (ii), we have

( Sa ) ( z ) - ( Sb ) ( z ) ≤ λ ∥ a - b ∥ + r .(4.8) Using (4.7), (4.8) and the fact that r is arbitrary, we deduce that

∥ Sa - Sb ∥ ≤ λ ∥ a - b ∥ .(4.9) Therefore, for all s > 0, we have M ( Sa , Sb , s ) = e - ∥ Sa - Sb ∥ s ≥ e - λ ∥ a - b ∥ s = ( e - ∥ a - b ∥ s ) λ = ( M ( a , b , s ) ) λ . Hence the mapping S is Θ _f -contractive with respect to Θ _f  (α) = α. Now, using Theorem 3, S has a unique fixed point so that the functional equation (4.1) admits a unique solution.□

Theorem 4.2. Assume that there exists λ ∈ (0, 1) such that | F ( z , w , a ( μ ( z , w ) ) ) - F ( z , w , b ( μ ( z , w ) ) ) | ≤ λ min { ∥ a - b ∥ , ∥ a - Sa ∥ , ∥ b - Sb ∥ } , for all (z, w) ∈ W × D and a, b ∈ B (W) where the operator S as given in (4.2). Then functional equation (4.1) has a unique bounded solution.

Proof. The proof can be done using similar arguments of the proof of Theorem 4.1 and applying Theorem 3.2.□

Motivated by the work of Jleli et al. [26, 27] in metric space, we proposed two types of contractive mappings: fuzzy Θ _f -contractive mapping and fuzzy Θ _f -weak contractive mapping which are weaker than the corresponding fuzzy contractive mapping due to Gregori and Sapena [9], and Tirado’s contraction [14]. Moreover, we established two fuzzy fixed point theorems in M-complete fuzzy metric spaces besides furnishing an application to dynamic programming. Our obtained results can be extend to the case of coupled fixed points in fuzzy metric spaces.