Fuzzy fixed points of fuzzy mappings via F-contractions and an application

1 Introduction and preliminaries

Banach contraction principle has been a very advantageous and e cacious means in nonlinear analysis. Nadler [20] established multi-valued version of Banach contraction principle utilizing the Hausdorff metric in the setting of complete metric space. Let ( S , σ ) be a metric space and CB ( S ) be the family of nonempty, closed and bounded subsets of S . For A , B ∈ CB ( S ) , define H ( A , B ) = max { sup ω ∈ A σ ( ω , B ) , sup ϱ ∈ B σ ( ϱ , A ) } where σ ( ω , A ) = inf ϱ ∈ A σ ( ω , ϱ ) .

Theorem 1. [20] Let ( S , σ ) be a complete metric space and J : S → CB ( S ) be a multi-valued mapping. If there exists k ∈ [0, 1) such that H ( J ω , J ϱ ) ≤ k σ ( ω , ϱ ) 1.1(1) for all ω , ϱ ∈ S , then there exists a fixed point of J .

In 1981, Heilpern [18] used the notion of fuzzy sets to give a class of fuzzy mappings, which is an extension of multivalued mappings. In [18], a result for fuzzy mappings in metric linear spaces has been established. It is worth mentioning that the theorem given by Heilpern [18] is a fuzzy generalization of the Banach contraction principle.

A fuzzy set in S is a mapping with domain S and range [0, 1], I S is the set of all fuzzy sets in S . If A is a fuzzy set and ω ∈ S , then A (ω)  is said to be the grade of membership of ω in A. The  α-level set of A is symbolized by  [A]  _α and is given as follows: [ A ] α = { ω : A ( ω ) ≥ α } if α ∈ ( 0 , 1 ] , [ A ] 0 = { ω : A ( ω ) > 0 } . ¯

We denote the set of all fuzzy sets in a metric space S by F ( S ) . For A , B ∈ F ( S ) , A ⊂ B means A (ω) ≤ B (ω) for each ω ∈ S . We denote the fuzzy set χ_{ω} by {ω} unless and until it is stated, where χ_{ω} is the characteristic function of the crisp set A. If there exists an α ∈ [0, 1] such that [ A ] α , [ B ] α ∈ CB ( S ) , then define p α ( A , B ) = inf ω ∈ [ A ] α , ϱ ∈ [ B ] α σ ( ω , ϱ ) , D α ( A , B ) = H ( [ A ] α , [ B ] α ) . If [ A ] α , [ B ] α ∈ CB ( S ) for each α ∈ [0, 1] , then define p ( A , B ) = sup α p α ( A , B ) , σ ∞ ( A , B ) = sup α D α ( A , B ) . We write p (ω, B) instead of p ({ω} , B) . A fuzzy set  A in a metric linear space V is said to be an approximate quantity if and only if  [A]  _α is compact and convex in V for each α ∈ [0, 1]  and sup ω ∈ V A ( ω ) = 1 . The set of entire approximate quantities in V is designated by W (V) . Let S 1 be an arbitrary set and S 2 be a metric space. A function Q is said to be a fuzzy mapping if Q : S 1 → F ( S 2 ) . A Q : S 1 → F ( S 2 ) is a fuzzy subset on S 1 × S 2 with membership Q ( ω ) ( ϱ ) . A function Q ( ω ) ( ϱ ) is the grade of membership of ϱ in Q ( ω ) . Later, many other researchers have studied the existence of fixed points of fuzzy mappings, see [1, 33].

Azam et al. [12] defined the notion of α- fuzzy fixed point and common α- fuzzy fixed point of fuzzy mappings in this way.

Definition 2. [12] Suppose P , Q : S → F ( S ) . A point u ∈ S is said to be an α- fuzzy fixed point of Q if ∃ α ∈ [0, 1] such that u ∈ [ Q u ] α and u ∈ S is said to be a common α- fuzzy fixed point of P and Q if ∃ α ∈ [0, 1] such that u ∈ [ P u ] α ∩ [ Q u ] α . When α = 1, it is called a common fixed point of fuzzy mappings.

On the other hand, a new access in the theory of fixed points was currently introduced by Wardowski [36]. In his paper, he developed and boosted the contraction and initiated the notion of F-contractions. He established a generalized fixed point result concerning F-contractions in the setting of complete metric spaces. Wardowski defined F-contractions in this way.

Definition 3. [36] Let ( S , σ ) be a metric space and Q : S → S be a self mapping. Then S is called an F-contraction if there exists τ > 0 such that for ω , ϱ ∈ S ; σ ( Q ω , Q ϱ ) > 0 ⇒ τ + F ( σ ( Q ω , Q ϱ ) ) ≤ F ( σ ( ω , ϱ ) ) 1.2(2) where F : ℝ + → ℝ satisfies the following assertions:

F is strictly increasing;

Subsequently, Altun et al. [5] modified the above definitions by adding a comprehensive condition (F₄) which is stated as:

F (inf A) = inf F (A) for all A ⊂ (0, ∞) with inf A > 0 .

We represent the set of all functions F : ℝ + → ℝ satisfying (F₁) - (F₄) as ϝ.

Many authors have obtained several fixed point theorems for F-contractions. We refer the readers to [2–5, 35] for common fixed point theorems and F-contractions.

Lemma 4.[20] Let ( S , σ ) be a metric space and A , B ∈ CB ( S ) . Then for each a ∈ A, σ ( a , B ) ≤ H ( A , B ) .

Lemma 5.[8] Let V be a metric linear space, Q : S → W ( V ) and ω 0 ∈ V . Then there exists ω 1 ∈ V such that { ω 1 } ⊂ Q ( ω 0 ) .

The purpose of this article is to prove some common α- fuzzy fixed points for fuzzy mappings P , Q : S → F ( S ) via F-contractions in the context of metric spaces. We also provide a non trivial example and an application to theoritical computer science.

Theorem 6. Let ( S , σ ) be a complete metric space and let P , Q : S → F ( S ) and for each ω, ϱ ∈ S , there exist α P ( ω ) , α Q ( ϱ ) ∈ ( 0 , 1 ] such that [ P ω ] α P ( ω ) , [ Q ϱ ] α Q ( ϱ ) ∈ CB ( S ) . Assume there exist some F∈ ϝ and τ > 0 such that τ + F ( H ( [ P ω ] α P ( ω ) , [ Q ϱ ] α Q ( ϱ ) ) ) ≤ F ( σ ( ω , ϱ ) )(3) for all ω, ϱ∈ S with H ( [ P ω ] α P ( ω ) , [ Q ϱ ] α Q ( ϱ ) ) > 0 . Then [ P u ] α P ( u ) and [ Q u ] α Q ( u ) have a common α- fuzzy fixed point.

Proof. Let ω₀ be an arbitrary point in S , then by assumption there exists α P ( ω 0 ) ∈ ( 0 , 1 ] such that [ P ω 0 ] α P ( ω 0 ) ∈ CB ( S ) . For convenience, we denote α P ( ω 0 ) by α₁ . Let ω 1 ∈ [ P ω 0 ] α P ( ω 0 ) . For this ω₁ there exists α Q ( ω 1 ) ∈ ( 0 , 1 ] such that [ Q ω 1 ] α Q ( ω 1 ) ∈ CB ( S ) . By Lemma 1, (F₁) and (1), we have τ + F ( σ ( ω 1 , [ Q ω 1 ] α Q ( ω 1 ) ) ) ≤ τ + F ( H ( [ P ω 0 ] α P ( ω 0 ) , [ Q ω 1 ] α Q ( ω 1 ) ) ) ≤ F ( σ ( ω 0 , ω 1 ) ) . From (F₄), we know that F ( σ ( ω 1 , [ Q ω 1 ] α Q ( ω 1 ) ) ) = inf ϱ ∈ [ Q ω 1 ] α Q ( ω 1 ) F ( σ ( ω 1 , ϱ ) ) .

Thus τ + inf ϱ ∈ [ Q ω 1 ] α Q ( ω 1 ) F ( σ ( ω 1 , ϱ ) ) ≤ [ F ( σ ( ω 0 , ω 1 ) ] .(4) Then, from (2), there exists ω 2 ∈ [ Q ω 1 ] α Q ( ω 1 ) such that τ + F ( σ ( ω 1 , ω 2 ) ) ≤ F ( σ ( ω 0 , ω 1 ) .(5) For this ω₂, there exists α P ( ω 2 ) ∈ ( 0 , 1 ] such that [ P ω 2 ] α P ( ω 2 ) ∈ CB ( S ) . By Lemma 1, (F₁) and (1), we have τ + F ( σ ( ω 2 , [ P ω 2 ] α P ( ω 2 ) ) ) ≤ τ + F ( H ( [ Q ω 1 ] α Q ( ω 1 ) , [ P ω 2 ] α P ( ω 2 ) ) ) = τ + F ( H ( [ P ω 2 ] α P ( ω 2 ) , [ Q ω 1 ] α Q ( ω 1 ) ) ) ≤ F ( σ ( ω 2 , ω 1 ) ) = F ( σ ( ω 1 , ω 2 ) ) . From (F₄), we know that F [ σ ( ω 2 , [ P ω 2 ] α P ( ω 2 ) ) ] = inf ϱ 1 ∈ [ P ω 2 ] α P ( ω 2 ) F ( σ ( ω 2 , ϱ 1 ) ) .

Thus, τ + inf ϱ 1 ∈ [ P ω 2 ] α P ( ω 2 ) F ( σ ( ω 2 , ϱ 1 ) ) ≤ F ( σ ( ω 1 , ω 2 ) ) . 2.4(6)

Then, from (2), there exists ω 3 ∈ [ P ω 2 ] α P ( ω 2 ) such that τ + F ( σ ( ω 2 , ω 3 ) ) ≤ F ( σ ( ω 1 , ω 2 ) ) .(7) Continuing in this way, we get {ω _n } in S such that ω 2 n + 1 ∈ [ P ω 2 n ] α P ( ω 2 n ) and ω 2 n + 2 ∈ [ Q ω 2 n + 1 ] α Q ( ω 2 n + 1 ) and τ + F ( σ ( ω 2 n + 1 , ω 2 n + 2 ) ) ≤ F ( σ ( ω 2 n , ω 2 n + 1 ) )(8) and τ + F ( σ ( ω 2 n + 2 , ω 2 n + 3 ) ) ≤ F ( σ ( ω 2 n + 1 , ω 2 n + 2 ) )(9) for all n ∈ ℕ . By (5) and (6), we get τ + F ( σ ( ω n , ω n + 1 ) ) ≤ F ( σ ( ω n - 1 , ω n ) ) . Therefore, F ( σ ( ω n , ω n + 1 ) ) ≤ F ( σ ( ω n - 1 , ω n ) ) - τ ≤ F ( σ ( ω n - 2 , ω n - 1 ) ) - 2 τ ≤ . . . ≤ F ( σ ( ω 0 , ω 1 ) ) - n τ .(10) Taking limit as n→ ∞ in (7), we get lim n → ∞ F ( σ ( ω n , ω n + 1 ) ) = - ∞ . Then by (F₂), we have lim n → ∞ σ ( ω n , ω n + 1 ) = 0 . Now, by (F₃), there exists h ∈ (0, 1) such that lim n → ∞ [ σ ( ω n , ω n + 1 ) ] h F ( σ ( ω n , ω n + 1 ) ) = 0 . From (2.8), we have [ σ ( ω n , ω n + 1 ) ] h F ( σ ( ω n , ω n + 1 ) ) - [ σ ( ω n , ω n + 1 ) ] h F ( σ ( ω 0 , ω n + 1 ) ) ≤ - n τ [ σ ( ω n , ω n + 1 ) ] h ≤ 0 . Letting n→ ∞, we get lim n → ∞ n [ σ ( ω n , ω n + 1 ) ] h = 0 . Thus lim n → ∞ n 1 h σ ( ω n , ω n + 1 ) = 0 and there exists n 1 ∈ ℕ such that n 1 h σ ( ω n , ω n + 1 ) ≤ 1 for all n ≥ n₁ . So we have σ ( ω n , ω n + 1 ) ≤ 1 n 1 / h(11) for all n ≥ n₁ . Now consider m , n ∈ ℕ such that m > n ≥ n₁, we have σ ( ω n , ω m ) ≤ σ ( ω n , ω n + 1 ) + σ ( ω n + 1 , ω n + 2 ) + . . . + σ ( ω m - 1 , ω m ) ≤ ∑ i = n ∞ 1 i 1 / h . Since ∑ i = 1 ∞ 1 i 1 / h is convergent, we have σ ( ω n , ω m ) → 0 as n, m→ ∞. Hence {ω _n } is a Cauchy sequence in S . Since ( S , σ ) is complete, there exists u ∈ S such that, lim n → ∞ ω n → u . Now, we show that u ∈ [ Q u ] α Q ( u ) . Assume on the contrary that u ∉ [ Q u ] α Q ( u ) , so there exist n 0 ∈ ℕ and a subsequence {ω _{n k} } of {ω _n } such that σ ( ω 2 n k + 1 , [ Q u ] α Q ( u ) ) > 0 for all n _k  ≥ n₀ . Since σ ( ω 2 n k + 1 , [ Q u ] α Q ( u ) ) > 0 for all n _k  ≥ n₀, so by Lemma 1, (F₁) and (1), we get

τ + F [ σ ( ω 2 n k + 1 , [ Q u ] α Q ( u ) ) ] ≤ τ + F [ H ( [ P ω 2 n k ] α P ( ω 2 n k ) , [ Q u ] α Q ( u ) ) ] ≤ F ( σ ( ω 2 n k , u ) ) . This implies that F [ σ ( ω 2 n k + 1 , [ Q u ] α Q ( u ) ) ] ≤ F ( σ ( ω 2 n k , u ) ) - τ < F ( σ ( ω 2 n k , u ) ) . By (F₁), we get σ ( ω 2 n k + 1 , [ Q u ] α Q ( u ) ) < σ ( ω 2 n k , u ) . Taking n → ∞ , we obtain σ ( u , [ Q u ] α Q ( u ) ) ≤ 0 . Thus u ∈ [ Q u ] α Q ( u ) . Similarly, we can easily show that u ∈ [ P u ] α P ( u ) . Thus u ∈ [ P u ] α P ( u ) ∩ [ Q u ] α Q ( u ) . ■

Corollary 7. Let ( S , σ ) be a complete metric space and let P : S → F ( S ) and for each ω ∈ S , there exist α P ( ω ) , α P ( ϱ ) ∈ ( 0 , 1 ] such that [ P ω ] α P ( ω ) , [ P ϱ ] α P ( ϱ ) ∈ CB ( S ) . Assume there exist some F∈ ϝ and τ > 0 such that τ + F ( H ( [ P ω ] α P ( ω ) , [ P ϱ ] α P ( ϱ ) ) ) ≤ F ( σ ( ω , ϱ ) )(12) for all ω, ϱ∈ S with H ( [ P ω ] α P ( ω ) , [ P ϱ ] α P ( ϱ ) ) > 0 . Then P has a fixed point.

Theorem 8. Let ( S , σ ) be a complete metric space and let J 1 , J 2 : S → CB ( S ) . Suppose there exist some F∈ ϝ and τ > 0 such that τ + F ( H ( J 1 ω , J 2 ϱ ) ) ≤ F ( σ ( ω , ϱ ) ) for all ω, ϱ∈ S with H (J₁ω, J₂ϱ) > 0. Then J₁ and J₂ have a common fixed point.

Proof. Consider α : S → ( 0 , 1 ] and P , Q : S → F ( S ) defined by P ( ω ) ( t ) = { α ( ω ) , if t ∈ J 1 ω , 0 , if t ∉ J 1 ω and Q ( ω ) ( t ) = { α ( ω ) , if t ∈ J 2 ω , 0 , if t ∉ J 2 ω . Then [ P ω ] α ( ω ) = { t : P ( ω ) ( t ) ≥ α ( ω ) } = J 1 ω and [ Q ω ] α ( ω ) = { t : Q ( ω ) ( t ) ≥ α ( ω ) } = J 2 ω . Thus, by Theorem 2, we get u ∈ S such that u ∈ [ P u ] α P ( u ) ∩ [ Q u ] α Q ( u ) = J 1 u ∩ J 2 u . ■

Corollary 9. [5] Let ( S , σ ) be a complete metric space and let J : S → CB ( S ) . Suppose there exist some F∈ ϝ and τ > 0 such that τ + F ( H ( J ω , J ϱ ) ) ≤ F ( σ ( ω , ϱ ) ) for all ω , ϱ ∈ S with H (Jω, Jϱ) > 0. Then J has fixed point.

Corollary 10. Let ( S , σ ) be a complete metric linear space and let P , Q : S → W ( S ) . Suppose there exist some F∈ ϝ and τ > 0 such that τ + F ( σ ∞ ( P ( ω ) , Q ( ϱ ) ) ) ≤ F ( σ ( ω , ϱ ) ) for all ω, ϱ∈ S with σ ∞ ( P ( ω ) , Q ( ϱ ) ) > 0 . Then there exists some u ∈ S such that { u } ⊂ P ( u ) and { u } ⊂ Q ( u ) .

Proof. Let ω ∈ S , then by Lemma 1, there exists ϱ ∈ S such that ϱ ∈ [ P ω ] 1 . Similarly, we can find z ∈ S such that z ∈ [ Q ω ] 1 . It follows that for each ω ∈ S , [ P ω ] α ( ω ) , [ Q ω ] α ( ω ) ∈ CB ( S ) . As α (ω) = α (ϱ) = 1, by the definition of a σ_∞-metric for fuzzy sets, we have H ( [ P ω ] α ( ω ) , [ Q ϱ ] α ( ϱ ) ) ≤ σ ∞ ( P ( ω ) , Q ( ϱ ) ) for all ω, ϱ∈ S . From (F₁), we have τ + F ( H ( [ P ω ] α ( ω ) , [ Q ϱ ] α ( ϱ ) ) ) ≤ τ + F ( σ ∞ ( P ( ω ) , Q ( ϱ ) ) ) ≤ F ( p ( ω , ϱ ) ) for all ω, ϱ∈ S . Since [ P ω ] 1 ⊆ [ P ω ] α for each α ∈ (0, 1] . Therefore, σ ( ω , [ P ω ] α ) ≤ σ ( ω , [ P ω ] 1 ) for each α ∈ (0, 1] . It implies that p ( ω , P ( ω ) ) ≤ σ ( ω , [ P ω ] 1 ) . Similarly, p ( ω , Q ( ω ) ) ≤ σ ( ω , [ Q ω ] 1 ) . This further implies that for all ω , ϱ ∈ S , τ + F ( H ( [ P ω ] 1 , [ Q ϱ ] 1 ) ) ≤ F ( σ ( ω , ϱ ) ) . By Theorem 2, we get u ∈ S such that u ∈ [ P u ] 1 ∩ [ Q u ] 1 , i.e., { u } ⊂ Q ( u ) and { u } ⊂ P ( u ) .□

Now, we consider results involving the multivalued mapping Q ˆ (for details see [33]), which is induced by Q : S → F ( S ) , i.e., Q ˆ ω = { ϱ : Q ( ω ) ( t ) = max t ∈ S Q ( ω ) ( t ) } .

Corollary 11. Let ( S , σ ) be a complete metric space and let P , Q : S → F ( S ) be such that for all ω ∈ S , P ˆ ( ω ) , Q ˆ ( ω ) ∈ CB ( S ) . Suppose there exist some F∈ ϝ and τ > 0 such that τ + F ( H ( P ˆ ( ω ) , Q ˆ ( ϱ ) ) ) ≤ F ( σ ( ω , ϱ ) ) for all ω, ϱ∈ S with H ( P ˆ ( ω ) , Q ˆ ( ϱ ) ) > 0 . Then there exists a point ω ∗ ∈ S such that P ( ω ∗ ) ( ω ∗ ) ≥ P ( ω ∗ ) ( ω ) and Q ( ω ∗ ) ( ω ∗ ) ≥ Q ( ω ∗ ) ( ω ) for all ω ∈ S .

Proof. By Theorem 2, there exists ω ∗ ∈ S such that ω ∗ ∈ P ˆ ω ∗ ∩ Q ˆ ω ∗ . Then by Lemma 1, we have P ( ω ∗ ) ( ω ∗ ) ≥ P ( ω ∗ ) ( ω ) and Q ( ω ∗ ) ( ω ∗ ) ≥ Q ( ω ∗ ) ( ω ) for all ω ∈ S . ■

Example 12. Let S = [ 0 , 1 ] and define σ : S × S → R + as follows: σ ( ω , ϱ ) = | ω - ϱ | Then ( S , σ ) is a complete metric space. Define P , Q : S → F ( S ) , for α ∈ [0, 1]  as follows:

For ω ∈ S , we have P ( ω ) ( t ) = { α if 0 ≤ t ≤ ω 40 α 2 if ω 40 < t ≤ ω 30 α 3 if ω 30 < t ≤ ω 20 α 5 if ω 20 < t ≤ 1 , and Q ( ω ) ( t ) = { α if 0 ≤ t ≤ ω 20 α 3 if ω 20 < t ≤ ω 10 α 4 if ω 10 < t ≤ ω 5 α 7 if ω 5 < t ≤ 1 , such that [ Q ω ] α = [ 0 , ω 20 ] , [ P ω ] α = [ 0 , ω 40 ] . Let F (t) = ln(t) . Then ∃ τ ∈ ( 0 , ln | ϱ - ω | | ϱ - ω 2 | 1 20 ) with ϱ ¬ = ω such that τ + F ( H ( [ P ω ] α P ( ω ) , [ Q ϱ ] α Q ( ϱ ) ) ) ≤ F ( σ ( ω , ϱ ) ) for all ω, ϱ∈ S with H ( [ P ω ] α P ( ω ) , [ Q ϱ ] α Q ( ϱ ) ) > 0 is satisfied to get 0 ∈ [ P 0 ] α ∩ [ Q 0 ] α .

Let Ω be a nonempty alphabet. Let Ω^∞ be the set of all finite and infinite sequences (“words”) over Ω, where we adopt the convention that the empty sequence ∅ is an element of Ω^∞. Moreover, on Ω^∞, we consider the prefix order ⊑ given by ω ⊑ ϱ if and only if ω is a prefix of ϱ . For each nonempty ω ∈ Ω^∞ denote by l (ω) the length of ω. Then l (ω) ∈ [0, ∞] , whenever ω¬ = ∅ and l (∅) =0 . For each ω, ϱ ∈ Ω^∞, let ω ⊓ ϱ be the common prefix of ω and ϱ. Clearly, ω = ϱ if and only if ω ⊑ ϱ and ϱ ⊑ ω and l (ω) = l (ϱ). Then, the Baire metric σ_⊑ defined on Ω^∞ × Ω^∞ by { σ ⊑ ( ω , ϱ ) = 0 , if ω = ϱ σ ⊑ ( ω , ϱ ) = 2 - l ( ω ⊓ ϱ ) , otherwise so that the metric space (Ω^∞, σ_⊑) is complete. Finally, we refer to the average case time complexity analysis of the Quicksort divide-and-conquer sorting algorithm in [29].

Precisely, we consider the following recurrence relation:

Q ( 1 ) = 0 and Q ( n ) = 2 ( n - 1 ) n + n + 1 n Q ( n - 1 ) , n ≥ 2 . 3.1(13)

Consider as an alphabet Ω the set of nonnegative real numbers, i.e., Ω = ℝ + . We associate to Q the functional Φ : Ω^∞ → Ω^∞ given by ( Φ ( ω ) ) 1 = Q ( 1 ) and ( Φ ( ω ) ) n = 2 ( n - 1 ) n + n + 1 n ω n - 1 for all n ≥ 2 (if ω ∈ Ω^∞ has length n < ∞ , we write ω : = ω 1 ω 2 . . . ω n , and if ω is an infinite word we write ω : = ω₁ω₂ . . . .) . It follows by the construction that l (Φ (ω)) = l (ω) +1 for all ω ∈ Ω^∞ and l (Φ (ω)) =+ ∞ whenever l (ω) = + ∞ . We will prove that the functional Φ has a fixed point by an application of Corollary 2. Let P : Ω ∞ → F ( Ω ∞ ) be the fuzzy mapping given by P ω = ( Φ ( ω ) ) α for all ω ∈ Ω ∞ . The following cases occur.

Case 01: If ω = ϱ, then we have H ⊑ ( ( Φ ( ω ) ) α , ( Φ ( ω ) ) α ) = 0 = σ ⊑ ( ω , ω ) . Case 02: If ω ¬ = ϱ, then we have H ⊑ ( ( Φ ( ω ) ) α , ( Φ ( ϱ ) ) α ) = σ ⊑ ( ( Φ ( ω ) ) α , ( Φ ( ϱ ) ) α ) = 2 - ( l ( Φ ( ω ) ) α ⊓ ( Φ ( ϱ ) ) α ) ) ≤ 2 - ( l ( Φ ( ω ⊓ ϱ ) ) ) = 2 - ( l ( ω ⊓ ϱ ) + 1 ) = 1 2 2 - l ( ω ⊓ ϱ ) = 1 2 σ ⊑ ( ω , ϱ ) . It is prompt to achieve that all the assumptions of Corollary 2 are satisfied with F (t) = ln (t) and e^-τ = 2^-1 . Thus, P : Ω ∞ → F ( Ω ∞ ) has a fuzzy fixed point u = u₁u₂ . . . ∈ Ω^∞ that is, u ∈ ( P u ) α . Also, by the definition of P , z is a fixed point of Φ. Thus u solves the recurrence relation (3.1), that is, u 1 = 0 , u n = 2 ( n - 1 ) n + n + 1 n u n - 1 , n ≥ 2 .

Conflict of interests

The authors declare that they have no competing interests.

Authors’ contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.