Fixed point theorems for fuzzy mappings with applications

1 Introduction

In 1981, Heilpern [18] used the concept of fuzzy set to introduce a class of fuzzy mappings, which is a generalization of the set-valued mappings, and proved a fixed point theorem for fuzzy contraction mappings in metric linear space. It is worth noting that the result announced by Heilpern [18] is a fuzzy extension of the Banach contraction principle. Subsequently, several other authors have studied existence of fixed points of fuzzy mappings, for example, Azam et al. [10, 11], Bose et al. [14], Chang et al. [15], Cho et al. [16], Qiu et al. [27], Rashwan et al. [29], Shi-sheng [31].

In the following we always suppose that (X, d) is a complete metric space. Moreover, we shall use the following notations which have been recorded from [1 , 30]:

Let CB (X) be the family of nonempty, closed and bounded subsets of X. For A, B ∈ CB (X), define H ( A , B ) = max { sup a ∈ A d ( a , B ) , sup b ∈ B d ( b , A ) } where d ( x , A ) = inf y ∈ A d ( x , y ) .

A fuzzy set in X is a function with domain X and values in [0, 1], I ^X is the collection of all fuzzy sets in X . If A is a fuzzy set and x ∈ X, then the function values A (x)  is called the grade of membership of x in A. The α -level set of A is denoted by  [A]  _α and is defined as follows: [ A ] α = { x : A ( x ) ≥ α } if α ∈ ( 0 , 1 ] , [ A ] 0 = { x : A ( x ) > 0 } . ¯

Here B ¯ denotes the closure of the set B. Let ℑ (X) be the collection of all fuzzy sets in a metric space X . For A, B ∈ ℑ (X) , A ⊂ B means A (x) ≤ B (x) for each x ∈ X . We denote the fuzzy set χ _{x} by {x} unless otherwise is stated, where χ _{x} is the characteristic function of the crisp set A. If there exists an α ∈ [0, 1] such that [A]  _α , [B]  _α  ∈ CB (X) , then define p α ( A , B ) = inf x ∈ [ A ] α , y ∈ [ B ] α d ( x , y ) ,

D α ( A , B ) = H ( [ A ] α , [ B ] α ) .

If [A]  _α , [B]  _α  ∈ CB (X) for each α ∈ [0, 1] , then define p ( A , B ) = sup α p α ( A , B ) , d ∞ ( A , B ) = sup α D α ( A , B ) . We write p (x, B) instead of p ({x} , 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 x ∈ V A ( x ) = 1 . The collection of all approximate quantities in V is denoted by W (V) . Let X be an arbitrary set, Y be a metric space. A mapping T is called fuzzy mapping if T is a mapping from X into ℑ (Y). A fuzzy mapping T is a fuzzy subset on X × Y with membership function T (x) (y). The function T (x) (y)  is the grade of membership of y in T (x) .

Definition 1.1. Let S, T be fuzzy mappings from X into ℑ (X) . A point u ∈ X is called an α- fuzzy fixed point of T if there exists α ∈ [0, 1] such that u ∈ [Tu]  _α  . The point u ∈ X is called a common α- fuzzy fixed point of S and T if there exists α ∈ [0, 1] such that u ∈ [Su]  _α  ∩ [Tu]  _α  . When α = 1, it is called a common fixed point of fuzzy mappings.

Example 1.2. Let X = [0, 1] and define the mappings S, T : X → (X) ,  for α ∈ [0, 1]  as follows: For x ∈ X, we have S ( x ) ( t ) = { α if 0 ≤ t ≤ x 30 α 2 if x 30 < t ≤ x 20 α 3 if x 20 < t ≤ x 10 α 5 if x 10 < t ≤ 1 , and T ( x ) ( t ) = { α if 0 ≤ t ≤ x 60 α 3 if x 60 < t ≤ x 40 α 4 if x 40 < t ≤ x 20 α 7 if x 20 < t ≤ 1 , such that [ Tx ] α = [ 0 , x 60 ] , [ Sx ] α = [ 0 , x 30 ] . Then there exists 0 such that 0 ∈ [S0]  _α  ∩ [T0]  _α .

Very recently, Jleli and Samet [22] introduced a new type of contraction called Θ-contraction and established some new fixed point theorems for such a contraction in the context of generalized metric spaces.

Definition 1.3. Let Θ : (0, ∞) → (1, ∞) be a function satisfying:

Θ is nondecreasing;

A mapping T : X → X is said to be Θ-contraction if there exist the function Θ satisfying (Θ ₁)-(Θ ₃) and a constant k ∈ (0, 1) such that for all x, y ∈ X, d ( Tx , Ty ) ¬ = 0 ⇒ Θ ( d ( Tx , Ty ) ) ≤ [ Θ ( d ( x , y ) ) ] k .(1.1)

We denote by ϝ, the set of all functions satisfying (Θ ₁) - (Θ ₃) .

Example 1.4. The following functions Θ : (0, ∞) → (1, ∞) are the elements of ϝ:

Θ ( t ) = e t ,

Theorem 1.5. [22]. Let (X, d) be a complete metric space and T : X → X be a Θ-contraction, then T has a unique fixed point.

Later on Hancer et al. [17] modified the above definitions by adding a general condition (Θ ₄) which is given in this way:

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

Following Hancer et al. [17], we represent the set of all continuous functions Θ : ℝ + → ( 1 , ∞ ) satisfying (Θ ₁) - (Θ ₄) conditions by Ω.

For more details on Θ-contractions, we refer the reader to [2–8 , 26].

For the sake of convenience, we first state some known results for subsequent use in the next section.

Lemma 1.6. [25]. Let (X, d) be a metric space and A, B ∈ CB (X) . Then for each a ∈ A, d ( a , B ) ≤ H ( A , B ) .

Lemma 1.7. [9]. Let V be a metric linear space, T : X → W (V) be a fuzzy mapping and x ₀ ∈ V . Then there exists x ₁ ∈ V such that {x ₁} ⊂ T (x ₀) .

Theorem 2.1. Let (X, d) be a complete metric space and let S, T be fuzzy mappings from X into ℑ (X) and for each x ∈ X, there exist α _S  (x) , α _T  (x) ∈ (0, 1] such that [Sx]  _{α S (x)},   [Ty]  _{α T (x)} are nonempty, closed and bounded subsets of X. Assume that there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( [ Sx ] α S ( x ) , [ Ty ] α T ( y ) ) ) ≤ Θ ( d ( x , y ) ) k(2.1) for all x, y∈ X with H ([Sx]  _{α S (x)}, [Ty]  _{α T (y)}) > 0. Then there exists some u ∈ X such that u ∈ [Su]  _{α S (u)} ∩ [Tu]  _{α T (u)} .

Proof. Let x ₀ be an arbitrary point in X, then by hypotheses there exists α _S  (x ₀) ∈ (0, 1] such that [Sx ₀]  _{α S (x 0)} is a nonempty, closed and bounded subset of X. For convenience, we denote α _S  (x ₀) by α ₁ . Let x ₁ ∈ [Sx ₀]  _{α S (x 0)} . For this x ₁, there exists α _T  (x ₁) ∈ (0, 1] such that [Tx ₁]  _{α T (x 1)} is a nonempty, closed and bounded subset of X . By Lemma 1.6, (Θ ₁) and (2.1), we have Θ ( d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) ) ≤ Θ ( H ( [ Sx 0 ] α S ( x 0 ) , [ Tx 1 ] α T ( x 1 ) ) ≤ Θ ( d ( x 0 , x 1 ) ) k . From (Θ ₄), we know that Θ ( d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) ) = inf y ∈ [ Tx 1 ] α T ( x 1 ) Θ ( d ( x 1 , y ) ) . Thus inf y ∈ [ Tx 1 ] α T ( x 1 ) Θ ( d ( x 1 , y ) ) ≤ [ Θ ( d ( x 0 , x 1 ) ] k .(2.2) Then, from (2.2), there exists x ₂ ∈ [Tx ₁]  _{α T (x 1)} such that Θ ( d ( x 1 , x 2 ) ) ≤ [ Θ ( d ( x 0 , x 1 ) ] k .(2.3) For this x ₂ there exists α _S  (x ₂) ∈ (0, 1] such that [Sx ₂]  _{α S (x 2)} is a nonempty, closed and bounded subset of X . By Lemma 1.6, (Θ ₁) and (2.1), we have Θ ( d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) ) ≤ Θ ( H ( [ Tx 1 ] α T ( x 1 ) , [ Sx 2 ] α S ( x 2 ) ) ) = Θ ( H ( [ Sx 2 ] α S ( x 2 ) , [ Tx 1 ] α T ( x 1 ) ) ) ≤ Θ ( d ( x 2 , x 1 ) ) k = Θ ( d ( x 1 , x 2 ) ) k

From (Θ ₄), we know that Θ [ d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) ] = inf y 1 ∈ [ Sx 2 ] α S ( x 2 ) Θ ( d ( x 2 , y 1 ) )

Thus inf y 1 ∈ [ Sx 2 ] α S ( x 2 ) Θ ( d ( x 2 , y 1 ) ) ≤ Θ [ d ( x 1 , x 2 ) ] k .(2.4) Then, from (2.4), there exists x ₃ ∈ [Sx ₂]  _{α S (x 2)} such that Θ ( d ( x 2 , x 3 ) ) ≤ [ Θ ( d ( x 1 , x 2 ) ] k .(2.5) So, continuing recursively, we obtain a sequence {x _n } in X such that x 2 n + 1 ∈ [ Sx 2 n ] α S ( x 2 n ) and x 2 n + 2 ∈ [ Tx 2 n + 1 ] α T ( x 2 n + 1 ) with Θ ( d ( x 2 n + 1 , x 2 n + 2 ) ) ≤ [ Θ ( d ( x 2 n , x 2 n + 1 ) ] k(2.6) and Θ ( d ( x 2 n + 2 , x 2 n + 3 ) ) ≤ [ Θ ( d ( x 2 n + 1 , x 2 n + 2 ) ] k(2.7) for all n ∈ ℕ . From (2.6) and (2.7), we have Θ ( d ( x n , x n + 1 ) ) ≤ [ Θ ( d ( x n - 1 , x n ) ] k(2.8) which further implies that Θ ( d ( x n , x n + 1 ) ) ≤ [ Θ ( d ( x n - 1 , x n ) ] k ≤ [ Θ ( d ( x n - 2 , x n - 1 ) ] k 2 Θ ( d ( x n , x n + 1 ) ) ≤ . . . ≤ [ Θ ( d ( x 0 , x 1 ) ] k n(2.9) for all n ∈ ℕ . Since Θ ∈ Ω, by taking limit as n→ ∞ in (2.9) we have, lim n → ∞ Θ ( d ( x n , x n + 1 ) ) = 1(2.10) which implies that lim n → ∞ d ( x n , x n + 1 ) = 0(2.11) by (Θ ₂). From the condition (Θ ₃), there exist 0 < r < 1 and l ∈ (0, ∞] such that lim n → ∞ Θ ( d ( x n , x n + 1 ) ) - 1 d ( x n , x n + 1 ) r = l .(2.12) Suppose that l < ∞ . In this case, let B = l 2 > 0 . From the definition of the limit, there exists n 0 ∈ ℕ such that | Θ ( d ( x n , x n + 1 ) ) - 1 d ( x n , x n + 1 ) r - l | ≤ B for all n > n ₀ . This implies that Θ ( d ( x n , x n + 1 ) ) - 1 d ( x n , x n + 1 ) r ≥ l - B = l 2 = B for all n > n ₀ . Then nd ( x n , x n + 1 ) r ≤ An [ Θ ( d ( x n , x n + 1 ) ) - 1 ](2.13) for all n > n ₀, where A = 1 B . Now we suppose that l = ∞ . Let B > 0 be an arbitrary positive number. From the definition of the limit, there exists n 0 ∈ ℕ such that B ≤ Θ ( d ( x n , x n + 1 ) ) - 1 d ( x n , x n + 1 ) r for all n > n ₀ . This implies that nd ( x n , x n + 1 ) r ≤ An [ Θ ( d ( x n , x n + 1 ) ) - 1 ] for all n > n ₀, where A = 1 B . Thus, in all cases, there exist A > 0 and n 0 ∈ ℕ such that nd ( x n , x n + 1 ) r ≤ An [ Θ ( d ( x n , x n + 1 ) ) - 1 ](2.14) for all n > n ₀ . Thus by (2.9) and (2.14), we get nd ( x n , x n + 1 ) r ≤ An ( [ ( Θ d ( x 0 , x 1 ) ) ] r n - 1 ) .(2.15) Letting n→ ∞ in the above inequality, we obtain lim n → ∞ nd ( x n , x n + 1 ) r = 0 . Thus, there exists n 1 ∈ ℕ such that d ( x n , x n + 1 ) ≤ 1 n 1 / r(2.16) for all n > n ₁ . Now we prove that {x _n } is a Cauchy sequence. For m > n > n ₁ we have, d ( x n , x m ) ≤ ∑ i = n m - 1 d ( x i , x i + 1 ) ≤ ∑ i = n m - 1 1 i 1 / r ≤ ∑ i = 1 ∞ 1 i 1 / r .(2.17) Since, 0 < r < 1, then ∑ i = 1 ∞ 1 i 1 / r converges. Therefore, d (x _n , x _m ) →0 as m, n → ∞ . Thus we proved that {x _n } is a Cauchy sequence in (X, d). The completeness of (X, d) ensures that there exists u ∈ X such that, lim n → ∞ x n → u . Now, we prove that u ∈ [Tu]  _{α T (u)} . We suppose on the contrary that u ∉ [Tu]  _{α T (u)}, then there exist an n 0 ∈ ℕ and a subsequence {x _{n k} } of {x _n } such that d (x _{2n k +1}, [Tu]  _{α T (u)}) >0 for all n _k  ≥ n ₀ . Since d (x _{2n k +1}, [Tu]  _{α T (u)}) >0 for all n _k  ≥ n ₀, so by (Θ ₁), we have Θ [ d ( x 2 n k + 1 , [ Tu ] α T ( u ) ) ] ≤ Θ [ H ( [ Sx 2 n k ] α S ( x 2 n k ) , [ Tu ] α T ( u ) ) ] ≤ [ Θ ( d ( x 2 n k , u ) ) ] k .

Letting k → ∞ , in the above inequality and using the continuity of Θ, we have Θ [ d ( u , [ Tu ] α T ( u ) ) ] ≤ 0 . Hence u ∈ [Tu]  _{α T (u)} . Similarly, one can easily prove that u ∈ [Su]  _{α S (u)} . Thus u∈ [Su]  _{α S (u)} ∩ [Tu]  _{α T (u)} . □

Theorem 2.2. Let (X, d) be a complete metric space and let S be a fuzzy mapping from X into ℑ (X) and for each x ∈ X, there exist α _S  (x) , α _T  (x) ∈ (0, 1] such that [Sx]  _{α S (x)},   [Sy]  _{α S (x)} are nonempty, closed and bounded subsets of X. Assume that there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( [ Sx ] α S ( x ) , [ Sy ] α T ( x ) ) ) ≤ Θ ( d ( x , y ) ) k for all x, y∈ X with H ([Sx]  _{α S (x)}, [Sy]  _{α T (x)}) >0. Then there exists some u ∈ [Su]  _{α S (u)} .

Now we state a common fixed point result for two multivalued mappings.

Corollary 2.3. Let (X, d) be a complete metric space and let F, G : X →CB (X) be multivalued mappings. Assume that there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( Fx , Gy ) ) ≤ Θ ( d ( x , y ) ) k for all x, y∈ X with H (Fx, Gy) > 0. Then there exists some u ∈ Fu ∩ Gu .

Proof. Consider a mapping α : X → (0, 1] and a pair of fuzzy mappings S, T : X → ℑ (X) defined by S ( x ) ( t ) = { α ( x ) , if t ∈ Fx , 0 , if t ∉ Fx and T ( x ) ( t ) = { α ( x ) , if t ∈ Gx , 0 , if t ∉ Gx . Then [ Sx ] α ( x ) = { t : S ( x ) ( t ) ≥ α ( x ) } = Fx and [ Tx ] α ( x ) = { t : T ( x ) ( t ) ≥ α ( x ) } = Gx . Thus, Theorem 2.1 can be applied to obtain u ∈ X such that u ∈ [ Su ] α S ( u ) ∩ [ Tu ] α T ( u ) = Fu ∩ Gu . □

Corollary 2.4. Let (X, d) be a complete metric space and let G : X →CB (X) be multivalued mappings. Assume that there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( Gx , Gy ) ) ≤ Θ ( d ( x , y ) ) k for all x, y∈ X with H (Gx, Gy) > 0. Then there exists some u ∈ X such that u ∈ Gu .

Corollary 2.5. Let (X, d) be a complete metric linear space and let S, T : X → W (X) be fuzzy mappings. Suppose that there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( d ∞ ( S ( x ) , T ( y ) ) ) ≤ Θ ( p ( x , y ) ) k for all x, y∈ X with d _∞ (S (x) , T (y)) > 0. Then there exists some u ∈ X such that {u} ⊂ S (u) and {u} ⊂ T (u) .

Proof. Let x ∈ X, then by Lemma 1.6 there exists y ∈ X such that y ∈ [Sx] ₁. Similarly, we can find z ∈ X such that z ∈ [Tx] ₁. It follows that for each x ∈ X, [Sx]  _α(x), [Tx]  _α(x) are nonempty, closed and bounded subsets of X. As α (x) = α (y) = 1, by the definition of a d _∞-metric for fuzzy sets, we have H ( [ Sx ] α ( x ) , [ Ty ] α ( x ) ) ≤ d ∞ ( S ( x ) , T ( y ) ) for all x, y∈ X . From (Θ ₁), we have Θ ( H ( [ Sx ] α ( x ) , [ Ty ] α ( x ) ) ) ≤ Θ ( d ∞ ( S ( x ) , T ( y ) ) ) ≤ [ Θ ( p ( x , y ) ) ] k for all x, y∈ X . Since [Sx] ₁ ⊆ [Sx]  _α for each α ∈ (0, 1] . Therefore d (x, [Sx]  _α ) ≤ d (x, [Sx] ₁) for each α ∈ (0, 1] . It implies that p (x, S (x)) ≤ d (x, [Sx] ₁) . Similarly, p (x, T (x)) ≤ d (x, [Tx] ₁) . Furthermore this implies that for all x, y ∈ X, Θ ( H ( [ Sx ] 1 , [ Ty ] 1 ) ) ≤ [ Θ ( d ( x , y ) ) ] k . Now, by Theorem 2.1, we obtain u ∈ X such that u ∈ [Su] ₁ ∩ [Tu] ₁, i.e., {u} ⊂ T (u) and {u} ⊂ S (u).

In the following, we suppose that T ˆ (for details, see [30, 31]) is the set-valued mapping induced by fuzzy mappings T : X → ℑ (X), i.e., T ˆ x = { y : T ( x ) ( t ) = max t ∈ X T ( x ) ( t ) } . □

Corollary 2.6. Let (X, d) be a complete metric space and let S, T : X → ℑ (X) be fuzzy mappings such that for all x ∈ X, S ˆ ( x ) , T ˆ ( x ) are nonempty, closed and bounded subsets of X. Assume that there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( S ˆ ( x ) , T ˆ ( y ) ) ) ≤ Θ ( d ( x , y ) ) k for all x, y∈ X with H ( S ˆ ( x ) , T ˆ ( y ) ) > 0 . Then there exists a point x ^∗ ∈ X such that S (x ^∗) (x ^∗) ≥ S (x ^∗) (x) and T (x ^∗) (x ^∗) ≥ T (x ^∗) (x) for all x ∈ X .

Proof. By Corollary 2.3, there exists x ^∗ ∈ X such that x ∗ ∈ S ˆ x ∗ ∩ T ˆ x ∗ . Then by Lemma 1.7, we have S ( x ∗ ) ( x ∗ ) ≥ S ( x ∗ ) ( x ) and T ( x ∗ ) ( x ∗ ) ≥ T ( x ∗ ) ( x ) for all x ∈ X .□

Example 2.7. Let X = [0, 1]  and define d : X × X → ℝ + as follows: d ( x , y ) = | x − y | . Then (X, d)  is a complete metric space. Define a pair of mappings S, T : X → ℑ (X) ,  for α ∈ [0, 1]  as follows:

For x ∈ X, we have S ( x ) ( t ) = { α if 0 ≤ t ≤ x 30 α 2 if x 30 < t ≤ x 20 α 3 if x 20 < t ≤ x 10 α 5 if x 10 < t ≤ 1 , and T ( x ) ( t ) = { α if 0 ≤ t ≤ x 15 α 3 if x 15 < t ≤ x 10 α 4 if x 10 < t ≤ x 5 α 7 if x 5 < t ≤ 1 . , such that [ Tx ] α = [ 0 , x 15 ] , [ Sx ] α = [ 0 , x 30 ] . Let Θ ( t ) = e t k . Then there exists some k = 1 15 ∈ ( 0 , 1 ) such that Θ ( H ( [ Sx ] α S ( x ) , [ Ty ] α T ( y ) ) ) ≤ Θ ( d ( x , y ) ) k for all x, y∈ X with H ([Sx]  _{α S (x)}, [Ty]  _{α T (y)}) >0 is satisfied to obtain 0 ∈ [S0]  _α  ∩ [T0]  _α  .

Let Σ be a nonempty alphabet. Let Σ ^∞ be the set of all finite and infinite sequences (“words”) over Σ, where we adopt the convention that the empty sequence ∅ is an element of Σ ^∞. Moreover, on Σ ^∞, we consider the prefix order ⊑ given by: x ⊑ y if and only if x is a prefix of y . For each nonempty x ∈ Σ ^∞ denote by l (x) the length of x. Then l (x) ∈ [0, ∞] , whenever x¬ = ∅ and l (∅) =0 . For each x, y ∈ Σ ^∞, let x ⊓ y be the common prefix of x and y. Clearly, x = y if and only if x ⊑ y and y ⊑ x and l (x) = l (y). Then, the the Baire metric d _⊑ is defined on Σ ^∞ × Σ ^∞ by { d ⊑ ( x , y ) = 0 , if x = y d ⊑ ( x , y ) = 2 - l ( x ⊓ y ) , otherwise

so that the metric space (Σ ^∞, d _⊑) is complete. Finally, we refer to the average case time complexity analysis of the Quicksort divide-and-conquer sorting algorithm in [30].

Precisely, we consider the following recurrence relation: T ( 1 ) = 0 and T ( n ) = 2 ( n - 1 ) n + n + 1 n T ( n - 1 ) , n ≥ 2 .(3.1) Consider as an alphabet Σ the set of nonnegative real numbers, i.e., Σ = ℝ + . We associate to T the functional Φ : Σ ^∞ → Σ ^∞ given by ( Φ ( x ) ) 1 = T ( 1 ) and ( Φ ( x ) ) n = 2 ( n - 1 ) n + n + 1 n x n - 1 for all n ≥ 2 (if x ∈ Σ ^∞ has length n < ∞ , we write x : = x ₁ x ₂ . . . x _n, and if x is an infinite word we write x : = x ₁ x ₂ . . . .). It follows by the construction that l (Φ (x)) = l (x) +1 for all x ∈ Σ ^∞ and l (Φ (x)) =+ ∞ whenever l (x) = + ∞ . We will show that the functional Φ has a fixed point by an application of Theorem 2.2. Let S : Σ ^∞ → ℑ (Σ ^∞) be the fuzzy mapping given by S x = ( Φ ( x ) ) α for all x ∈ Σ ∞ and distinguish the following two cases:

Case 01: If x = y, then we write H ⊑ ( ( Φ ( x ) ) α , ( Φ ( x ) ) α ) = 0 = d ⊑ ( x , x ) . Case 02: If x ¬ = y, then we write H ⊑ ( ( Φ ( x ) ) α , ( Φ ( y ) ) α ) = d ⊑ ( ( Φ ( x ) ) α , ( Φ ( y ) ) α ) = 2 - ( l ( Φ ( x ) ) α ⊓ ( Φ ( y ) ) α ) ) ≤ 2 - ( l ( Φ ( x ⊓ y ) ) ) = 2 - ( l ( x ⊓ y ) + 1 ) = 1 2 2 - l ( x ⊓ y ) = ( 1 2 ) 2 d ⊑ ( x , y ) . It is immediate to conclude that all the conditions of Theorem 2.2 are satisfied with Θ ( t ) = e t k and k = 1 2 . Consequently, the fuzzy mapping S has a fuzzy fixed point u = u ₁ u ₂ . . . ∈ Σ ^∞ that is, u ∈ (S _u )  _α . Also, in view of the definition of S, z is a fixed point of Φ, and hence, u solves the recurrence relation (3.1). We have u 1 = 0 , u n = 2 ( n - 1 ) n + n + 1 n u n - 1 , n ≥ 2 .

In this paper, we obtained the common fuzzy fixed points of fuzzy mappings satisfying Θ-contraction in the context of metric space. In the process, we generalize several well known recent and classical results. Decision Making problems are most important for humans in physical sciences and social sciences as well. Our results can be further extended for Neutrosophic soft mappings to apply in decision making problems [28], moreover we can use more advanced techniques in decision making problems (see. [24 , 33–35]).

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.