Generalized Θ-contractive fuzzy mappings

1 Introduction and Preliminaries

Fixed point theorems are very important tools for providing evidence of the existence and uniqueness of solutions to various mathematical models. The literature of the last four decades flourishes with results which discover fixed points of self and nonself nonlinear operators in a metric space. The Banach contraction theorem plays a fundamental role in fixed point theory and has become even more important because being based on iteration, it can be easily implemented on a computer. Fixed point theory studies are done in two aspects in fuzyy mathematics. One is to investigate the fixed points of single and multivalued mappings on fuzzy metric space (one may refer to [22–24]) and the other is to investigate the fixed points of fuzzy mappings on metric linear space. Weiss [26] and Butnariu [6] initiated the study of fixed point theorems in fuzzy mathematics. Heilpern [12] first used the concept of fuzzy mappings to prove the Banach contraction principle for fuzzy (approximate quantity-valued) mappings on complete metric linear spaces. Subsequently more than a few authors (e.g. [4, 21]) studied Heilpern fixed point results of fuzzy mappings satisfying a contractive type condition by using the d_∞-metric for fuzzy sets.

Let (X, d) be a metric space and 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 on 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 F (X) be the collection of all fuzzy sets in a metric space X. For A, B ∈ F (X), A ⊂ B means A (x) ≤ B (x) for each x ∈ X. We denote the fuzzy set χ_{x} by {x} unless and until it 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 F (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. Let S, T be fuzzy mappings from X into F (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.

Very recently, Jleli and Samet [14] introduced a new type of contraction called Θ-contraction and established some new fixed point theorems for such contraction in the context of generalized metric spaces.

Definition 2. Let Θ : (0, ∞) → (1, ∞) be a function satisfying:

Θ is nondecreasing;

A mapping S : X → X is said to be Θ-contraction if there exists the function Θ satisfying (Θ₁)-(Θ₃) and a constant k ∈ (0, 1) such that for all x, y ∈ X, d ( Sx , Sy ) > 0 ⇒ Θ ( d ( Sx , Sy ) ) ≤ [ Θ ( d ( x , y ) ) ] k .(1)

Theorem 1. ([14]) Let (X, d) be a complete metric space and S : X → X be a Θ-contraction, then S has a unique fixed point.

Samet et al. [14], established that any Banach contraction is a particular case of Θ-contraction while there are Θ-contractions which are not Banach contractions. To be consistent with Samet et al. [14], we denote by the Ψ set of all functions Θ : (0, ∞) → (1, ∞) satisfying the above conditions (Θ₁)-(Θ₃).

Later on Altun et al. [11] modified the above definitions by adding a general condition (Θ₄) which is given in this way:

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

Following Altun et al. [11], we represent the set of all continuous functions Θ : R⁺ → R satisfying (Θ₁)-(Θ₄) conditions by Ω.

For more details on Θ-contraction, one may refer to [1, 25].

Among various developments of fuzzy set theory, a progressive development has been made to find the fuzzy analogues of fixed point results of the classical fixed point theorems.

In this paper, we use a generalized Θ-contraction to obtain common α-fuzzy fixed points for fuzzy mappings in a metric space to generalize some known results of literature.

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

Lemma 1. Let (X, d) be a metric space and A, B ∈ CB (X), then for each a ∈ A, d ( a , B ) ≤ H ( A , B ) .

Lemma 2. ([3]) Let V be a metric linear space, T : X → W (V) be a fuzzy mapping and x₀ ∈ V. Then there exists x₁ ∈ V such that {x₁} ⊂ T (x₀).

Theorem 2. Let (X, d) be a complete metric space and let S, T be fuzzy mappings from X into F (X) and for each x ∈ X, there exist α _S (x), α _T (y) ∈ (0, 1] such that [Sx] _{α S (x)},   [Ty] _{α T (y)} are nonempty closed bounded subsets of X. If there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( [ Sx ] α S ( x ) , [ Ty ] α T ( y ) ) ) ≤ [ Θ ( M ( x , y ) ) ] k(2) for all x, y∈ X with H ([Sx] _{α S (x)}, [Ty] _{α T (y)}) > 0, where M ( x , y ) = max { d ( x , y ) , d ( x , [ Sx ] α S ( x ) ) , d ( y , [ Ty ] α T ( y ) ) , 1 2 [ d ( x , [ Ty ] α T ( y ) ) + d ( y , [ Sx ] α S ( x ) ) ] } . Then there exists some u ∈ [Su] _{α S (u)} ∩ [Tu] _{α T (u)}.

Proof. Let x₀ be an arbitrary point in X, then by hypotheses there exists α _S (x₀) ∈ (0, 1] such that [Sx₀] _{α S (x0)} is a nonempty closed bounded subset of X. For convenience, we denote α _S (x₀) by α₁. Let x₁ ∈ [Sx₀] _{α S (x0)}. For this x₁ there exists α _T (x₁) ∈ (0, 1] such that [Tx₁] _{α T (x1)} is a nonempty, closed and bounded subset of X. By Lemma 1, (Θ₁) and (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 ( M ( x 0 , x 1 ) ) k = [ Θ ( max { d ( x 0 , x 1 ) , d ( x 0 , [ Sx 0 ] α S ( x 0 ) ) , d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) , 1 2 [ d ( x 0 , [ Tx 1 ] α T ( x 1 ) ) + d ( x 1 , [ Sx 0 ] α S ( x 0 ) ) ] } ) ] k ≤ [ Θ ( max { d ( x 0 , x 1 ) , d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) } ) ] k Let d ( x 0 , x 1 ) ≤ d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) . Then from the above inequality, we get Θ ( d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) ) ≤ [ Θ ( d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) ) ] k ≤ [ Θ ( d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) ) ] , which is a contradiction. So, d ( x 0 , x 1 ) > d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) . Then we have Θ ( d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) ) ≤ [ Θ ( d ( x 0 , x 1 ) ] k .(3) From (Θ₄), 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 from (4), we get inf y ∈ [ Tx 1 ] α T ( x 1 ) Θ ( d ( x 1 , y ) ) ≤ [ Θ ( d ( x 0 , x 1 ) ] k .(4) Then, from (5), there exists x₂ ∈ [Tx₁] _{α T (x1)} such that Θ ( d ( x 1 , x 2 ) ) ≤ [ Θ ( d ( x 0 , x 1 ) ] k . For this x₂ there exists α _S (x₂) ∈ (0, 1] such that [Sx₂] _{α S (x2)} is a nonempty closed bounded subset of X. By Lemma 1, (Θ₁) and (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 ( M ( x 2 , x 1 ) ) k = [ Θ ( max { d ( x 2 , x 1 ) , d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) , d ( x 1 , [ Tx 1 ] α T ( x 1 ) ) , 1 2 [ d ( x 2 , [ Tx 1 ] α T ( x 1 ) ) + d ( x 1 , [ Sx 2 ] α S ( x 2 ) ) ] } ) ] k ≤ [ Θ ( max { d ( x 2 , x 1 ) , d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) } ) ] k = [ Θ ( max { d ( x 1 , x 2 ) , d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) } ) ] k . Let d ( x 1 , x 2 ) ≤ d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) . Then from the above inequality, we get Θ [ d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) ] ≤ Θ [ d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) ] k ≤ Θ [ d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) ] which is a contradiction. So, d ( x 1 , x 2 ) > d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) Then Θ [ d ( x 2 , [ Sx 2 ] α S ( x 2 ) ) ] ≤ Θ [ d ( x 1 , x 2 ) ] k . From (Θ₄), 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 .(5) Then, from (9), there exists x₃ ∈ [Sx₂] _{α S (x2)} such that Θ ( d ( x 2 , x 3 ) ) ≤ [ Θ ( d ( x 1 , x 2 ) ] k . So, continuing recursively, we obtain a sequence {x _n } in X such that x_2n+1 ∈ [Sx_2n] _{α S (x2n)} and x_2n+2 ∈ [Tx_2n+1] _{α T (x2n+1)} and Θ ( d ( x 2 n + 1 , x 2 n + 2 ) ) ≤ [ Θ ( d ( x 2 n , x 2 n + 1 ) ] k(6) and Θ ( d ( x 2 n + 2 , x 2 n + 3 ) ) ≤ [ Θ ( d ( x 2 n + 1 , x 2 n + 2 ) ] k(7) for all n ∈ N. From (11) and (12), we have Θ ( d ( x n , x n + 1 ) ) ≤ [ Θ ( d ( x n - 1 , x n ) ] k which further implies

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 0 , x 1 ) ] k n(8) for all n ∈ N. Since Θ ∈ Ω, so by taking limit as n→ ∞ in (14) we have, lim n → ∞ Θ ( d ( x n , x n + 1 ) ) = 1 which implies lim n → ∞ d ( x n , x n + 1 ) = 0 by (Θ₂). From the condition (Θ₃), 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 . Suppose that l < ∞. In this case, let B = l 2 > 0 . From the definition of the limit, there exists n₀ ∈ N such that | Θ ( d ( x n , x n + 1 ) ) - 1 d ( x n , x n + 1 ) r - l | ≤ B for all n > n₀. This implies Θ ( d ( x n , x n + 1 ) ) - 1 d ( x n , x n + 1 ) r ≥ l - B = l 2 = B for all n > n₀. Then nd ( x n , x n + 1 ) r ≤ An [ Θ ( d ( x n , x n + 1 ) ) - 1 ] for all n > n₀, where A = 1 B . Next suppose that l = ∞. Let B > 0 be an arbitrary positive number. From the definition of the limit, there exists n₀ ∈ N such that B ≤ Θ ( d ( x n , x n + 1 ) ) - 1 d ( x n , x n + 1 ) r for all n > n₀. This implies nd ( x n , x n + 1 ) r ≤ An [ Θ ( d ( x n , x n + 1 ) ) - 1 ] for all n > n₀, where A = 1 B . Thus, in all cases, there exist A > 0 and n₀ ∈ N such that nd ( x n , x n + 1 ) r ≤ An [ Θ ( d ( x n , x n + 1 ) ) - 1 ](9) for all n > n₀. Thus by (14) and (19), we get nd ( x n , x n + 1 ) r ≤ An ( [ ( Θ d ( x 0 , x 1 ) ) ] r n - 1 ) . Letting n→ ∞ in the above inequality, we obtain lim n → ∞ nd ( x n , x n + 1 ) r = 0 . Thus, there exists n₁ ∈ N such that d ( x n , x n + 1 ) ≤ 1 n 1 / r for all n > n₁. Now we prove that {x _n } is a Cauchy sequence. For m > n > n₁ 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 . 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 {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 a n₀ ∈ 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 ≥ n₀. Since d (x_{2n k +1}, [Tu] _{α T (u)}) >0 for all n _k ≥ n₀, so by (Θ₁), 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 ( M ( x 2 n k , u ) ) ] k = [ Θ ( max { d ( x 2 n k , u ) , d ( x 2 n k , [ Sx 2 n k ] α S ( x 2 n k ) ) , d ( u , [ Tu ] α T ( u ) ) , 1 2 [ d ( x 2 n k , [ Tu ] α T ( u ) ) + d ( u , [ Sx 2 n k ] α S ( x 2 n k ) ) ] } ) ] k ≤ [ Θ ( max { d ( x 2 n k , u ) , d ( x 2 n k , x 2 n k + 1 ) , d ( u , [ Tu ] α T ( u ) ) , 1 2 [ d ( x 2 n k , [ Tu ] α T ( u ) ) + d ( u , x 2 n k + 1 ) ] } ) ] k . Letting n → ∞, in above inequality and using the continuity of Θ, we have Θ [ d ( u , [ Tu ] α T ( u ) ) ] ≤ [ Θ ( d ( u , [ Tu ] α T ( u ) ) ) ] k which is a conradiction because k ∈ (0, 1). Hence u ∈ [Tu] _αT(u). Similarly, one can easily prove that u ∈ [Su] _αS(u). Thus u ∈ [Su] _αS(u) ∩ [Tu] _αT(u). The following result is a direct consequence of above theorem.

Theorem 3. Let (X, d) be a complete metric space and let S be fuzzy mapping from X into F (X) and for each x ∈ X, there exist α _S (x), α _S (y) ∈ (0, 1] such that [Sx] _αS(x),   [Sy] _αS(y) are nonempty closed bounded subsets of X. If there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( [ Sx ] α S ( x ) , [ Sy ] α S ( y ) ) ) ≤ Θ ( M ( x , y ) ) k(10) for all x, y∈ X with H ([Sx] _{α S (x)}, [Sy] _{α S (y)}) > 0, where M ( x , y ) = max { d ( x , y ) , d ( x , [ Sx ] α S ( x ) ) , d ( y , [ Sy ] α S ( y ) ) , 1 2 [ d ( x , [ Sy ] α S ( y ) ) + d ( y , [ Sx ] α S ( x ) ) ] } .(11) Then there exists some u ∈ [Su] _αS(u).

Corollary 1. Let (X, d) be a complete metric space and let F, G:X →CB (X) be multivalued mappings. If there exists some Θ ∈ Ω and k ∈ (0, 1) such that M Θ ( H ( Fx , Gy ) ) ≤ Θ ( M ( x , y ) ) k for all x, y∈ X with H (Fx, Gy) > 0, where M ( x , y ) = max { d ( x , y ) , d ( x , Fx ) , d ( y , Gy ) , 1 2 [ d ( x , Gy ) + d ( y , Fx ) ] } . Then there exists some u ∈ Fu ∩ Gu.

Proof. Consider a mapping α : X → [0, 1] and a pair of fuzzy mappings S, T : X → F (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 can be applied to obtain u ∈ X such that u ∈ [ Su ] α S ( u ) ∩ [ Tu ] α T ( u ) = Fu ∩ Gu .

Corollary 2. ([10]) Let (X, d) be a complete metric space and let F : X →CB (X) be multivalued mappings. If there exists some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( Fx , Fy ) ) ≤ Θ ( M ( x , y ) ) k for all x, y∈ X with H (Fx, Fy) > 0, where M ( x , y ) = max { d ( x , y ) , d ( x , Fx ) , d ( y , Fy ) , 1 2 [ d ( x , Fy ) + d ( y , Fx ) ] } . Then there exists some u ∈ Fu.

Corollary 3. Let (X, d) be a complete metric linear space and let S, T : X → W (X) be fuzzy mappings. If there exists some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( d ∞ ( S ( x ) , T ( y ) ) ) ≤ Θ ( M ( x , y ) ) k for all x, y∈ X with d_∞ (S (x), T (y)) > 0, where M ( x , y ) = max { p ( x , y ) , p ( x , S ( x ) ) , p ( y , T ( y ) ) , 1 2 [ p ( x , T ( y ) ) + p ( y , S ( x ) ) ] } . Then there exists some u ∈ X such that {u} ⊂ S (u) and {u} ⊂ T (u).

Proof. Let x ∈ X, then by Lemma 1 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 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. This implies Θ ( H ( [ Sx ] α ( x ) , [ Ty ] α ( x ) ) ) ≤ Θ ( d ∞ ( S ( x ) , T ( y ) ) ) ≤ [ Θ ( max { p ( x , y ) , p ( x , S ( x ) ) , p ( y , T ( y ) ) , 1 2 [ p ( x , T ( y ) ) + p ( y , S ( x ) ) ] } ) ] 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]. This implies p (x, S (x)) ≤ d (x, [Sx] ₁). Similarly, p (x, T (x)) ≤ d (x, [Tx] ₁). This further implies that for all x, y ∈ X,

Θ ( H ( [ Sx ] 1 , [ Ty ] 1 ) ) ≤ [ Θ ( max { d ( x , y ) , d ( x , [ Sx ] 1 ) , d ( y , [ Tx ] 1 ) , 1 2 [ d ( x , [ Tx ] 1 ) + d ( y , [ Sx ] 1 ) ] } ) ] k . Now, by Theorem 2, we obtain u ∈ X such that u ∈ [Su] ₁ ∩ [Tu] ₁, i.e, {u} ⊂ T (u) and {u} ⊂ S (u).

Corollary 4. Let (X, d) be a complete metric linear space and let S : X → W (X) be fuzzy mapping. If there exists some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( d ∞ ( S ( x ) , S ( y ) ) ) ≤ Θ ( M ( x , y ) ) k for all x, y∈ X with d_∞ (S (x), S (y)) > 0, where M ( x , y ) = max { p ( x , y ) , p ( x , S ( x ) ) , p ( y , S ( y ) ) , 1 2 [ p ( x , S ( y ) ) + p ( y , S ( x ) ) ] } . Then there exists some u ∈ X such that {u} ⊂ S (u).

In the following, we suppose that T ˆ (for the details, one may refer to [20, 21]) is the set-valued mapping induced by fuzzy mappings T : X → F (X), i.e, T ˆ x = { y : T ( x ) ( t ) = max t ∈ X T ( x ) ( t ) } .

Corollary 5. Let (X, d) be a complete metric space and let S, T : X → F (X) be fuzzy mappings such that for all x ∈ X, S ˆ ( x ) , T ˆ ( x ) are nonempty closed bounded subsets of X. If there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( S ˆ ( x ) , T ˆ ( y ) ) ) ≤ Θ ( M ( x , y ) ) k for all x, y∈ X with H ( S ˆ ( x ) , T ˆ ( y ) ) > 0 , where M ( x , y ) = max { d ( x , y ) , d ( x , S ˆ ( x ) ) , d ( y , T ˆ ( y ) ) , 1 2 [ d ( x , T ˆ ( y ) ) + d ( y , S ˆ ( x ) ) ] } . 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 1, there exists x^∗ ∈ X such that x ∗ ∈ S ˆ x ∗ ∩ T ˆ x ∗ . Then by Lemma 2, we have S ( x ∗ ) ( x ∗ ) ≥ S ( x ∗ ) ( x ) and T ( x ∗ ) ( x ∗ ) ≥ T ( x ∗ ) ( x ) for all x ∈ X.

Corollary 6. Let (X, d) be a complete metric space and let S : X → F (X) be fuzzy mappings such that for all x ∈ X, S ˆ ( x ) is a nonempty closed bounded subsets of X. If there exist some Θ ∈ Ω and k ∈ (0, 1) such that Θ ( H ( S ˆ ( x ) , S ˆ ( y ) ) ) ≤ Θ ( M ( x , y ) ) k for all x, y∈ X with H ( S ˆ ( x ) , S ˆ ( y ) ) > 0 , where M ( x , y ) = max { d ( x , y ) , d ( x , S ˆ ( x ) ) , d ( y , S ˆ ( y ) ) , 1 2 [ d ( x , S ˆ ( y ) ) + d ( y , S ˆ ( x ) ) ] } . Then there exists a point x^∗ ∈ X such that S (x^∗) (x^∗) ≥ S (x^∗) (x) for all x ∈ X.

Example Let X = {0, 1, 2} and d : X × X → R be the metric defined by d ( x , y ) = { 0 if x = y , 10 if x ¬ = y and x , y ∈ X - { 2 } 6 if x ¬ = y and x , y ∈ X - { 1 } 16 if x ¬ = y and x , y ∈ X - { 0 } . Then (X, d)  is a complete metric space. Let Θ ( t ) = e t k ∈ Ω for t > 0. Let α ∈ ( 0 , 1 3 ) , define a fuzzy mapping S : X → F (X) as follows: S ( 0 ) ( t ) { α if t = 0 0 if t = 1 α 2 if t = 2 , S ( 1 ) ( t ) { α if t = 0 α 3 if t = 1 0 if t = 2 , S ( 2 ) ( t ) { α 2 if t = 0 α if t = 1 α 3 if t = 2 . Then, for α ( x ) = 1 4 , we have [ Sx ] α ( x ) = { t : T ( x ) ( t ) = α ( x ) } = { { 0 } if x ¬ = 2 { 1 } if x = 2 . Note that, for x, y ∈ {0, 1} or x = y, we have H ( [ Sx ] α S ( x ) , [ Sy ] α S ( y ) ) = 0 . On the other hand, we get H ( [ Sx ] α S ( x ) , [ Sy ] α S ( y ) ) = 10 . Now we have the following two cases:

Case 1: If x = 0, y = 2, then H ( [ Sx ] α S ( x ) , [ Sy ] α S ( y ) ) = 10 > 0 and max { d ( x , y ) , d ( x , [ Sx ] α S ( x ) ) , d ( y , [ Sy ] α S ( y ) ) , 1 2 [ d ( x , [ Sy ] α S ( y ) ) + d ( y , [ Sx ] α S ( x ) ) ] } = 16 . Then there exists some k = 9 10 ∈ ( 0 , 1 ) such that the inequalities (23) and (24) are satisfied.

Case 2: H ( [ Sx ] α S ( x ) , [ Sy ] α S ( y ) ) = 10 > 0 and max { d ( x , y ) , d ( x , [ Sx ] α S ( x ) ) , d ( y , [ Sy ] α S ( y ) ) , 1 2 [ d ( x , [ Sy ] α S ( y ) ) + d ( y , [ Sx ] α S ( x ) ) ] } = 16 . Then there exists some k = 9 10 ∈ ( 0 , 1 ) such that the inequalities (23) and (24) are satisfied.

Hence all the conditions of Theorem 3 are satisfied to obtain 0 ∈ [ S 0 ] 1 4 .