Fixed points of F -contractive type fuzzy mappings

1 Introduction and preliminaries

After the introduction of the concept of a fuzzy set by Zadeh [19], several researches were conducted on the generalizations of the concept of a fuzzy set. Heilpern [9] introduced the concept of fuzzy mapping and proved a fixed point theorem for fuzzy contraction mappings which is a generalization of the fixed point theorem for multivalued mappings of Nadler [14]. Then, Estruch and Vidal [6], Frigon and O’Regan [7], Türkoğlu and Rhoades [16] and many researches obtained some fixed point theorems for fuzzy contraction mappings over a complete metric spaces which is a generalization of the given Heilpern’s fixed point theorem. Also, Gregori and Romaguera [8] studied on quasi metric space for fixed points of fuzzy mappings. In this paper we will present a general fixed point theorem for fuzzy mappings on a complete metric space.

Let (X, d) be a metric space. A fuzzy set A in X is a function with domain X and values in I = [0, 1]. If A is a fuzzy set and x ∈ X, then the function value A (x) is called the grade of membership of x in X to the fuzzy set A. The α-level set of A, denoted by A_α, is defined as A α = { x : A ( x ) ≥ α } if α ∈ ( 0 , 1 ] , A 0 = { x : A ( x ) > 0 } ¯ , where B ¯ is the closure of the non-fuzzy set B in the metric space X.

A fuzzy set A in X said to be an approximate quantity if and only if A_α is compact in X for each α ∈ (0, 1] and sup x ∈ X A ( x ) = 1 . We denote by W ( X ) the family of all approximate quantities in X. When A ∈ W ( X ) and A (x₀) =1 for some x₀ ∈ X, we will identify A with an approximation of x₀. Let A , B ∈ W ( X ) , then A is said to be more accurate than B, denoted by A ⊂ B, if and only if A (x) ≤ B (x) for each x ∈ X. It is easy to see that relation ⊂ is a partial order determined on the family W ( X ) . Also, it is easy to see that if 0 ≤ r ≤ s ≤ 1, then A _s ⊆ A _r .

For A , B ∈ W ( X ) , α ∈ [0, 1], define p α ( A , B ) = inf { d ( x , y ) : x ∈ A α , y ∈ B α } , D α ( A , B ) = H ( A α , B α ) , where H is the Pompeiu-Hausdorff distance [5], that is, H ( A α , B α ) = max { sup x ∈ A a d ( x , B α ) , sup y ∈ B a d ( y , A α ) } .

Also, D ( A , B ) = sup α D α ( A , B ) , p ( A , B ) = sup α p α ( A , B ) .

Note that, since A_α and B_α are compact sets in the metric space (X, d) for all α ∈ [0, 1], then p (A, B)< ∞ and D (A, B)< ∞ for all A , B ∈ W ( X ) .

The function p_α is called α-space, D_α is a α-distance, and D is a distance between A and B. We note that p_α is nondecrasing function of α.

Let X be an arbitrary set, Y be a metric space. A mapping T is called a fuzzy mapping if T is a mapping from X into W ( Y ) , that is, Tx ∈ W ( Y ) for each x in X. Thus, if we characterize a fuzzy set Tx in a metric space Y by a membership function Tx, then (Tx) (y) is the grade of membership of y in Tx. Therefore, a fuzzy mapping T is a fuzzy subset on X × Y with membership function (Tx) (y).

Corresponding to each α ∈ [0, 1] and x ∈ X, the fuzzy point x_α of X is the fuzzy set x_α : X → [0, 1] given by x α ( y ) = { α , if x = y 0 , otherwise . For α = 1, we have x 1 ( y ) = { 1 , if x = y 0 , otherwise = { x } , that is, {x} be a fuzzy set with membership function equal a characteristic function of set {x}. A fuzzy point x_α in X is called a fixed fuzzy point of the fuzzy mapping T if x_α ⊂ Tx, that is, (Tx) (x) ≥ α or x ∈ (Tx) _α. That is, the fixed degree of x in Tx is at least α. If {x} ⊂ Tx, then x is a fixed point of a fuzzy mapping T.

The following lemmas are needed in the sequel [6, 11]. We will present the proofs of some of them because of importance.

Lemma 1. [8] Let (X, d) be a metric space. Then, for each A ∈ W ( X ) there exists p ∈ X such thatA (p) =1.

Proof. Since A ∈ W ( X ) , then sup x ∈ X A ( x ) = 1 . Choose a strictly increasing (r _n ) _n∈N in (0, 1] such that r _n → 1. For each n ∈ N choose y _n ∈ A _{r n} . Since A _{r n} ⊆ A₀ for all n ∈ N, the sequence (y _n ) has a subsequence (y_n(k)) _k∈N which converges to a pointp ∈ A₀. Suppose that A (p) = r < 1. Choose an m ∈ N such that r _m > r. Then, (y_n(k)) _k∈N has a cluster point a ∈ A _{r m} . Hence a = p. So, A (p) ≥ r _m , a contradiction. We conclude that A (p) =1.

Corollary 1. [8, 11] Let (X, d) be a metric space, T be a fuzzy mapping from X into W ( X ) and x₀ ∈ X. Then, there exists x₁ ∈ X such that {x₁ } ⊂ Tx₀ (that is (Tx₀) (x₁) =1).

Lemma 2. [8] Let (X, d) be a metric space, A , B ∈ W ( X ) and x ∈ A₁ (such an x exists by Lemma 1). Then, there is y ∈ B₁ such that d (x, y) ≤ D₁ (A, B).

Proof. Since D₁ (A, B) = H (A₁, B₁) and x ∈ A₁, we have d (x, B₁) ≤ D₁ (A, B). Since B₁ is compact in (X, d), there exists y ∈ B₁ such that d (x, y) = d (x, B₁). Therefore, d (x, y) ≤ D₁ (A, B).

Lemma 3. [8] Let (X, d) be a metric space and let A , B ∈ W ( X ) . Then, p (A, B) = p₁ (A, B).

Proof. Since A _r ⊆ A _s and B _r ⊆ B _s whenever 0 ≤ s ≤ r ≤ 1, it follows that p _r (A, B) ≤ p₁ (A, B) for all r ∈ [0, 1]. Hence, p (A, B) = p₁ (A, B).

Lemma 4. [8] Let (X, d) be a metric space, A ∈ W ( X ) and y ∈ A₁. Then, for each x ∈ X, p (x, A) ≤ d (x, y).

Proof. We obtain from Lemma 3.

Lemma 5. [6, 9] Let (X, d) be a metric space and let A ∈ W ( X ) . Then, for each x, y ∈ X and each α ∈ [0, 1], p_α (x, A) ≤ d (x, y) + p_α (y, A).

Proof. For each x, y ∈ X, p α ( x , A ) = inf { d ( x , z ) : z ∈ A α } ≤ inf { d ( x , y ) + d ( y , z ) : z ∈ A α } = d ( x , y ) + inf { d ( y , z ) : z ∈ A α } = d ( x , y ) + p α ( y , A ) .

Lemma 6. [6, 9] Let (X, d) be a metric space, A ∈ W ( X ) and x ∈ A₁. Then, for each B ∈ W ( X ) and each α ∈ [0, 1], p_α (x, B) ≤ D_α (A, B).

Proof. Clearly, x ∈ A_α for all α ∈ [0, 1]. Hence, p α ( x , B ) = d ( x , B α ) ≤ H ( A α , B α ) = D α ( A , B ) for all B ∈ W ( X ) and α ∈ [0, 1].

Lemma 7. [6, 9] Let (X, d) be a metric space, A ∈ W ( X ) . If x ∈ A₁ (that is {x} ⊂ A) then p_α (x, A) =0 for each α ∈ [0, 1].

Proof. Let x ∈ A₁. Then, x ∈ A_α for all α ∈ [0, 1]. Hence, for each α ∈ [0, 1] we obtain p α ( x , A ) = inf y ∈ A α d ( x , y ) = 0 .

Lemma 8. [8] Let (X, d) be a metric space, A ∈ W ( X ) . If p (x, A) =0, then x ∈ A₁ (that is A (x) =1).

Proof. Let p (x, A) =0. By Lemma 3, we obtain p₁ (x, A) = d (x, A₁) =0. Then we have x ∈ A₁.

Lemma 9. [6, 9] Let (X, d) be a metric space, A ∈ W ( X ) and x ∈ X. Then, x_α ⊂ A if and only if p_α (x, A) =0.

Proof. If x_α ⊂ A, then x ∈ A_α. Hence, p α ( x , A ) = inf y ∈ A α d ( x , y ) = 0 .

If p_α (x, A) =0, then p_α (x, A) = d (x, A_α) =0. Hence, we have x ∈ A_α.

Lemma 10. [6, 9] Let (X, d) be a metric space and let A ∈ W ( X ) . If x_α ⊂ A, then p_α (x, B) ≤ D_α (A, B) for each B ∈ W ( X ) .

Proof. If x_α ⊂ A, then x ∈ A_α. Hence, for each B ∈ W ( X ) , p α ( x , B ) = d ( x , B α ) ≤ H ( A α , B α ) = D α ( A , B ) .

In this paper, we present a fixed point theorem for new type contractive fuzzy mappings over a complete metric space. Our results are based on a new approach to contraction mapping, which is called F-contraction. The concept of F-contraction for single valued maps on complete metric space was introduced by Wardowski [18]. First, we recall this new concept and some related results.

Let F : (0, ∞) → R be a function. For the sake of completeness, we will consider the following conditions:

(F1) F is strictly increasing, i.e., for all α, β ∈ (0, ∞) such that α < β, F (α) < F (β),

(F2) For each sequence {α _n } of positive numbers lim n → ∞ α n = 0 if and only if lim n → ∞ F ( α n ) = - ∞ ,

(F3) There exists k ∈ (0, 1) such that lim α → 0 + α k F ( α ) = 0 ,

We denote by F be the set of all functions F satisfying (F1)-(F3). Some examples of the functions belonging F are F₁ (α) = ln α, F₂ (α) = α + ln α, F 3 ( α ) = - 1 α and F₄ (α) = ln(α² + α).

Definition 1. [18] Let (X, d) be a metric space and T : X → X be a mapping. Then, we say that T is an F-contraction if F ∈ F and there exists τ > 0 such that for all x, y ∈ X, d (Tx, Ty) >0 implies τ + F ( d ( Tx , Ty ) ) ≤ F ( d ( x , y ) ) .(1)

If we take F (α) = ln α in Definition 1, the inequality (1) turns to d ( Tx , Ty ) ≤ e - τ d ( x , y ) ,(2) for all x, y ∈ X, Tx ≠ Ty. It is clear that for x, y ∈ X such that Tx = Ty, the inequality d (Tx, Ty) ≤ e^-τd (x, y) also holds. Thus T is an ordinary contraction with contractive constant e^-τ. Therefore every ordinary contraction is also F-contraction with F (α) = ln α, but the converse may not be true as shown in the Example 2.5 of [18]. If we choose F (α) = α + ln α, the inequality (1) turns to d ( Tx , Ty ) d ( x , y ) e d ( Tx , Ty ) - d ( x , y ) ≤ e - τ ,(3) for all x, y ∈ X, Tx ≠ Ty. In addition, Wardowski showed that every F-contraction T is a contractive mapping, i.e., d ( Tx , Ty ) < d ( x , y ) , for all x , y ∈ X , Tx ≠ Ty . Thus, every F-contraction is a continuous map. Also, Wardowski concluded that if F 1 , F 2 ∈ F with F₁ (α) ≤ F₂ (α) for all α > 0 and G = F₂ - F₁ is nondecreasing, then every F₁ -contraction T is an F₂-contraction. He noted that for the mappings F₁ (α) = ln α and F₂ (α) = α + ln α, F₁ < F₂ and the mapping F₂ - F₁ is strictly increasing. Hence, every Banach contraction satisfies the contractive condition (3). On the other side, Example 2.5 in [18] shows that the mapping T is not F₁-contraction (Banach Contraction), but still is an F₂-contraction. Thus, Wardowski proved that every F-contraction on a complete metric space has a unique fixed point, which is a proper generalization of Banach Contraction Principle.

Following Wardowski [18], Altun et al. [3] introduced the multivalued version of F-contraction as follows: Let (X, d) be a metric space T : X → K (X) (be the family of all nonempty compact subsets of X) be a multivalued map and F ∈ F . If there exists τ > 0 such that τ + F ( H ( Tx , Ty ) ) ≤ F ( d ( x , y ) ) for all x, y ∈ X with H (Tx, Ty) >0, then T is called multivalued F-contraction. Then they proved every multivalued F-contraction on complete metric space has a fixed point. It can be find more informations and results on F-contractions for both single valued and multivalued mappings in [1, 17].

Definition 2. Let (X, d) be a metric space, F ∈ F and T be fuzzy mapping from X to W ( X ) . Then T is said to be a F-contractive type fuzzy mapping if there exists τ > 0 such that τ + F ( D ( Tx , Ty ) ) ≤ F ( d ( x , y ) ) ,(4) for x, y ∈ X with D (Tx, Ty) >0.

Theorem 1. Let (X, d) be a complete metric space and T be a F-contractive type fuzzy mapping. Then T has a fixed point, that is, there exists u ∈ X such that (Tu) (u) =1.

Proof. Assume that (Tx) (x) ≠1 for all x ∈ X. Then by Lemma 8, p (x, Tx) >0 for all x ∈ X. Let x₀ ∈ X. By Lemma 1, there exists an x₁ ∈ X such that (Tx₀) (x₁) =1. By Lemma 2, there exists an x₂ ∈ X such that (Tx₁) (x₂) =1 and d (x₁, x₂) ≤ D₁ (Tx₀, Tx₁). Again, we can find an x₃ ∈ X such that (Tx₂) (x₃) =1 and d (x₂, x₃) ≤ D₁ (Tx₁, Tx₂). Following the process we construct a sequence (x _n ) _n∈N in X such that (Tx _n ) (x_n+1) =1 and d (x _n , x_n+1) ≤ D₁ (Tx_n-1, Tx _n ). Note that, x _n ≠ x_n-1 for all n ∈ N. Then, we get 0 < d ( x n , x n + 1 ) ≤ D 1 ( Tx n - 1 , Tx n ) ≤ D ( Tx n - 1 , Tx n ) .

On the other hand, from (F1) we have

τ + F ( d ( x n , x n + 1 ) ) ≤ τ + F ( D ( Tx n - 1 , Tx n ) ) ≤ F ( d ( x n - 1 , x n ) ) ,(5) for all n ∈ N. Denote a _n = d (x _n , x_n+1) for n ∈ N. Then a _n > 0 for all n ∈ N and, using (5), the following holds:

F ( a n ) ≤ F ( a n - 1 ) - τ ≤ F ( a n - 2 ) - 2 τ ⋮ ≤ F ( a 0 ) - n τ .(6) From (6), we get lim n → ∞ F ( a n ) = - ∞ . Thus, from (F2), we have lim n → ∞ a n = 0 . From (F3) there exists k ∈ (0, 1) such that lim n → ∞ a n k F ( a n ) = 0 . By (6), the following holds for all n ∈ N a n k F ( a n ) - a n k F ( a 0 ) ≤ - a n k n τ ≤ 0 .(7) Letting n→ ∞ in (7), we obtain that lim n → ∞ na n k = 0 .(8) From (8), there exits n₁ ∈ N such that na n k ≤ 1 for all n ≥ n₁. So, we have, for all n ≥ n₁ a n ≤ 1 n 1 / k .(9)

In order to show that {x _n } is a Cauchy sequence consider m, n ∈ N such that m > n ≥ n₁. Using the triangular inequality for the metric and from (9), we have d ( x n , x m ) ≤ d ( x n , x n + 1 ) + d ( x n + 1 , x n + 2 ) + ⋯ + d ( x m - 1 , x m ) = a n + a n + 1 + ⋯ + a m - 1 = ∑ i = n m - 1 a i ≤ ∑ i = n ∞ a i ≤ ∑ i = n ∞ 1 i 1 / k By the convergence of the series ∑ i = 1 ∞ 1 i 1 / k , passing to limit n → ∞, we get d (x _n , x _m )   →0. This yields that {x _n } is a Cauchy sequence in (X, d). Since (X, d) is a complete metric space, the sequence {x _n } converges to some point z ∈ X, that is, lim n → ∞ x n = z .

From (4), for all x, y ∈ X with D (Tx, Ty) >0, we get D ( Tx , Ty ) < d ( x , y ) and so D ( Tx , Ty ) ≤ d ( x , y ) for all x, y ∈ X. Then by Lemmas 3, 5, 6, it follows that p ( z , Tz ) ≤ d ( z , x n ) + D ( Tx n - 1 , Tz ) ≤ d ( z , x n ) + d ( x n - 1 , z ) . Passing to limit n → ∞, we obtain p (z, Tz) =0, and so from Lemma 8 we get (Tz) (z) =1, which is a contradiction. Consequently, there exists an u ∈ X such that (Tu) (u) =0, that is, T has a fixed point.

If (X, d) is a metric space and A , B ∈ W ( X ) , we define δ α ( A , B ) = sup { d ( a , b ) : a ∈ A α , b ∈ B α } , δ ( A , B ) = sup α ∈ [ 0 , 1 ] δ α ( A , B ) .

Clearly, D (A, B) ≤ δ (A, B) for all A , B ∈ W ( X ) . Hence, we immediately deduce from Theorem 1 the following.

Theorem 2. Let (X, d) be a complete metric space, F ∈ F and T be fuzzy mapping from X to W ( X ) . If there exists τ > 0 such that τ + F ( δ ( Tx , Ty ) ) ≤ F ( d ( x , y ) ) , for x, y ∈ X with δ (Tx, Ty) >0 and d (x, y) >0, then T has a fixed point.

By taking F (α) = ln α in Theorem 1, we obtain the following corollary, which is main result of Heilpern [9].

Corollary 2. Let (X, d) be a complete metric space and T be a fuzzy mapping from X to W ( X ) satisfying the following condition: there exists α ∈ (0, 1) such that D ( Tx , Ty ) ≤ α d ( x , y ) for all x, y ∈ X. Then T has a fixed point in X.

Again, by taking F (α) = α + ln α in Theorem 1, we obtain the following corollary:

Corollary 3. Let (X, d) be a complete metric space and T be a fuzzy mapping from X to W ( X ) satisfying the following condition: there exists τ > 0 D ( Tx , Ty ) d ( x , y ) e D ( Tx , Ty ) - d ( x , y ) ≤ e - τ for all x, y ∈ X with D (Tx, Ty) >0. Then T has a fixed point in X.