In this paper, taking into account the recent contractive technique, which was introduced by Wardowski [18], we present a fixed point theorem for F-contractive type fuzzy mappings over a complete metric space.
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 Aα, is defined as
where 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 Aα is compact in X for each α ∈ (0, 1] and . We denote by the family of all approximate quantities in X. When and A (x0) =1 for some x0 ∈ X, we will identify A with an approximation of x0. Let , 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 . Also, it is easy to see that if 0 ≤ r ≤ s ≤ 1, then As ⊆ Ar.
For α ∈ [0, 1], define
where H is the Pompeiu-Hausdorff distance [5], that is,
Also,
Note that, since Aα and Bα are compact sets in the metric space (X, d) for all α ∈ [0, 1], then p (A, B)< ∞ and D (A, B)< ∞ for all .
The function pα is called α-space, Dα is a α-distance, and D is a distance between A and B. We note that pα 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 that is, 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 xα of X is the fuzzy set xα : X → [0, 1] given by
For α = 1, we have
that is, {x} be a fuzzy set with membership function equal a characteristic function of set {x}. A fuzzy point xα in X is called a fixed fuzzy point of the fuzzy mapping T if xα ⊂ 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 there exists p ∈ X such thatA (p) =1.
Proof. Since then . Choose a strictly increasing (rn) n∈N in (0, 1] such that rn → 1. For each n ∈ N choose yn ∈ Arn. Since Arn ⊆ A0 for all n ∈ N, the sequence (yn) has a subsequence (yn(k)) k∈N which converges to a pointp ∈ A0. Suppose that A (p) = r < 1. Choose an m ∈ N such that rm > r. Then, (yn(k)) k∈N has a cluster point a ∈ Arm. Hence a = p. So, A (p) ≥ rm, 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 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, 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 . Then, p (A, B) = p1 (A, B).
Proof. Since Ar ⊆ As and Br ⊆ Bs whenever 0 ≤ s ≤ r ≤ 1, it follows that pr (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, 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 . Then, for each x, y ∈ X and each α ∈ [0, 1], pα (x, A) ≤ d (x, y) + pα (y, A).
Proof. For each x, y ∈ X,
Lemma 6.[6, 9] Let (X, d) be a metric space, and x ∈ A1. Then, for each and each α ∈ [0, 1], pα (x, B) ≤ Dα (A, B).
Proof. Clearly, x ∈ Aα for all α ∈ [0, 1]. Hence,
for all and α ∈ [0, 1].
Lemma 7.[6, 9] Let (X, d) be a metric space, . If x ∈ A1 (that is {x} ⊂ A) then pα (x, A) =0 for each α ∈ [0, 1].
Proof. Let x ∈ A1. Then, x ∈ Aα for all α ∈ [0, 1]. Hence, for each α ∈ [0, 1] we obtain
Lemma 8.[8] Let (X, d) be a metric space, . 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, and x ∈ X. Then, xα ⊂ A if and only if pα (x, A) =0.
Proof. If xα ⊂ A, then x ∈ Aα. Hence,
If pα (x, A) =0, then pα (x, A) = d (x, Aα) =0. Hence, we have x ∈ Aα.
Lemma 10.[6, 9] Let (X, d) be a metric space and let . If xα ⊂ A, then pα (x, B) ≤ Dα (A, B) for each
Proof. If xα ⊂ A, then x ∈ Aα. Hence, for each
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 {αn} of positive numbers
(F3) There exists k ∈ (0, 1) such that
We denote by be the set of all functions F satisfying (F1)-(F3). Some examples of the functions belonging are F1 (α) = ln α, F2 (α) = α + ln 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 and there exists τ > 0 such that for all x, y ∈ X, d (Tx, Ty) >0 implies
If we take F (α) = ln α in Definition 1, the inequality (1) turns to
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
for all x, y ∈ X, Tx ≠ Ty. In addition, Wardowski showed that every F-contraction T is a contractive mapping, i.e.,
Thus, every F-contraction is a continuous map. Also, Wardowski concluded that if 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 . If there exists τ > 0 such that
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].
Main results
Definition 2. Let (X, d) be a metric space, and T be fuzzy mapping from X to Then T is said to be a F-contractive type fuzzy mapping if there exists τ > 0 such that
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 (xn) n∈N in X such that (Txn) (xn+1) =1 and d (xn, xn+1) ≤ D1 (Txn-1, Txn). Note that, xn ≠ xn-1 for all n ∈ N. Then, we get
On the other hand, from (F1) we have
for all n ∈ N. Denote an = d (xn, xn+1) for n ∈ N. Then an > 0 for all n ∈ N and, using (5), the following holds:
From (6), we get Thus, from (F2), we have
From (F3) there exists k ∈ (0, 1) such that
By (6), the following holds for all n ∈ N
Letting n→ ∞ in (7), we obtain that
From (8), there exits n1 ∈ N such that for all n ≥ n1. So, we have, for all n ≥ n1
In order to show that {xn} 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
By the convergence of the series passing to limit n → ∞, we get d (xn, xm) →0. This yields that {xn} is a Cauchy sequence in (X, d). Since (X, d) is a complete metric space, the sequence {xn} converges to some point z ∈ X, that is,
From (4), for all x, y ∈ X with D (Tx, Ty) >0, we get
and so
for all x, y ∈ X. Then by Lemmas 3, 5, 6, it follows that
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 , we define
Clearly, D (A, B) ≤ δ (A, B) for all . Hence, we immediately deduce from Theorem 1 the following.
Theorem 2.Let (X, d) be a complete metric space, and T be fuzzy mapping from X to If there exists τ > 0 such thatfor 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 satisfying the following condition: there exists α ∈ (0, 1) such thatfor 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 satisfying the following condition: there exists τ > 0for all x, y ∈ X with D (Tx, Ty) >0. Then T has a fixed point in X.
Footnotes
Acknowledgements
This research was supported by Kirikkale University project number 2016/117, Turkey. The authors are thankful to the referees for making valuable suggestions leading to the better presentations of the paper.
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