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On φ-contractions in probabilistic and fuzzy metric spaces
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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
On ϕ-contractions in probabilistic and fuzzy metric spaces Jin-Xuan Fang School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Received 8 October 2013; received in revised form 22 June 2014; accepted 23 June 2014
Abstract In this paper, by means of weakening conditions of the gauge function ϕ, a new fixed point theorem for probabilistic ϕ-contraction in Menger probabilistic metric spaces with a t-norm of H -type is established. This theorem improves and gener´ c (2010) [4], Jachymski (2010) [12] and Xiao et al. (2013) [25]. By using the theorem, we also alizes the recent results of Ciri´ obtain some important corollaries and the fixed point theorems for ϕ-contraction in fuzzy metric spaces. © 2014 Published by Elsevier B.V. Keywords: Fixed point; Menger probabilistic metric space; Fuzzy metric space; ϕ-contraction
1. Introduction The probabilistic version of the classical Banach Contraction Principle was first studied in 1972 by Sehgal and Bharucha-Reid [21]. They proved the following theorem. Theorem 1.1. (See Sehgal and Bharucha-Reid [22].) Let (X, F, M ) be a complete Menger probabilistic metric space with t -norm M defined by M (a, b) = min{a, b}. If T : X → X is a mapping such that for every x, y ∈ X, FT x,T y (kt) ≥ Fx,y (t)
for all t > 0,
(1.1)
where k ∈ (0, 1), then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. The above mapping T : X → X satisfying condition (1.1) is usually called a probabilistic k-contraction. A natural generalization of probabilistic k-contraction is the so-called probabilistic ϕ-contraction. A mapping T : X → X is called a probabilistic ϕ-contraction (or a ϕ-contraction in probabilistic metric space) if it satisfies FT x,T y ϕ(t) ≥ Fx,y (t) for all x, y ∈ X and t > 0, (1.2) where ϕ : R+ → R+ is a gauge function satisfying certain conditions. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2014.06.013 0165-0114/© 2014 Published by Elsevier B.V.
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Fixed point theorems for probabilistic ϕ-contractions have been studied by many authors. But, many results are n (t) < ∞ for all t > 0 and some other conditions (see, e.g. obtained under the assumption that ϕ satisfies ∞ ϕ n=1 ´ c [4] has pointed out, the condition “ ∞ ϕ n (t) < ∞ for all t > 0” is very strong and [2,3,5–7,10,16,23]). As Ciri´ n=1 difficult for testing in practice. And then, a natural question arises, whether this condition can be omitted or improved. ´ c tried to solve this question. Using the introduced condition (CBW), he gives the following theorem In [4], Ciri´ (i.e., Theorem 12 in [4]). ´ c [4].) Let (X, F, ) be a complete Menger probabilistic metric space with a continuous Theorem 1.2. (See Ciri´ t -norm of H -type, and let ϕ : R+ → R+ be a function satisfying the condition (CBW): ϕ(0) = 0,
ϕ(t) < t
and
lim inf ϕ(t) < t r→t +
for all t > 0.
If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. In [4], the proof of Theorem 12 depends strongly on Lemma 10. Unfortunately, Lemma 10 in [4] is false. Jachymski [12] gives a counterexample (see Example 1 in [12]) to show that there exists a probabilistic ϕ-contraction satisfying the condition (CBW), but it has no fixed point, and so Theorem 12 is also incorrect. He provides a corrected version of Theorem 12. Theorem 1.3. (See Jachymski [12].) Let (X, F, ) be a complete Menger probabilistic metric space with a continuous t-norm of H -type, and let ϕ : R+ → R+ be a function satisfying conditions: 0 < ϕ(t) < t
and
lim ϕ n (t) = 0 for all t > 0.
n→∞
(1.3)
If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. ´ c had found an error in the proof of this Remark 1.1. Soon after the paper [12] was accepted for publication, Ciri´ Lemma 10, and pointed out that the conclusion of Lemma 10 and Theorem 12 in [4] would be correct if we modified condition (CBW) to the Boyd–Wong condition, by substituting lim sup for lim inf (see the Added in proof in [12]). It is not difficult to see that the revised Theorem 12 in this way is a special case of Theorem 1.3. Can the condition (1.3) in Theorem 1.3 be weakened further? It goes without saying that this question is worth studying. In this paper, we will give an affirmative answer to this question. This paper is organized as follows. In Section 2, we present some basic concepts and the relevant lemmas about probabilistic metric spaces and fuzzy metric spaces. In Section 3, we prove the main theorem in this paper, i.e., a new fixed point theorem for probabilistic ϕ-contraction in Menger probabilistic metric spaces, where the gauge function ϕ only needs to satisfy the following condition: for each t > 0 there exists r ≥ t such that lim ϕ n (r) = 0. n→∞
(1.4)
´ c [4], Jachymski [12] We also give several important corollaries. Our results improve and generalize the results of Ciri´ and Xiao et al. [25]. As applications of the main theorem, in Section 4, we consider the fixed point theorems for ϕ-contraction in fuzzy metric spaces. In Section 5, an example is given to illustrate the usability of our main result. 2. Preliminaries Throughout this paper, let R = (−∞, ∞), R+ = [0, ∞), and N be the set of all natural numbers. A mapping F : R → R+ is called a distribution if it is non-decreasing left-continuous with supt∈R F (t) = 1 and inft∈R F (t) = 0. The set of all distribution functions is denoted by D , and D + = {F | F ∈ D, F (0) = 0}. A special element H of D + is defined by 0, t ≤ 0 H (t) = (2.1) 1, t > 0.
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A mapping : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (for short, a t -norm) if the following conditions are satisfied: (a, 1) = a; (a, b) = (b, a); a ≥ b, c ≥ d ⇒ (a, c) ≥ (b, d); (a, (b, c)) = ((a, b), c). Definition 2.1. (See Hadžíc [10], Hadžíc and Pap [11].) A t -norm is said to be of H -type if the family of functions {m (t)}m∈N is equicontinuous at t = 1, where 1 (t) = (t, t), m (t) = t, m−1 (t) , m = 1, 2, · · · , t ∈ [0, 1] 0 (t) = t . The t -norm M = min is a trivial example of t -norm of H -type, but there are many t -norms of H -type with = M (see, e.g., [11] and [25]). The following example shows that this class is very large. Example 2.1. (See Roldán et al. [18].) Let δ ∈ (0, 1] be a real number and let be a t -norm. Define δ as δ (x, y) = (x, y), if max{x, y} ≤ 1 − δ, and δ (x, y) = min{x, y}, if max{x, y} > 1 − δ. Then δ is a t -norm of H -type. Definition 2.2. (See Menger [14], Schweizer and Sklar [19].) A triplet (X, F, ) is called a Menger probabilistic metric space (for short, Menger space) if X is a nonempty set, is a t -norm and F is a mapping from X × X into D + satisfying the following conditions (for x, y ∈ X, we denote F (x, y) by Fx,y ): (PM-1) Fx,y (t) = H (t) for all t ∈ R if and only if x = y; (PM-2) Fx,y (t) = Fy,x (t) for all t ∈ R; (PM-3) Fx,y (t + s) ≥ (Fx,z (t), Fz,y (s)) for all x, y, z ∈ X and t, s > 0. Lemma 2.1. (See Schweizer et al. [20].) Let (X, F, ) be a Menger space. If the t -norm satisfies sup0 0, λ ∈ (0, 1]} (x ∈ X), where Ux (ε, λ) = {y ∈ X | Fx,y (ε) > 1 − λ}. According to the lemma, we can introduce the following concepts in this kind of Menger spaces: a sequence {xn} T
in X is said to be T -convergent to x ∈ X (we write xn → x) if limn→∞ Fxn ,x (t) = 1 for all t > 0; {xn } is called a T -Cauchy sequence if for any given ε > 0 and λ ∈ (0, 1], there exists N = N (ε, λ) such that Fxn ,xm (ε) > 1 − λ, whenever n, m ≥ N ; (X, F, ) is said to be complete, if each T -Cauchy sequence in X is T -convergent to some point in X. Definition 2.3. (cf. Kramosil and Michálek [13].) A fuzzy metric space in the sense of Kramosil and Michálek (briefly, a KM-fuzzy metric space) is a triple (X, M, ) where X is a nonempty set, is a t -norm and M is a fuzzy set on X2 × [0, ∞) satisfying the following conditions for all x, y, z ∈ X and s, t > 0: (FM-1) (FM-2) (FM-3) (FM-4) (FM-5)
M(x, y, 0) = 0; M(x, y, t) = 1, for all t > 0 if and only if x = y; M(x, y, t) = M(y, x, t); M(x, z, t + s) ≥ (M(x, y, t), M(y, z, s)); M(x, y, ·) : R+ → [0, 1] is left continuous.
Remark 2.1. A slight difference between Definition 2.3 and the original definition in [13] is that in [13], is continuous. From (FM-4) and (FM-2), it is easy to show that M(x, y, ·) is nondecreasing for all x, y ∈ X (see [8]). So, by Definition 2.2 and Definition 2.3, it is easy to obtain the following lemma. Lemma 2.2. If (X, M, ) is a KM-fuzzy metric space satisfying the condition (FM-6) lim M(x, y, t) = 1 for all x, y ∈ X, t→∞
then (X, F, ) is a Menger space, where F is defined by
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Fx,y (t) =
M(x, y, t), 0,
t ≥ 0, t < 0.
(2.2)
On the other hand, if (X, F, ) is a Menger space, then (X, M, ) is a KM-fuzzy space with (FM-6), where M is defined by M(x, y, t) = Fx,y (t) for t ≥ 0. Definition 2.4. (See George and Veeramani [9], Mihe¸t [15].) Let (X, M, ) be a KM-fuzzy metric space. A sequence {xn }n∈N in X is said to be convergent to x ∈ X if lim M(xn , x, t) = 0 for all t > 0.
n→∞
A sequence {xn }n∈N in X is said to an M-Cauchy sequence, if for each ε ∈ (0, 1) and t > 0 there exists n0 ∈ N such that M(xn , xm , t) > 1 − ε for all m, n ≥ n0 . A fuzzy metric space is called complete if every M-Cauchy sequence is convergent in X. 3. Main results We first introduce two classes of functions. Let denote the class of all functions ϕ : R+ → R+ satisfying limn→∞ ϕ n (t) = 0 for all t > 0, and let w denote the class of all functions ϕ : R+ → R+ satisfying the condition (1.4), i.e., for each t > 0 there exists r ≥ t such that limn→∞ ϕ n (r) = 0. Obviously, the condition limn→∞ ϕ n (t) = 0 for all t > 0 implies the condition (1.4). The following example shows that the reverse is not true in general. Hence is a proper subclass of w . Example 3.1. Let the function ϕ : R+ → R+ be defined by ⎧ t , 0 ≤ t < 1, ⎪ ⎨ 1+t ϕ(t) = − 3t + 43 , 1 ≤ t ≤ 2, ⎪ ⎩ 2 < t < ∞. t − 43 ,
(3.1)
Notice that ϕ(1) = ϕ( 73 ) = 1 and limn→∞ ϕ n (1) = limn→∞ ϕ n ( 73 ) = 1 = 0. This shows that the function ϕ does not satisfy the condition limn→∞ ϕ n (t) = 0 for all t > 0. On the other hand, we can prove that ϕ satisfies the condition (1.4). t In fact, for each 0 < t < 1, by (3.1) we have limn→∞ ϕ n (t) = limn→∞ 1+nt = 0. In order to consider the case of t ≥ 1, we first show by induction that
n 4k lim φ = 0 for all k ∈ N. (3.2) n→∞ 3 Clearly, (3.2) holds for k = 1 since ϕ( 43 ) = 89 ∈ (0, 1). Assume that (3.2) holds for some k = p. Since 4(p+1) > 2, it 3 4(p+1) 4p 4(p+1) 4p n n−1 follows from (3.1) that ϕ( 3 ) = 3 , and so limn→∞ ϕ ( 3 ) = limn→∞ ϕ ( 3 ) = 0, which shows that (3.2) also holds for k = p + 1. Hence (3.2) holds for all k ∈ N. For each t ≥ 1, we can choose large enough k0 ∈ N such that r = 4k30 > t. By (3.2), we get limn→∞ ϕ n (r) = 0. This shows that the ϕ satisfies the condition (1.4). In order to obtain our main theorem, we need the following lemmas. Lemma 3.1. Let ϕ ∈ w , then for each t > 0 there exists r ≥ t such that ϕ(r) < t. Proof. Suppose that there exists some t0 > 0 such that ϕ(r) ≥ t0 for all r ≥ t0 . By induction, we obtain that ϕ n (r) ≥ t0 for all n ∈ N. This implies that limn→∞ ϕ n (r) = 0 for all r ≥ t0 , which contradicts ϕ ∈ w . 2 Lemma 3.2. Let (X, F, ) be a Menger space with a t -norm of H -type. Let {xn } be a sequence in (X, F, ). If there exists a function ϕ ∈ w such that
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(1) ϕ(t) > 0 for all t > 0; (2) Fxn ,xm (ϕ(t)) ≥ Fxn−1 ,xm−1 (t) for all n, m ∈ N and t > 0, then {xn } is a T -Cauchy sequence in X. Proof. It is evident that the condition (1) implies that ϕ n (t) > 0 for all n ∈ N and t > 0, and the condition (2) implies that Fxn ,xn+1 ϕ(t) ≥ Fxn−1 ,xn (t) for all n ∈ N and t > 0. (3.3) By induction, it follows from (3.3) that Fxn ,xn+1 ϕ n (t) ≥ Fx0 ,x1 (t) for all n ∈ N and t > 0.
(3.4)
We now prove that lim Fxn ,xn+1 (t) = 1 for all t > 0.
(3.5)
n→∞
Owing to Fx0 ,x1 (t) → 1 as t → ∞, for any ε ∈ (0, 1] there exists t1 > 0 such that Fx0 ,x1 (t1 ) > 1 − ε. Since ϕ ∈ w , there exists t0 ≥ t1 such that limn→∞ ϕ n (t0 ) = 0. Thus, for each t > 0, there exists n0 ∈ N such that ϕ n (t0 ) < t for all n ≥ n0 . By (3.4) and the monotonicity of distribution functions, we get Fxn ,xn+1 (t) ≥ Fxn ,xn+1 ϕ n (t0 ) ≥ Fx0 ,x1 (t0 ) ≥ Fx0 ,x1 (t1 ) > 1 − ε for all n ≥ n0 . Hence (3.5) holds. Since ϕ ∈ w , by Lemma 3.1, for any t > 0 there exists r ≥ t such that ϕ(r) < t. Let n ∈ N be given. We can show by induction that for any k ∈ N, (3.6) Fxn ,xn+k (t) ≥ k−1 Fxn ,xn+1 t − ϕ(r) . This is obvious for k = 1, since Fxn ,xn+1 (t) ≥ Fxn ,xn+1 (t − ϕ(r)) = 0 (Fxn ,xn+1 (t − ϕ(r))). Assume that (3.6) holds for some k. By (PM-3), the condition (2) and the monotonicity of , we have Fxn ,xn+k+1 (t) = Fxn ,xn+k+1 t − ϕ(r) + ϕ(r) ≥ Fxn ,xn+1 t − ϕ(r) , Fxn+1 ,xn+k+1 ϕ(r) ≥ Fxn ,xn+1 t − ϕ(r) , Fxn ,xn+k (r) ≥ Fxn ,xn+1 t − ϕ(r) , Fxn ,xn+k (t) ≥ Fxn ,xn+1 t − ϕ(r) , k−1 Fxn ,xn+1 t − ϕ(r) = k Fxn ,xn+1 t − ϕ(r) , which completes the induction. Hence (3.6) holds for all k ∈ N. Next, we show that {xn } is a T -Cauchy sequence. Let 0 < ε < 1. By the hypothesis, {n (t)} is equi-continuous at t = 1 and n (1) = 1, so there exists δ > 0 such that n (s) > 1 − ε
for all s ∈ (1 − δ, 1] and n ∈ N.
(3.7)
By (3.5), we know that limn→∞ Fxn ,xn+1 (t − ϕ(r)) = 1. Hence there exists n0 ∈ N such that Fxn ,xn+1 (t − ϕ(r)) > 1 − δ for all n ≥ n0 . It follows from (3.6) and (3.7) that Fxn ,xn+k (t) ≥ k−1 Fxn ,xn+1 t − ϕ(r) > 1 − ε for all n ≥ n0 and k ∈ N. This shows that {xn } is a T -Cauchy sequence.
2
Lemma 3.3. Let (X, F, ) be a Menger space and x, y ∈ X. If there exists a function ϕ ∈ w such that Fx,y ϕ(t) ≥ Fx,y (t) for all t > 0, then x = y.
(3.8)
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Proof. It is easy to show that the condition (3.8) implies that ϕ(t) > 0 for all t > 0. In fact, if there is some t0 > 0 such that ϕ(t0 ) = 0, then it follows from (3.8) that 0 = Fx,x (ϕ(t0 )) ≥ Fx,x (t0 ) = 1, which is a contradiction. Hence ϕ(t) > 0 for all t > 0, and so we have ϕ n (t) > 0 for all n ∈ N and t > 0. By induction, it follows from (3.8) that (3.8)
Fx,y ϕ n (t) ≥ Fx,y (t) for all n ∈ N and t > 0, In order to prove that x = y, we only need to prove that Fx,y (t) = 1 for all t > 0. Suppose that there exists some t0 > 0 such that Fx,y (t0 ) < 1. Since limt→∞ Fx,y (t) = 1, there exists t1 > t0 such that Fx,y (t) > Fx,y (t0 )
for all t ≥ t1 .
(3.9)
Because ϕ ∈ w , there exists t2 ≥ t1 such that limn→∞ 2 ) = 0, and so we can choose a large enough n0 ∈ N such that ϕ n0 (t2 ) < t0 . By the monotonicity of Fx,y (·), it follows from (3.8) and (3.9) that Fx,y (t0 ) ≥ Fx,y ϕ n0 (t2 ) ≥ Fx,y (t2 ) > Fx,y (t0 ), ϕ n (t
which is a contradiction. Therefore Fx,y (t) = 1 for all t > 0, i.e., x = y.
2
The following theorem is our main result. Theorem 3.1. Let (X, F, ) be a complete Menger space with a t -norm of H -type. If T : X → X is a probabilistic ϕ-contraction, i.e., it satisfies the condition (1.2), where ϕ ∈ w , then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. It is not hard to see that the condition (1.2) implies that ϕ(t) > 0 for all t > 0. In fact, if there exists some t0 > 0 such that ϕ(t0 ) = 0, by the condition (1.2), we have 0 = FT x,T x (ϕ(t0 )) ≥ Fx,x (t0 ) = 1, which is a contradiction. Let x0 ∈ X and xn = T xn−1 , n ∈ N. By (1.2), Fxn ,xm ϕ(t) = FT xn−1 ,T xm−1 ϕ(t) ≥ Fxn−1 ,xm−1 (t) for all n, m ∈ N and t > 0 So, by Lemma 3.2, we conclude that {xn } is a T -Cauchy sequence of (X, F, ). Since X is complete, we can assume that xn → x∗ ∈ X. Now, we show that x∗ is a fixed point of T . Owing to ϕ ∈ w , for each t > 0 there exists r ≥ t such that ϕ(r) < t by Lemma 3.1. Using (1.2) and monotonicity of , we get Fx∗ ,T x∗ (t) ≥ Fx∗ ,xn+1 t − ϕ(r) , FT xn ,T x∗ ϕ(r) (3.10) ≥ Fx∗ ,xn+1 t − ϕ(r) , Fxn ,x∗ (r) ≥ (an , an ), where an = min{Fx∗ ,xn+1 (t − ϕ(r)), Fxn ,x∗ (r)}. Notice that an → 1 as n → ∞ and (t, t) is continuous at t = 1. Letting n → ∞ in (3.10), we get Fx∗ ,T x∗ (t) = 1 for all t > 0. Hence T x∗ = x∗ . Finally, we show the uniqueness of fixed point. Suppose that y∗ is a fixed point of T , i.e., T y∗ = y∗ . By (1.2), we have Fx∗ ,y∗ ϕ(t) = FT x∗ ,T y∗ ϕ(t) ≥ Fx∗ ,y∗ (t) for all t > 0. From Lemma 3.3, we can conclude that x∗ = y∗ . This completes the proof.
2
Remark 3.1. Theorem 3.1 is an improvement and generalization of Theorem 1 in [12] (i.e., Theorem 1.3). Our result shows that the continuity condition of and the condition “0 < ϕ(t) < t for all t > 0” in [12, Theorem 1] are unnecessary, and the condition “limn→∞ ϕ n (t) = 0 for all t > 0” can be weakened to “for each t > 0 there exists r ≥ t such that limn→∞ ϕ n (r) = 0”. Corollary 3.1. Let (X, F, ) be a complete Menger space with a t -norm of H -type. Let T0 , T1 : X → X be two mappings such that FT0 x,T0 y ϕ(t) ≥ Fx,y (t), FT1 x,T1 y (t) ≥ Fx,y (t) for all x, y ∈ X and t > 0, (3.11) where ϕ ∈ w . If T0 is commutative with T1 , then there exists a unique common fixed point of T0 and T1 .
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Proof. We define T = T0 T1 . From (3.11), it is easy to see that T is a probabilistic ϕ-contraction, where ϕ ∈ w . By Theorem 3.1, we conclude that T has a unique fixed point z ∈ X. Since T0 is commutative with T1 , i.e., T0 T1 = T1 T0 , we have T (T0 z) = T0 T1 (T0 z) = T0 (T1 T0 z) = T0 (T z) = T0 z and T (T1 z) = T1 T0 (T1 z) = T1 (T z) = T1 z. By the uniqueness of fixed point of T , we have T0 z = T1 z = z, i.e., z is a common fixed point of T0 and T1 . It is clear that z is a unique common fixed point of T0 and T1 . This completes the proof. 2 n Remark 3.2. Notice that the condition “ ∞ n=1 ϕ (t) < ∞ for all t > 0” implies that ϕ ∈ w . Hence Corollary 3.1 is an improvement and generalization of Theorem 4.1 in [25]. In order to give another corrected version of Theorem 1.2 (i.e., Theorem 12 in [4]), we need the following definition and lemma: Definition 3.1. A function ϕ : R+ → R+ is said to be right-locally monotone if for each t ∈ [0, ∞) there exists δ > 0 such that ϕ is monotone on [t, t + δ). Lemma 3.4. Let ϕ : R+ → R+ be a right-locally monotone function satisfying the condition lim infr→t + ϕ(r) < t for all t > 0, and let {αn } be a sequence in R+ , αn > t0 > 0 and αn → t0 as n → ∞. There exists n0 ∈ N such that ϕ(αn0 ) < t0 . Proof. Because ϕ is right-locally monotone, there exists δ0 > 0 such that ϕ is monotone on [t0 , t0 + δ0 ). We consider the following two cases, respectively: Case 1: ϕ is monotonically increasing on [t0 , t0 + δ0 ). Since lim infr→t + ϕ(r) = supδ>0 infr∈(t0 ,t0 +δ) ϕ(r) < t0 , 0 there exists r ∈ (t0 , t0 + δ0 ) such that φ(r) < t0 . Since αn > t0 and αn → t0 as n → ∞, there exists n0 ∈ N such that t0 < αn0 < r, and so φ(αn0 ) ≤ φ(r) < t0 . Case 2: ϕ is monotonically decreasing on [t0 , t0 + δ0 ). Since lim infr→t + ϕ(r) = supδ>0 infr∈(t0 ,t0 +δ) ϕ(r) < t0 , for 0
each n ∈ N, there exists βn ∈ (t0 , t0 + n1 ) such that ϕ(βn ) < t0 . Note that αn , βn > t0 and αn , βn → t0 as n → ∞. We can choose n0 , m0 ∈ N such that t0 < βm0 < αn0 < t0 + δ0 , and so ϕ(αn0 ) ≤ ϕ(βm0 ) < t0 . This completes the proof. 2 Theorem 3.2. Let (X, F, ) be a complete Menger space with a t -norm of H -type, and let ϕ : R+ → R+ be a right-locally monotone function satisfying the condition (CBW). If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. We shall prove that if ϕ satisfies the conditions in the theorem, then ϕ ∈ . Since T is a probabilistic ϕ-contraction, we can prove that ϕ(t) > 0 for all t > 0 as in the proof of Theorem 3.1, and so we have ϕ n (t) > 0 for all n ∈ N and t > 0. Notice that ϕ(t) < t for all t > 0. We have 0 < ϕ n+1 (t) = ϕ(ϕ n (t)) < ϕ n (t) for all n ∈ N and t > 0. This shows that for each t > 0, the sequence {ϕ n (t)} is strictly decreasing and bounded from below, and so limn→∞ φ n (t) = α(t) ≥ 0 and ϕ n (t) > α(t) for all n ∈ N and t > 0. We need to show that α(t) ≡ 0. Suppose that there exists some t0 > 0 such that limn→∞ ϕ n (t0 ) = α(t0 ) = α0 > 0. Note that ϕ is right-locally monotone and lim infr→α + φ(r) = supδ>0 infr∈(α0 ,α0 +δ) φ(r) < α0 . Then, by Lemma 3.4, there exists n0 ∈ N such that 0
ϕ(ϕ n0 (t0 )) < α0 . So, we have α0 ≤ ϕ n0 +1 (t0 ) < α0 , which is a contradiction. Hence, limn→∞ ϕ n (t) = α(t) = 0 for all t > 0. This shows that ϕ ∈ . Note that ⊂ w . By Theorem 3.1, we conclude that the conclusion of Theorem 3.2 holds. 2 From Theorem 3.1, we also can obtain the following corollary, which is a probabilistic version of the Boyd–Wong fixed point theorem [1]. Corollary 3.2. Let (X, F, ) be a complete Menger space with a t -norm of H -type, and let ϕ : R+ → R+ be a function satisfying the following conditions:
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ϕ(t) < t
and
lim sup ϕ(r) < t r→t +
for each t > 0.
(BW)
If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. We only need to prove that the condition (BW) implies limn→∞ ϕ n (t) = 0 for all t > 0. According to the same reason as in the proof of Theorem 3.2, we know that ϕ n (t) > 0 for all n ∈ N and t > 0. Notice that ϕ(t) < t for all t > 0. It follows that 0 < ϕ n+1 (t) < ϕ n (t) for all n ∈ N and t > 0. This shows that for each t > 0, the sequence {ϕ n (t)} is strictly decreasing and bounded from below. So, for each t > 0 there exists β(t) ≥ 0 such that limn→∞ ϕ n (t) = β(t). Suppose that there is some t0 > 0, such that limn→∞ ϕ n (t0 ) = β(t0 ) = β0 > 0. By (BW), we have lim sup ϕ(r) = inf r→β0+
sup
δ>0 β0
ϕ(r) < β0 .
So, there exists δ0 > 0 such that ϕ(r) < β0 for all r ∈ (β0 , β0 + δ0 ). Because ϕ n (t0 ) > β0 for all n ∈ N and limn→∞ ϕ n (t0 ) = β0 , there exists n0 ∈ N such that β0 < ϕ n (t0 ) < β0 + δ0 for all n ≥ n0 . Thus we have β0 < ϕ n+1 (t0 ) = ϕ ϕ n (t0 ) < β0 for all n ≥ n0 , which is a contradiction. This shows that limn→∞ ϕ n (t) = 0 for all t > 0, and so ϕ ∈ w . Therefore, the conclusion of Corollary 3.2 follows from Theorem 3.1 immediately. 2 Remark 3.3. In essence, the above probabilistic version of the Boyd–Wong fixed point theorem (i.e., Corollary 3.2) ´ c (see Remark 1.1). In addition, there is a slight difference between Corollary 3.2 and Corollary 6.3 in is given by Ciri´ [24]. Corollary 3.2 shows that in Corollary 6.3 of [24], the assumption “ϕ({0}) = {0}” is not necessary and condition “lim supq→r ϕ(q) < r for all r > 0” can be weakened to “lim supq→r + ϕ(q) < r for all r > 0”. ´ c also obtained the following theorem, In [4], as a direct consequence of Theorem 1.2 (i.e., Theorem 12 in [4]), Ciri´ which is a probabilistic version of the Rakotch fixed point theorem [17]. ´ c [4].) Let (X, F, ) be a complete Menger probabilistic metric space with a continuous t -norm Theorem C. (See Ciri´ of H -type, and α : (0, ∞) → [0, 1) be a monotonically decreasing function. If a mapping T : X → X satisfies condition: FT x,T y α(t)t ≥ Fx,y (t) for all x, y ∈ X and t > 0, (CR) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Remark 3.4. In [4], the proof of the above theorem depends strongly on Theorem 1.2. Jachymski [12] has pointed out that Theorem 1.2 is incorrect. One may wonder whether Theorem C (i.e., Theorem 1.3 in [4]) is true. This problem is worth studying. Here, we will use Theorem 3.1 to prove a more general result than Theorem C, and so also give an affirmative answer to the question. To this end, we introduce the following concept: Definition 3.2. A function f on R is said to be piecewise monotone if there exists a partition of [0, ∞): 0 = x0 < x1 < · · · < xn−1 < xn = ∞ such that f is monotonic on each subinterval [xi−1 , xi ) (i = 1, · · · , n). These subintervals are called monotone subintervals of f . Obviously, a monotone function on R+ is necessarily a piecewise monotone function, however the reverse is not true in general.
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Example 3.2. Let α : R+ → [0, 1) be defined by ⎧ 0 ≤ t < 1, ⎨ x, α(t) = 1 − 12 t, 1 ≤ t ≤ 2, ⎩ 0, 2 < t < ∞. Then α(t) is a piecewise monotone function, [0, 1) and [1, ∞) are the monotone intervals of α(t), but it is not a monotone function. Theorem 3.3. Let (X, F, ) be a complete Menger space with a t -norm of H -type, and let α : R+ → [0, 1) be a piecewise monotone function. If a mapping T : X → X satisfies the condition FT x,T y α(t)t ≥ Fx,y (t) for all x, y ∈ X and t > 0, (CR) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. Set ϕ(t) = α(t)t . By (CR), it is easy to see that T is a ϕ-probabilistic contraction. So, we know that ϕ n (t) > 0 for all n ∈ N and t > 0. Next, we will prove that ϕ ∈ . Notice that ϕ n+1 (t) = ϕ ϕ n (t) = α ϕ n (t) ϕ n (t) < ϕ n (t) for all n ∈ N and t > 0, which shows that the sequence {ϕ n (t)} is strictly decreasing and bounded from below. Hence limn→∞ ϕ n (t) = s(t) ≥ 0 for all t > 0 and ϕ n (t) > s(t) for all n ∈ N and t > 0. Now we show that s(t) = 0 for all t > 0. Suppose that there is some t0 > 0 such that limn→∞ ϕ n (t0 ) = s(t0 ) = s0 > 0. Note that α(t) is a piecewise monotone function, there exist a monotone subinterval I0 = [xi−1 , xi ) of α(t) and a large enough n0 ∈ N such that s0 , ϕ n (t0 ) ∈ I0 for all n ≥ n0 . We consider respectively the following two cases: Case 1: α(t) is monotonically decreasing on I0 . In this case, ϕ n (t0 ) > s0 for all n ≥ n0 implies that α(ϕ n (t0 )) ≤ α(s0 ) for all n ≥ n0 , and so we have s0 < ϕ n+1 (t0 ) = ϕ φ n (t0 ) = α ϕ n (t0 ) ϕ n (t0 ) ≤ α(s0 )ϕ n (t0 ) for all n ≥ n0 . Letting n → ∞, we get s0 ≤ α(s0 )s0 < s0 , which is a contradiction. Case 2: α(t) is monotonically increasing on I0 . In this case, owing to that the sequence {ϕ n (t0 )} is strictly decreasing, we can infer that {α(ϕ n (t0 ))}∞ n=n0 is monotonically decreasing and bounded from below. So, there exists β ∈ [0, 1) such that limn→∞ α(ϕ n (t0 )) = β. We choose an ε > 0 satisfying β + ε < 1. Then there exists n1 ≥ n0 such that α(ϕ n (t0 )) < β + ε for all n ≥ n1 . Thus, we have s0 ≤ ϕ n+1 (t0 ) = α ϕ n (t0 ) ϕ n (t0 ) < (β + ε)ϕ n (t0 ) for all n ≥ n1 . Letting n → ∞, we get s0 ≤ (β + ε)s0 < s0 , which is a contradiction. This shows that limn→∞ ϕ n (t) = 0 for all t > 0. Hence ϕ ∈ w . Therefore, the conclusion of this theorem follows from Theorem 3.1 immediately. 2 Remark 3.5. Obviously, Theorem 3.3 is a generalization of Theorem C (i.e., Theorem 13 in [4]). This shows that the conclusion of Theorem 13 in [4] is true. As direct consequences of the above results, we can obtain the corresponding fixed point theorems in usual metric spaces. Lemma 3.5. Let (X, d) be a metric space. Define a mapping F : X × X → D + by 0, t ≤ 0 or d(x, y) > t > 0, F (x, y)(t) = Fx,y (t) = for x, y ∈ X. 1, d(x, y) ≤ t (t > 0).
(3.12)
Then (X, F, M ) is a Menger space. It is called the induced Menger space by (X, d) and it is complete iff (X, d) is complete.
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Proof. It is easy to see that the F defined by (3.12) satisfies (PM-1) and (PM-2) in Definition 2.2. We now prove that (PM-3) also holds. In fact, for any given t, s > 0, if d(x, z) > t or d(z, y) > s, then Fx,y (t + s) ≥ 0 = min{Fx,z (t), Fz,y (s)}. If d(x, z) ≤ t and d(z, y) ≤ s, then d(x, y) ≤ d(x, z) + d(z, y) ≤ t + s. By (3.12), we get Fx,y (t + s) = 1 = min{Fx,z (t), Fz,y (s)}. This shows that (PM-3) holds. Therefore (X, T , M ) is a Menger space. By (3.12), we know that for any ε > 0, d(x, y) ≤ ε iff Fx,y (ε) = 1. Thus, it is not difficult to prove that {xn } is d
T
a Cauchy sequence in (X, d) iff {xn } is a T -Cauchy sequence in (X, F, M ), and xn → x iff xn → x. Therefore (X, F, M ) is complete iff (X, d) is complete. 2 Remark 3.6. There is a slight difference between Lemma 3.5 and Lemma 2.1 in [7]. In [7, Lemma 2.1], F is defined by Fx,y (t) = H (t − d(x, y)). Corollary 3.3. Let (X, d) be a complete metric space, and let ϕ : R+ → R+ be a nondecreasing function in w and satisfy ϕ(t) > 0 for all t > 0. If a mapping T : X → X satisfies d(T x, T y) ≤ ϕ d(x, y) for all x, y ∈ X, (3.13) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. We define a mapping F : X × X → D + by (3.12). Since (X, d) is a complete metric space, by Lemma 3.5 we know that (X, F, M ) is a complete Menger space. Note that ϕ ∈ w . Next, we only need to prove that the condition (3.13) implies (1.2). For any given t > 0, if Fx,y (t) = 0, then it is evident that (1.2) holds. If Fx,y (t) = 1, then by (3.12) we know that d(x, y) ≤ t. Since ϕ is nondecreasing, it follows from (3.13) that d(T x, T y) ≤ ϕ(d(x, y)) ≤ ϕ(t). Notice that ϕ(t) > 0 for all t > 0. From (3.12), we have FT x,T y (ϕ(t)) = 1 = Fx,y (t). Hence (1.2) holds, i.e., T is a probabilistic ϕ-contraction. Therefore, the conclusion of Corollary 3.3 follows from Theorem 3.1 immediately. 2 Corollary 3.4. Let (X, d) be a complete metric space, and let α : R+ → [0, 1) be a piecewise monotone function. If a mapping T : X → X satisfies the condition d(T x, T y) ≤ α d(x, y) d(x, y) for all x, y ∈ X, (3.14) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. Since α : R+ → [0, 1) is a piecewise monotone function, there exists a partition of [0, ∞): 0 = t0 < t1 < · · · < tn−1 < tn = ∞, such that α(t) is monotonic on each subinterval Ii = [ti−1 , ti ) (i = 1, · · · , n). We define a mapping β : R+ → [0, 1) as follows: α(t), if t ∈ Ii = [ti−1 , ti ) (Ii is a monotone increasing interval of α), β(t) = α(tj −1 ), if t ∈ Ij = [tj −1 , tj ) (Ij is a monotone decreasing interval of α). Obviously, β(t) is also a piecewise monotone function and α(t) ≤ β(t) for all t > 0. So, it follows from (3.14) that d(T x, T y) ≤ β d(x, y) d(x, y) for all x, y ∈ X. (3.15) On the other hand, from the proof of Corollary 3.3, we know that (X, F, M ) is a complete Menger space, where F : X × X → D + is defined by (3.12). In addition, as in the proof of Corollary 3.3, we can prove that the condition (3.15) implies FT x,T y β(t)t ≥ Fx,y (t) for all x, y ∈ X and t > 0. This shows that T satisfies the condition (CR). Therefor, the conclusion of Corollary 3.4 follows from Theorem 3.3. 2 Remark 3.7. Corollary 3.4 is a generalization of the Rakotch fixed point theorem [17]. In Rakotch’s theorem, α : (0, ∞) → [0, 1) is a monotone decreasing function.
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4. Fixed point theorems for ϕ-contraction in fuzzy metric spaces In this section, we shall apply the results in Section 3 to obtain the corresponding fixed point theorems for ϕ-contraction in KM-fuzzy metric spaces. Lemma 4.1. Let (X, M, ) be a KM-fuzzy metric space, where the t -norm is continuous at (1, 1). Suppose that there exist x0 , x1 ∈ X such that limt →∞ M(x0 , x1 , t) = 1. Define Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}. Then (Y0 , F, ) is a Menger space, where F is defined by (2.2). If (X, M, ) is complete, then (Y0 , F, ) is a complete Menger space. Proof. It is evident that Y0 is a nonempty subset of X, It is not difficult to show that M satisfies the condition (FM-6) in Y0 , i.e., limt→∞ M(y, z, t) = 1 for all y, z ∈ Y0 . In fact, if y, z ∈ Y0 , then we have limt→∞ M(x0 , y, t) = 1 and limt→∞ M(x0 , z, t) = 1. Note that M(y, z, t) ≥ (M(y, x0 , t/2), M(x0 , z, t/2)). Letting t → ∞, by the continuity of at (1, 1), it follows that limt→∞ M(y, z, t) = 1. Thus, by Lemma 2.2, we conclude that (Y0 , F, ) is a Menger space, where F is defined by (2.2). Next, we show that (Y0 , M, ) is a closed subspace of (X, M, ). Let {yn } be a sequence in Y0 such that limn→∞ yn = y ∈ X. We need to prove that y ∈ Y0 . By the continuity of at (1, 1), for any ε ∈ (0, 1), there exists δ ∈ (0, 1) such that (1 − δ, 1 − δ) > 1 − ε. Taking t0 > 0, because limn→∞ M(yn , y, t0 /2) = 1 there exists n0 ∈ N such that M(yn0 , y, t0 /2) > 1 − δ. Note that yn0 ∈ Y0 , i.e., limt→∞ M(x0 , yn0 , t/2) = 1. Hence there exists t1 ≥ t0 such that M(x0 , yn0 , t/2) > 1 − δ for all t ≥ t1 . Thus, we have M(x0 , y, t) ≥ M(x0 , yn0 , t/2), M(yn0 , y, t/2) ≥ M(x0 , yn0 , t/2), M(yn0 , y, t0 /2) ≥ (1 − δ, 1 − δ) > 1 − ε
for all t ≥ t1 ,
which shows that limt→∞ M(x0 , y, t) = 1, i.e., y ∈ Y0 . Hence (Y0 , M, ) is a closed subspace of (X, M, ). By the completeness of (X, M, ), we infer that (Y0 , M, ) is also complete, and so (Y0 , F, ) is a complete Menger space. 2 Theorem 4.1. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type. Let T : X → X be a mapping such that M T x, T y, ϕ(t) ≥ M(x, y, t) for all x, y ∈ X and t > 0, (4.1) where ϕ ∈ w . Suppose that there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1. Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . Proof. We define a mapping F : Y0 × Y0 → D + by (2.2). Since (X, M, ) is a complete KM-fuzzy metric space and there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1, by Lemma 4.1 we know that (Y0 , F, ) is a complete Menger space. We can prove that (4.1) implies that M(T x, T y, t) ≥ M(x, y, t) for all x, y ∈ X and t > 0.
(4.2)
In fact, since ϕ ∈ w , for each t > 0 there exists r ≥ t such that ϕ(r) < t by Lemma 3.1. So, by (4.1) and the monotony of M(u, v, ·), we have M(T x, T y, t) ≥ M T x, T y, ϕ(r) ≥ M(x, y, r) ≥ M(x, y, t). Eq. (4.2) is proved. It is not difficult to prove that T is a mapping of Y0 into itself. In fact, if y ∈ Y0 , then limt→∞ M(x0 , y, t/2) = 1. By the hypothesis, limt→∞ M(x0 , T x0 , t/2) = 1. In addition, using (FM-4) and (4.2), we get M(x0 , T y, t) ≥ M(x0 , T x0 , t/2), M(T x0 , T y, t/2) ≥ M(x0 , T x0 , t/2), M(x0 , y, t/2) .
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Letting t → ∞ in the above inequality, from the continuity of at (1, 1), we obtain limt→∞ M(x0 , T y, t) = 1, i.e., T y ∈ Y0 . This shows that T is a mapping of Y0 into itself. Clearly, (4.1) implies that FT x,T y (ϕ) ≥ Fx,y (t)
for all x, y ∈ Y0 and t > 0,
where F is defined by (2.2). This shows that T is a probabilistic ϕ-contraction in (Y0 , F, ). Thus, by Theorem 3.1, we conclude that T has a unique fixed point x∗ in Y0 , and {T n (y0 )} converges to x∗ for each y0 ∈ Y . In particular, {T n (x0 )} converges to x∗ . This completes the proof. 2 Remark 4.1. From the above proof, it is easy to see that Theorem 3.1 implies Theorem 4.1. On the other hand, by Lemma 2.2 we know that if (X, F, ) is a complete Menger space, then (X, M, ) is a complete KM-fuzzy metric space with (FM-6), where M is defined by M(x, y, t) = Fx,y (t) for t ≥ 0. So, it is not difficult to prove that Theorem 4.1 implies Theorem 3.1. This shows that Theorem 4.1 is equivalent to Theorem 3.1. That is to say, Theorem 4.1 is an equivalent type of Theorem 3.1 in KM-fuzzy metric spaces. In the same way, from Theorem 3.2, Corollary 3.2 and Theorem 3.3, we can prove the following theorems, respectively. Theorem 4.2. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type, and let T : X → X be a fuzzy ϕ-contraction, i.e., it satisfies (4.1). Suppose that there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1, and ϕ : R+ → R+ is a right-locally monotone function satisfying the condition (CBW). Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . Theorem 4.3. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type, and let T : X → X be a fuzzy ϕ-contraction, i.e., it satisfies (4.1). Suppose that there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1, and ϕ : R+ → R+ is a function satisfying the following conditions: ϕ(t) < t
and
lim sup ϕ(r) < t r→t +
for each t > 0.
(BW)
Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . Theorem 4.4. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type, and let α : R+ → [0, 1) be a piecewise monotone function. Suppose that T : X → X is a mapping satisfying the condition M T x, T y, α(t)t ≥ M(x, y, t) for all x, y ∈ X and > 0, (4.3) and there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1. Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . 5. An example The following example shows that Theorem 3.1 is more general than Jachymski’s theorem (i.e. Theorem 1 in [12]). Example 5.1. Let X = [0, ∞) and define F : X × X → D + as follows: t , if |x − y| ≥ t, F (x, y)(t) = Fx,y (t) = t+|x−y| 1, if |x − y| < t. Then (X, F, M ) is a Menger space (see Example in [4]).
(5.1)
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To prove that (X, F, M ) is complete, we first notice that the metric space (X, d) is complete, where d is defined by d(x, y) = |x − y|. Next, we show that if {xn } is a T -Cauchy sequence in (X, F, M ), then {xn } is also a Cauchy sequence in (X, d). In fact, if {xn } is not a Cauchy sequence in (X, d), then there are ε0 > 0 and two sequences {ni } and {mi } such that mi > ni ≥ i and d(xmi , xni ) ≥ ε0 for all i ∈ N. Taking λ0 ∈ (0, 1/2], by (5.1) we have Fxmi ,xni (ε0 ) =
ε0 1 ≤ ≤ 1 − λ0 ε0 + |xmi − xni | 2
for all i ∈ N,
which contradicts that {xn } is a T -Cauchy sequence in (X, F, M ). Hence {xn } is also a Cauchy sequence in (X, d), by the completeness of (X, d), there exists x ∈ X such that limn→∞ xn = x, and so for any ε > 0 there exists n0 ∈ N such that |xn − x| < ε for all n ≥ n0 . Thus, for any ε > 0 and λ ∈ (0, 1], by (5.1) we have Fxn ,x (ε) = 1 > 1 − λ for all n ≥ n0 . This shows that {xn } is T -convergent to x ∈ X. Therefore, (X, F, M ) is complete. x Let T : X → X be defined by T x = 1+x , and let ϕ : R+ → R+ be defined by ⎧ t , 0 ≤ t < 1, ⎪ ⎨ 1+t t 4 (5.2) ϕ(t) = − 3 + 3 , 1 ≤ t ≤ 2, ⎪ ⎩ 4 t − 3, 2 < t < ∞. By Example 3.1, we know that ϕ ∈ w but ϕ ∈ / . We now show that T is a probabilistic ϕ-contraction, i.e., T satisfies (1.2). If |T x − T y| < ϕ(t), then we have FT x,T y (ϕ(t)) = 1 ≥ Fx,y (t), (1.2) holds. Suppose that |T x − T y| ≥ ϕ(t). From (5.2), it is easy to see that ϕ(t) ≥
t 1+t
for all t ≥ 0,
and so |T x − T y| ≥ |T x − T y| =
t 1+t .
Note that
|x − y| |x − y| |x − y| = ≤ . 1 + x + y + xy 1 + |x − y| + 2 min{x, y} + xy 1 + |x − y|
|x−y| t We get 1+t ≤ 1+|x−y| , which implies that |x − y| ≥ t since the function f (u) = t . So, By (5.1), we have Fx,y (t) = t+|x−y|
FT x,T y ϕ(t) =
ϕ(t) ≥ ϕ(t) + |T x − T y|
t 1+t |x−y| t 1+t + 1+|x−y|
≥
u 1+u
is strictly increasing on [0, ∞).
t = Fx,y (t), t + |x − y|
i.e., (1.2) holds. This shows that all the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, we conclude that T has a unique fixed point x∗ in X. Indeed, x = 0 is a unique fixed point of T . However, Theorem 1 in [12] cannot be applied to this example because the ϕ defined by (5.2) does not meet the conditions “ϕ(t) < t and limn→∞ ϕ n (t) = 0 for all t > 0”. 6. Conclusion We have proved a new fixed point theorem for probabilistic ϕ-contraction in Menger spaces with a t -norm of H -type, namely Theorem 3.1. This is a main theorem in this paper. It is worth noting that in the theorem, the contractive gauge function ϕ only needs to meet (1.4), i.e., for each t > 0 there exists r ≥ t such that lim ϕ n (r) = 0, n→∞
which is a quite weak condition. Therefore, this theorem improves and generalizes some important fixed point theo´ c [4], Jachymski [12] and Xiao et al. [25]. As consequences of the theorem, rems, including the recent results of Ciri´ a corrected version of the main theorem in [4], the probabilistic versions of Boyd–Wong’s fixed point theorem [1] and Rakotch’s fixed point theorem [17] and some corresponding fixed point theorems in general metric spaces are
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obtained. Using Theorem 3.1, we also considered the fixed point theorems for ϕ-contraction in fuzzy metric space. An equivalent type of Theorem 3.1 in KM-fuzzy metric spaces is given. Can the condition (1.4) and the reasoning methods and techniques of Theorem 3.1 be applied to generalize some other types of fixed point theorems? This is a question that we want to study in the future. Can we replace the condition (1.4) in Theorem 3.1 by a more weak condition? This question is also worthy of further investigation. References [1] D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20 (1969) 458–464. [2] S.S. Chang, Y.J. Cho, S.M. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publisher, New York, 2001. [3] Y.J. Cho, K.S. Park, S.S. Chang, Fixed point theorems in metric spaces and probabilistic metric spaces, Int. J. Math. Math. Sci. 19 (1996) 243–252. ´ c, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Anal. 72 (2010) [4] L. Ciri´ 2009–2018. [5] M.F. Duh, Y.Y. Huang, Common fixed points for mappings on complete Menger spaces, Far East J. Math. Sci. 6 (2) (1998) 169–193. [6] J.-X. Fang, A note on fixed point theorems of Hadži´c, Fuzzy Sets Syst. 48 (1992) 391–395. [7] J.-X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Anal. 71 (2009) 1833–1843. [8] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (3) (1988) 385–389. [9] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399. [10] O. Hadžíc, Fixed point theorems for multi-valued mappings in probabilistic metric spaces, Mat. Vesn. 3 (1979) 125–133. [11] O. Hadži´c, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, 2001. [12] J. Jachymski, On probabilistic ϕ-contractions on Menger spaces, Nonlinear Anal. 73 (2010) 2199–2203. [13] I. Kramosil, J. Michálek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975) 336–344. [14] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28 (1942) 535–537. [15] D. Mihe¸t, A class of contractions in fuzzy metric spaces, Fuzzy Sets Syst. 161 (2010) 1131–1137. [16] D. O’Regan, R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput. 195 (2008) 86–93. [17] E. Rakotch, A note on contractive mappings, Proc. Am. Math. Soc. 13 (1962) 459–465. [18] A. Roldán, J. Martínez-Moreno, C. Roldán, Tripled fixed point theorem in fuzzy metric spaces and applications, Fixed Point Theory Appl. 2013 (2013). Article ID 29. [19] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983. [20] B. Schweizer, A. Sklar, E. Thorp, The metrization of statistical metric spaces, Pac. J. Math. 10 (1960) 673–675. [21] V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings on PM-spaces, Math. Syst. Theory 6 (1972) 97–102. [22] J.Z. Xiao, X.H. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst. 125 (2002) 153–161. [23] J.Z. Xiao, X.H. Zhu, Y.F. Cao, Common coupled fixed point results for probabilistic φ-contractions in Menger spaces, Nonlinear Anal., Theory Methods Appl. 74 (2011) 4589–4600. [24] J.Z. Xiao, X.H. Zhu, X. Jin, Fixed point theorems for nonlinear contractions in Kaleva–Seikkala’s type fuzzy metric spaces, Fuzzy Sets Syst. 200 (2012) 65–83. [25] J.Z. Xiao, X.H. Zhu, X.Y. Liu, An alternative characterization of probabilistic Menger spaces with H -type triangular norms, Fuzzy Sets Syst. 227 (2013) 107–114.
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Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
On ϕ-contractions in probabilistic and fuzzy metric spaces Jin-Xuan Fang School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, China Received 8 October 2013; received in revised form 22 June 2014; accepted 23 June 2014
Abstract In this paper, by means of weakening conditions of the gauge function ϕ, a new fixed point theorem for probabilistic ϕ-contraction in Menger probabilistic metric spaces with a t-norm of H -type is established. This theorem improves and gener´ c (2010) [4], Jachymski (2010) [12] and Xiao et al. (2013) [25]. By using the theorem, we also alizes the recent results of Ciri´ obtain some important corollaries and the fixed point theorems for ϕ-contraction in fuzzy metric spaces. © 2014 Published by Elsevier B.V. Keywords: Fixed point; Menger probabilistic metric space; Fuzzy metric space; ϕ-contraction
1. Introduction The probabilistic version of the classical Banach Contraction Principle was first studied in 1972 by Sehgal and Bharucha-Reid [21]. They proved the following theorem. Theorem 1.1. (See Sehgal and Bharucha-Reid [22].) Let (X, F, M ) be a complete Menger probabilistic metric space with t -norm M defined by M (a, b) = min{a, b}. If T : X → X is a mapping such that for every x, y ∈ X, FT x,T y (kt) ≥ Fx,y (t)
for all t > 0,
(1.1)
where k ∈ (0, 1), then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. The above mapping T : X → X satisfying condition (1.1) is usually called a probabilistic k-contraction. A natural generalization of probabilistic k-contraction is the so-called probabilistic ϕ-contraction. A mapping T : X → X is called a probabilistic ϕ-contraction (or a ϕ-contraction in probabilistic metric space) if it satisfies FT x,T y ϕ(t) ≥ Fx,y (t) for all x, y ∈ X and t > 0, (1.2) where ϕ : R+ → R+ is a gauge function satisfying certain conditions. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.fss.2014.06.013 0165-0114/© 2014 Published by Elsevier B.V.
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Fixed point theorems for probabilistic ϕ-contractions have been studied by many authors. But, many results are n (t) < ∞ for all t > 0 and some other conditions (see, e.g. obtained under the assumption that ϕ satisfies ∞ ϕ n=1 ´ c [4] has pointed out, the condition “ ∞ ϕ n (t) < ∞ for all t > 0” is very strong and [2,3,5–7,10,16,23]). As Ciri´ n=1 difficult for testing in practice. And then, a natural question arises, whether this condition can be omitted or improved. ´ c tried to solve this question. Using the introduced condition (CBW), he gives the following theorem In [4], Ciri´ (i.e., Theorem 12 in [4]). ´ c [4].) Let (X, F, ) be a complete Menger probabilistic metric space with a continuous Theorem 1.2. (See Ciri´ t -norm of H -type, and let ϕ : R+ → R+ be a function satisfying the condition (CBW): ϕ(0) = 0,
ϕ(t) < t
and
lim inf ϕ(t) < t r→t +
for all t > 0.
If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. In [4], the proof of Theorem 12 depends strongly on Lemma 10. Unfortunately, Lemma 10 in [4] is false. Jachymski [12] gives a counterexample (see Example 1 in [12]) to show that there exists a probabilistic ϕ-contraction satisfying the condition (CBW), but it has no fixed point, and so Theorem 12 is also incorrect. He provides a corrected version of Theorem 12. Theorem 1.3. (See Jachymski [12].) Let (X, F, ) be a complete Menger probabilistic metric space with a continuous t-norm of H -type, and let ϕ : R+ → R+ be a function satisfying conditions: 0 < ϕ(t) < t
and
lim ϕ n (t) = 0 for all t > 0.
n→∞
(1.3)
If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. ´ c had found an error in the proof of this Remark 1.1. Soon after the paper [12] was accepted for publication, Ciri´ Lemma 10, and pointed out that the conclusion of Lemma 10 and Theorem 12 in [4] would be correct if we modified condition (CBW) to the Boyd–Wong condition, by substituting lim sup for lim inf (see the Added in proof in [12]). It is not difficult to see that the revised Theorem 12 in this way is a special case of Theorem 1.3. Can the condition (1.3) in Theorem 1.3 be weakened further? It goes without saying that this question is worth studying. In this paper, we will give an affirmative answer to this question. This paper is organized as follows. In Section 2, we present some basic concepts and the relevant lemmas about probabilistic metric spaces and fuzzy metric spaces. In Section 3, we prove the main theorem in this paper, i.e., a new fixed point theorem for probabilistic ϕ-contraction in Menger probabilistic metric spaces, where the gauge function ϕ only needs to satisfy the following condition: for each t > 0 there exists r ≥ t such that lim ϕ n (r) = 0. n→∞
(1.4)
´ c [4], Jachymski [12] We also give several important corollaries. Our results improve and generalize the results of Ciri´ and Xiao et al. [25]. As applications of the main theorem, in Section 4, we consider the fixed point theorems for ϕ-contraction in fuzzy metric spaces. In Section 5, an example is given to illustrate the usability of our main result. 2. Preliminaries Throughout this paper, let R = (−∞, ∞), R+ = [0, ∞), and N be the set of all natural numbers. A mapping F : R → R+ is called a distribution if it is non-decreasing left-continuous with supt∈R F (t) = 1 and inft∈R F (t) = 0. The set of all distribution functions is denoted by D , and D + = {F | F ∈ D, F (0) = 0}. A special element H of D + is defined by 0, t ≤ 0 H (t) = (2.1) 1, t > 0.
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A mapping : [0, 1] × [0, 1] → [0, 1] is called a triangular norm (for short, a t -norm) if the following conditions are satisfied: (a, 1) = a; (a, b) = (b, a); a ≥ b, c ≥ d ⇒ (a, c) ≥ (b, d); (a, (b, c)) = ((a, b), c). Definition 2.1. (See Hadžíc [10], Hadžíc and Pap [11].) A t -norm is said to be of H -type if the family of functions {m (t)}m∈N is equicontinuous at t = 1, where 1 (t) = (t, t), m (t) = t, m−1 (t) , m = 1, 2, · · · , t ∈ [0, 1] 0 (t) = t . The t -norm M = min is a trivial example of t -norm of H -type, but there are many t -norms of H -type with = M (see, e.g., [11] and [25]). The following example shows that this class is very large. Example 2.1. (See Roldán et al. [18].) Let δ ∈ (0, 1] be a real number and let be a t -norm. Define δ as δ (x, y) = (x, y), if max{x, y} ≤ 1 − δ, and δ (x, y) = min{x, y}, if max{x, y} > 1 − δ. Then δ is a t -norm of H -type. Definition 2.2. (See Menger [14], Schweizer and Sklar [19].) A triplet (X, F, ) is called a Menger probabilistic metric space (for short, Menger space) if X is a nonempty set, is a t -norm and F is a mapping from X × X into D + satisfying the following conditions (for x, y ∈ X, we denote F (x, y) by Fx,y ): (PM-1) Fx,y (t) = H (t) for all t ∈ R if and only if x = y; (PM-2) Fx,y (t) = Fy,x (t) for all t ∈ R; (PM-3) Fx,y (t + s) ≥ (Fx,z (t), Fz,y (s)) for all x, y, z ∈ X and t, s > 0. Lemma 2.1. (See Schweizer et al. [20].) Let (X, F, ) be a Menger space. If the t -norm satisfies sup0
in X is said to be T -convergent to x ∈ X (we write xn → x) if limn→∞ Fxn ,x (t) = 1 for all t > 0; {xn } is called a T -Cauchy sequence if for any given ε > 0 and λ ∈ (0, 1], there exists N = N (ε, λ) such that Fxn ,xm (ε) > 1 − λ, whenever n, m ≥ N ; (X, F, ) is said to be complete, if each T -Cauchy sequence in X is T -convergent to some point in X. Definition 2.3. (cf. Kramosil and Michálek [13].) A fuzzy metric space in the sense of Kramosil and Michálek (briefly, a KM-fuzzy metric space) is a triple (X, M, ) where X is a nonempty set, is a t -norm and M is a fuzzy set on X2 × [0, ∞) satisfying the following conditions for all x, y, z ∈ X and s, t > 0: (FM-1) (FM-2) (FM-3) (FM-4) (FM-5)
M(x, y, 0) = 0; M(x, y, t) = 1, for all t > 0 if and only if x = y; M(x, y, t) = M(y, x, t); M(x, z, t + s) ≥ (M(x, y, t), M(y, z, s)); M(x, y, ·) : R+ → [0, 1] is left continuous.
Remark 2.1. A slight difference between Definition 2.3 and the original definition in [13] is that in [13], is continuous. From (FM-4) and (FM-2), it is easy to show that M(x, y, ·) is nondecreasing for all x, y ∈ X (see [8]). So, by Definition 2.2 and Definition 2.3, it is easy to obtain the following lemma. Lemma 2.2. If (X, M, ) is a KM-fuzzy metric space satisfying the condition (FM-6) lim M(x, y, t) = 1 for all x, y ∈ X, t→∞
then (X, F, ) is a Menger space, where F is defined by
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Fx,y (t) =
M(x, y, t), 0,
t ≥ 0, t < 0.
(2.2)
On the other hand, if (X, F, ) is a Menger space, then (X, M, ) is a KM-fuzzy space with (FM-6), where M is defined by M(x, y, t) = Fx,y (t) for t ≥ 0. Definition 2.4. (See George and Veeramani [9], Mihe¸t [15].) Let (X, M, ) be a KM-fuzzy metric space. A sequence {xn }n∈N in X is said to be convergent to x ∈ X if lim M(xn , x, t) = 0 for all t > 0.
n→∞
A sequence {xn }n∈N in X is said to an M-Cauchy sequence, if for each ε ∈ (0, 1) and t > 0 there exists n0 ∈ N such that M(xn , xm , t) > 1 − ε for all m, n ≥ n0 . A fuzzy metric space is called complete if every M-Cauchy sequence is convergent in X. 3. Main results We first introduce two classes of functions. Let denote the class of all functions ϕ : R+ → R+ satisfying limn→∞ ϕ n (t) = 0 for all t > 0, and let w denote the class of all functions ϕ : R+ → R+ satisfying the condition (1.4), i.e., for each t > 0 there exists r ≥ t such that limn→∞ ϕ n (r) = 0. Obviously, the condition limn→∞ ϕ n (t) = 0 for all t > 0 implies the condition (1.4). The following example shows that the reverse is not true in general. Hence is a proper subclass of w . Example 3.1. Let the function ϕ : R+ → R+ be defined by ⎧ t , 0 ≤ t < 1, ⎪ ⎨ 1+t ϕ(t) = − 3t + 43 , 1 ≤ t ≤ 2, ⎪ ⎩ 2 < t < ∞. t − 43 ,
(3.1)
Notice that ϕ(1) = ϕ( 73 ) = 1 and limn→∞ ϕ n (1) = limn→∞ ϕ n ( 73 ) = 1 = 0. This shows that the function ϕ does not satisfy the condition limn→∞ ϕ n (t) = 0 for all t > 0. On the other hand, we can prove that ϕ satisfies the condition (1.4). t In fact, for each 0 < t < 1, by (3.1) we have limn→∞ ϕ n (t) = limn→∞ 1+nt = 0. In order to consider the case of t ≥ 1, we first show by induction that
n 4k lim φ = 0 for all k ∈ N. (3.2) n→∞ 3 Clearly, (3.2) holds for k = 1 since ϕ( 43 ) = 89 ∈ (0, 1). Assume that (3.2) holds for some k = p. Since 4(p+1) > 2, it 3 4(p+1) 4p 4(p+1) 4p n n−1 follows from (3.1) that ϕ( 3 ) = 3 , and so limn→∞ ϕ ( 3 ) = limn→∞ ϕ ( 3 ) = 0, which shows that (3.2) also holds for k = p + 1. Hence (3.2) holds for all k ∈ N. For each t ≥ 1, we can choose large enough k0 ∈ N such that r = 4k30 > t. By (3.2), we get limn→∞ ϕ n (r) = 0. This shows that the ϕ satisfies the condition (1.4). In order to obtain our main theorem, we need the following lemmas. Lemma 3.1. Let ϕ ∈ w , then for each t > 0 there exists r ≥ t such that ϕ(r) < t. Proof. Suppose that there exists some t0 > 0 such that ϕ(r) ≥ t0 for all r ≥ t0 . By induction, we obtain that ϕ n (r) ≥ t0 for all n ∈ N. This implies that limn→∞ ϕ n (r) = 0 for all r ≥ t0 , which contradicts ϕ ∈ w . 2 Lemma 3.2. Let (X, F, ) be a Menger space with a t -norm of H -type. Let {xn } be a sequence in (X, F, ). If there exists a function ϕ ∈ w such that
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(1) ϕ(t) > 0 for all t > 0; (2) Fxn ,xm (ϕ(t)) ≥ Fxn−1 ,xm−1 (t) for all n, m ∈ N and t > 0, then {xn } is a T -Cauchy sequence in X. Proof. It is evident that the condition (1) implies that ϕ n (t) > 0 for all n ∈ N and t > 0, and the condition (2) implies that Fxn ,xn+1 ϕ(t) ≥ Fxn−1 ,xn (t) for all n ∈ N and t > 0. (3.3) By induction, it follows from (3.3) that Fxn ,xn+1 ϕ n (t) ≥ Fx0 ,x1 (t) for all n ∈ N and t > 0.
(3.4)
We now prove that lim Fxn ,xn+1 (t) = 1 for all t > 0.
(3.5)
n→∞
Owing to Fx0 ,x1 (t) → 1 as t → ∞, for any ε ∈ (0, 1] there exists t1 > 0 such that Fx0 ,x1 (t1 ) > 1 − ε. Since ϕ ∈ w , there exists t0 ≥ t1 such that limn→∞ ϕ n (t0 ) = 0. Thus, for each t > 0, there exists n0 ∈ N such that ϕ n (t0 ) < t for all n ≥ n0 . By (3.4) and the monotonicity of distribution functions, we get Fxn ,xn+1 (t) ≥ Fxn ,xn+1 ϕ n (t0 ) ≥ Fx0 ,x1 (t0 ) ≥ Fx0 ,x1 (t1 ) > 1 − ε for all n ≥ n0 . Hence (3.5) holds. Since ϕ ∈ w , by Lemma 3.1, for any t > 0 there exists r ≥ t such that ϕ(r) < t. Let n ∈ N be given. We can show by induction that for any k ∈ N, (3.6) Fxn ,xn+k (t) ≥ k−1 Fxn ,xn+1 t − ϕ(r) . This is obvious for k = 1, since Fxn ,xn+1 (t) ≥ Fxn ,xn+1 (t − ϕ(r)) = 0 (Fxn ,xn+1 (t − ϕ(r))). Assume that (3.6) holds for some k. By (PM-3), the condition (2) and the monotonicity of , we have Fxn ,xn+k+1 (t) = Fxn ,xn+k+1 t − ϕ(r) + ϕ(r) ≥ Fxn ,xn+1 t − ϕ(r) , Fxn+1 ,xn+k+1 ϕ(r) ≥ Fxn ,xn+1 t − ϕ(r) , Fxn ,xn+k (r) ≥ Fxn ,xn+1 t − ϕ(r) , Fxn ,xn+k (t) ≥ Fxn ,xn+1 t − ϕ(r) , k−1 Fxn ,xn+1 t − ϕ(r) = k Fxn ,xn+1 t − ϕ(r) , which completes the induction. Hence (3.6) holds for all k ∈ N. Next, we show that {xn } is a T -Cauchy sequence. Let 0 < ε < 1. By the hypothesis, {n (t)} is equi-continuous at t = 1 and n (1) = 1, so there exists δ > 0 such that n (s) > 1 − ε
for all s ∈ (1 − δ, 1] and n ∈ N.
(3.7)
By (3.5), we know that limn→∞ Fxn ,xn+1 (t − ϕ(r)) = 1. Hence there exists n0 ∈ N such that Fxn ,xn+1 (t − ϕ(r)) > 1 − δ for all n ≥ n0 . It follows from (3.6) and (3.7) that Fxn ,xn+k (t) ≥ k−1 Fxn ,xn+1 t − ϕ(r) > 1 − ε for all n ≥ n0 and k ∈ N. This shows that {xn } is a T -Cauchy sequence.
2
Lemma 3.3. Let (X, F, ) be a Menger space and x, y ∈ X. If there exists a function ϕ ∈ w such that Fx,y ϕ(t) ≥ Fx,y (t) for all t > 0, then x = y.
(3.8)
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Proof. It is easy to show that the condition (3.8) implies that ϕ(t) > 0 for all t > 0. In fact, if there is some t0 > 0 such that ϕ(t0 ) = 0, then it follows from (3.8) that 0 = Fx,x (ϕ(t0 )) ≥ Fx,x (t0 ) = 1, which is a contradiction. Hence ϕ(t) > 0 for all t > 0, and so we have ϕ n (t) > 0 for all n ∈ N and t > 0. By induction, it follows from (3.8) that (3.8)
Fx,y ϕ n (t) ≥ Fx,y (t) for all n ∈ N and t > 0, In order to prove that x = y, we only need to prove that Fx,y (t) = 1 for all t > 0. Suppose that there exists some t0 > 0 such that Fx,y (t0 ) < 1. Since limt→∞ Fx,y (t) = 1, there exists t1 > t0 such that Fx,y (t) > Fx,y (t0 )
for all t ≥ t1 .
(3.9)
Because ϕ ∈ w , there exists t2 ≥ t1 such that limn→∞ 2 ) = 0, and so we can choose a large enough n0 ∈ N such that ϕ n0 (t2 ) < t0 . By the monotonicity of Fx,y (·), it follows from (3.8) and (3.9) that Fx,y (t0 ) ≥ Fx,y ϕ n0 (t2 ) ≥ Fx,y (t2 ) > Fx,y (t0 ), ϕ n (t
which is a contradiction. Therefore Fx,y (t) = 1 for all t > 0, i.e., x = y.
2
The following theorem is our main result. Theorem 3.1. Let (X, F, ) be a complete Menger space with a t -norm of H -type. If T : X → X is a probabilistic ϕ-contraction, i.e., it satisfies the condition (1.2), where ϕ ∈ w , then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. It is not hard to see that the condition (1.2) implies that ϕ(t) > 0 for all t > 0. In fact, if there exists some t0 > 0 such that ϕ(t0 ) = 0, by the condition (1.2), we have 0 = FT x,T x (ϕ(t0 )) ≥ Fx,x (t0 ) = 1, which is a contradiction. Let x0 ∈ X and xn = T xn−1 , n ∈ N. By (1.2), Fxn ,xm ϕ(t) = FT xn−1 ,T xm−1 ϕ(t) ≥ Fxn−1 ,xm−1 (t) for all n, m ∈ N and t > 0 So, by Lemma 3.2, we conclude that {xn } is a T -Cauchy sequence of (X, F, ). Since X is complete, we can assume that xn → x∗ ∈ X. Now, we show that x∗ is a fixed point of T . Owing to ϕ ∈ w , for each t > 0 there exists r ≥ t such that ϕ(r) < t by Lemma 3.1. Using (1.2) and monotonicity of , we get Fx∗ ,T x∗ (t) ≥ Fx∗ ,xn+1 t − ϕ(r) , FT xn ,T x∗ ϕ(r) (3.10) ≥ Fx∗ ,xn+1 t − ϕ(r) , Fxn ,x∗ (r) ≥ (an , an ), where an = min{Fx∗ ,xn+1 (t − ϕ(r)), Fxn ,x∗ (r)}. Notice that an → 1 as n → ∞ and (t, t) is continuous at t = 1. Letting n → ∞ in (3.10), we get Fx∗ ,T x∗ (t) = 1 for all t > 0. Hence T x∗ = x∗ . Finally, we show the uniqueness of fixed point. Suppose that y∗ is a fixed point of T , i.e., T y∗ = y∗ . By (1.2), we have Fx∗ ,y∗ ϕ(t) = FT x∗ ,T y∗ ϕ(t) ≥ Fx∗ ,y∗ (t) for all t > 0. From Lemma 3.3, we can conclude that x∗ = y∗ . This completes the proof.
2
Remark 3.1. Theorem 3.1 is an improvement and generalization of Theorem 1 in [12] (i.e., Theorem 1.3). Our result shows that the continuity condition of and the condition “0 < ϕ(t) < t for all t > 0” in [12, Theorem 1] are unnecessary, and the condition “limn→∞ ϕ n (t) = 0 for all t > 0” can be weakened to “for each t > 0 there exists r ≥ t such that limn→∞ ϕ n (r) = 0”. Corollary 3.1. Let (X, F, ) be a complete Menger space with a t -norm of H -type. Let T0 , T1 : X → X be two mappings such that FT0 x,T0 y ϕ(t) ≥ Fx,y (t), FT1 x,T1 y (t) ≥ Fx,y (t) for all x, y ∈ X and t > 0, (3.11) where ϕ ∈ w . If T0 is commutative with T1 , then there exists a unique common fixed point of T0 and T1 .
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Proof. We define T = T0 T1 . From (3.11), it is easy to see that T is a probabilistic ϕ-contraction, where ϕ ∈ w . By Theorem 3.1, we conclude that T has a unique fixed point z ∈ X. Since T0 is commutative with T1 , i.e., T0 T1 = T1 T0 , we have T (T0 z) = T0 T1 (T0 z) = T0 (T1 T0 z) = T0 (T z) = T0 z and T (T1 z) = T1 T0 (T1 z) = T1 (T z) = T1 z. By the uniqueness of fixed point of T , we have T0 z = T1 z = z, i.e., z is a common fixed point of T0 and T1 . It is clear that z is a unique common fixed point of T0 and T1 . This completes the proof. 2 n Remark 3.2. Notice that the condition “ ∞ n=1 ϕ (t) < ∞ for all t > 0” implies that ϕ ∈ w . Hence Corollary 3.1 is an improvement and generalization of Theorem 4.1 in [25]. In order to give another corrected version of Theorem 1.2 (i.e., Theorem 12 in [4]), we need the following definition and lemma: Definition 3.1. A function ϕ : R+ → R+ is said to be right-locally monotone if for each t ∈ [0, ∞) there exists δ > 0 such that ϕ is monotone on [t, t + δ). Lemma 3.4. Let ϕ : R+ → R+ be a right-locally monotone function satisfying the condition lim infr→t + ϕ(r) < t for all t > 0, and let {αn } be a sequence in R+ , αn > t0 > 0 and αn → t0 as n → ∞. There exists n0 ∈ N such that ϕ(αn0 ) < t0 . Proof. Because ϕ is right-locally monotone, there exists δ0 > 0 such that ϕ is monotone on [t0 , t0 + δ0 ). We consider the following two cases, respectively: Case 1: ϕ is monotonically increasing on [t0 , t0 + δ0 ). Since lim infr→t + ϕ(r) = supδ>0 infr∈(t0 ,t0 +δ) ϕ(r) < t0 , 0 there exists r ∈ (t0 , t0 + δ0 ) such that φ(r) < t0 . Since αn > t0 and αn → t0 as n → ∞, there exists n0 ∈ N such that t0 < αn0 < r, and so φ(αn0 ) ≤ φ(r) < t0 . Case 2: ϕ is monotonically decreasing on [t0 , t0 + δ0 ). Since lim infr→t + ϕ(r) = supδ>0 infr∈(t0 ,t0 +δ) ϕ(r) < t0 , for 0
each n ∈ N, there exists βn ∈ (t0 , t0 + n1 ) such that ϕ(βn ) < t0 . Note that αn , βn > t0 and αn , βn → t0 as n → ∞. We can choose n0 , m0 ∈ N such that t0 < βm0 < αn0 < t0 + δ0 , and so ϕ(αn0 ) ≤ ϕ(βm0 ) < t0 . This completes the proof. 2 Theorem 3.2. Let (X, F, ) be a complete Menger space with a t -norm of H -type, and let ϕ : R+ → R+ be a right-locally monotone function satisfying the condition (CBW). If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. We shall prove that if ϕ satisfies the conditions in the theorem, then ϕ ∈ . Since T is a probabilistic ϕ-contraction, we can prove that ϕ(t) > 0 for all t > 0 as in the proof of Theorem 3.1, and so we have ϕ n (t) > 0 for all n ∈ N and t > 0. Notice that ϕ(t) < t for all t > 0. We have 0 < ϕ n+1 (t) = ϕ(ϕ n (t)) < ϕ n (t) for all n ∈ N and t > 0. This shows that for each t > 0, the sequence {ϕ n (t)} is strictly decreasing and bounded from below, and so limn→∞ φ n (t) = α(t) ≥ 0 and ϕ n (t) > α(t) for all n ∈ N and t > 0. We need to show that α(t) ≡ 0. Suppose that there exists some t0 > 0 such that limn→∞ ϕ n (t0 ) = α(t0 ) = α0 > 0. Note that ϕ is right-locally monotone and lim infr→α + φ(r) = supδ>0 infr∈(α0 ,α0 +δ) φ(r) < α0 . Then, by Lemma 3.4, there exists n0 ∈ N such that 0
ϕ(ϕ n0 (t0 )) < α0 . So, we have α0 ≤ ϕ n0 +1 (t0 ) < α0 , which is a contradiction. Hence, limn→∞ ϕ n (t) = α(t) = 0 for all t > 0. This shows that ϕ ∈ . Note that ⊂ w . By Theorem 3.1, we conclude that the conclusion of Theorem 3.2 holds. 2 From Theorem 3.1, we also can obtain the following corollary, which is a probabilistic version of the Boyd–Wong fixed point theorem [1]. Corollary 3.2. Let (X, F, ) be a complete Menger space with a t -norm of H -type, and let ϕ : R+ → R+ be a function satisfying the following conditions:
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ϕ(t) < t
and
lim sup ϕ(r) < t r→t +
for each t > 0.
(BW)
If T : X → X is a probabilistic ϕ-contraction, then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. We only need to prove that the condition (BW) implies limn→∞ ϕ n (t) = 0 for all t > 0. According to the same reason as in the proof of Theorem 3.2, we know that ϕ n (t) > 0 for all n ∈ N and t > 0. Notice that ϕ(t) < t for all t > 0. It follows that 0 < ϕ n+1 (t) < ϕ n (t) for all n ∈ N and t > 0. This shows that for each t > 0, the sequence {ϕ n (t)} is strictly decreasing and bounded from below. So, for each t > 0 there exists β(t) ≥ 0 such that limn→∞ ϕ n (t) = β(t). Suppose that there is some t0 > 0, such that limn→∞ ϕ n (t0 ) = β(t0 ) = β0 > 0. By (BW), we have lim sup ϕ(r) = inf r→β0+
sup
δ>0 β0
ϕ(r) < β0 .
So, there exists δ0 > 0 such that ϕ(r) < β0 for all r ∈ (β0 , β0 + δ0 ). Because ϕ n (t0 ) > β0 for all n ∈ N and limn→∞ ϕ n (t0 ) = β0 , there exists n0 ∈ N such that β0 < ϕ n (t0 ) < β0 + δ0 for all n ≥ n0 . Thus we have β0 < ϕ n+1 (t0 ) = ϕ ϕ n (t0 ) < β0 for all n ≥ n0 , which is a contradiction. This shows that limn→∞ ϕ n (t) = 0 for all t > 0, and so ϕ ∈ w . Therefore, the conclusion of Corollary 3.2 follows from Theorem 3.1 immediately. 2 Remark 3.3. In essence, the above probabilistic version of the Boyd–Wong fixed point theorem (i.e., Corollary 3.2) ´ c (see Remark 1.1). In addition, there is a slight difference between Corollary 3.2 and Corollary 6.3 in is given by Ciri´ [24]. Corollary 3.2 shows that in Corollary 6.3 of [24], the assumption “ϕ({0}) = {0}” is not necessary and condition “lim supq→r ϕ(q) < r for all r > 0” can be weakened to “lim supq→r + ϕ(q) < r for all r > 0”. ´ c also obtained the following theorem, In [4], as a direct consequence of Theorem 1.2 (i.e., Theorem 12 in [4]), Ciri´ which is a probabilistic version of the Rakotch fixed point theorem [17]. ´ c [4].) Let (X, F, ) be a complete Menger probabilistic metric space with a continuous t -norm Theorem C. (See Ciri´ of H -type, and α : (0, ∞) → [0, 1) be a monotonically decreasing function. If a mapping T : X → X satisfies condition: FT x,T y α(t)t ≥ Fx,y (t) for all x, y ∈ X and t > 0, (CR) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Remark 3.4. In [4], the proof of the above theorem depends strongly on Theorem 1.2. Jachymski [12] has pointed out that Theorem 1.2 is incorrect. One may wonder whether Theorem C (i.e., Theorem 1.3 in [4]) is true. This problem is worth studying. Here, we will use Theorem 3.1 to prove a more general result than Theorem C, and so also give an affirmative answer to the question. To this end, we introduce the following concept: Definition 3.2. A function f on R is said to be piecewise monotone if there exists a partition of [0, ∞): 0 = x0 < x1 < · · · < xn−1 < xn = ∞ such that f is monotonic on each subinterval [xi−1 , xi ) (i = 1, · · · , n). These subintervals are called monotone subintervals of f . Obviously, a monotone function on R+ is necessarily a piecewise monotone function, however the reverse is not true in general.
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Example 3.2. Let α : R+ → [0, 1) be defined by ⎧ 0 ≤ t < 1, ⎨ x, α(t) = 1 − 12 t, 1 ≤ t ≤ 2, ⎩ 0, 2 < t < ∞. Then α(t) is a piecewise monotone function, [0, 1) and [1, ∞) are the monotone intervals of α(t), but it is not a monotone function. Theorem 3.3. Let (X, F, ) be a complete Menger space with a t -norm of H -type, and let α : R+ → [0, 1) be a piecewise monotone function. If a mapping T : X → X satisfies the condition FT x,T y α(t)t ≥ Fx,y (t) for all x, y ∈ X and t > 0, (CR) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. Set ϕ(t) = α(t)t . By (CR), it is easy to see that T is a ϕ-probabilistic contraction. So, we know that ϕ n (t) > 0 for all n ∈ N and t > 0. Next, we will prove that ϕ ∈ . Notice that ϕ n+1 (t) = ϕ ϕ n (t) = α ϕ n (t) ϕ n (t) < ϕ n (t) for all n ∈ N and t > 0, which shows that the sequence {ϕ n (t)} is strictly decreasing and bounded from below. Hence limn→∞ ϕ n (t) = s(t) ≥ 0 for all t > 0 and ϕ n (t) > s(t) for all n ∈ N and t > 0. Now we show that s(t) = 0 for all t > 0. Suppose that there is some t0 > 0 such that limn→∞ ϕ n (t0 ) = s(t0 ) = s0 > 0. Note that α(t) is a piecewise monotone function, there exist a monotone subinterval I0 = [xi−1 , xi ) of α(t) and a large enough n0 ∈ N such that s0 , ϕ n (t0 ) ∈ I0 for all n ≥ n0 . We consider respectively the following two cases: Case 1: α(t) is monotonically decreasing on I0 . In this case, ϕ n (t0 ) > s0 for all n ≥ n0 implies that α(ϕ n (t0 )) ≤ α(s0 ) for all n ≥ n0 , and so we have s0 < ϕ n+1 (t0 ) = ϕ φ n (t0 ) = α ϕ n (t0 ) ϕ n (t0 ) ≤ α(s0 )ϕ n (t0 ) for all n ≥ n0 . Letting n → ∞, we get s0 ≤ α(s0 )s0 < s0 , which is a contradiction. Case 2: α(t) is monotonically increasing on I0 . In this case, owing to that the sequence {ϕ n (t0 )} is strictly decreasing, we can infer that {α(ϕ n (t0 ))}∞ n=n0 is monotonically decreasing and bounded from below. So, there exists β ∈ [0, 1) such that limn→∞ α(ϕ n (t0 )) = β. We choose an ε > 0 satisfying β + ε < 1. Then there exists n1 ≥ n0 such that α(ϕ n (t0 )) < β + ε for all n ≥ n1 . Thus, we have s0 ≤ ϕ n+1 (t0 ) = α ϕ n (t0 ) ϕ n (t0 ) < (β + ε)ϕ n (t0 ) for all n ≥ n1 . Letting n → ∞, we get s0 ≤ (β + ε)s0 < s0 , which is a contradiction. This shows that limn→∞ ϕ n (t) = 0 for all t > 0. Hence ϕ ∈ w . Therefore, the conclusion of this theorem follows from Theorem 3.1 immediately. 2 Remark 3.5. Obviously, Theorem 3.3 is a generalization of Theorem C (i.e., Theorem 13 in [4]). This shows that the conclusion of Theorem 13 in [4] is true. As direct consequences of the above results, we can obtain the corresponding fixed point theorems in usual metric spaces. Lemma 3.5. Let (X, d) be a metric space. Define a mapping F : X × X → D + by 0, t ≤ 0 or d(x, y) > t > 0, F (x, y)(t) = Fx,y (t) = for x, y ∈ X. 1, d(x, y) ≤ t (t > 0).
(3.12)
Then (X, F, M ) is a Menger space. It is called the induced Menger space by (X, d) and it is complete iff (X, d) is complete.
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Proof. It is easy to see that the F defined by (3.12) satisfies (PM-1) and (PM-2) in Definition 2.2. We now prove that (PM-3) also holds. In fact, for any given t, s > 0, if d(x, z) > t or d(z, y) > s, then Fx,y (t + s) ≥ 0 = min{Fx,z (t), Fz,y (s)}. If d(x, z) ≤ t and d(z, y) ≤ s, then d(x, y) ≤ d(x, z) + d(z, y) ≤ t + s. By (3.12), we get Fx,y (t + s) = 1 = min{Fx,z (t), Fz,y (s)}. This shows that (PM-3) holds. Therefore (X, T , M ) is a Menger space. By (3.12), we know that for any ε > 0, d(x, y) ≤ ε iff Fx,y (ε) = 1. Thus, it is not difficult to prove that {xn } is d
T
a Cauchy sequence in (X, d) iff {xn } is a T -Cauchy sequence in (X, F, M ), and xn → x iff xn → x. Therefore (X, F, M ) is complete iff (X, d) is complete. 2 Remark 3.6. There is a slight difference between Lemma 3.5 and Lemma 2.1 in [7]. In [7, Lemma 2.1], F is defined by Fx,y (t) = H (t − d(x, y)). Corollary 3.3. Let (X, d) be a complete metric space, and let ϕ : R+ → R+ be a nondecreasing function in w and satisfy ϕ(t) > 0 for all t > 0. If a mapping T : X → X satisfies d(T x, T y) ≤ ϕ d(x, y) for all x, y ∈ X, (3.13) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. We define a mapping F : X × X → D + by (3.12). Since (X, d) is a complete metric space, by Lemma 3.5 we know that (X, F, M ) is a complete Menger space. Note that ϕ ∈ w . Next, we only need to prove that the condition (3.13) implies (1.2). For any given t > 0, if Fx,y (t) = 0, then it is evident that (1.2) holds. If Fx,y (t) = 1, then by (3.12) we know that d(x, y) ≤ t. Since ϕ is nondecreasing, it follows from (3.13) that d(T x, T y) ≤ ϕ(d(x, y)) ≤ ϕ(t). Notice that ϕ(t) > 0 for all t > 0. From (3.12), we have FT x,T y (ϕ(t)) = 1 = Fx,y (t). Hence (1.2) holds, i.e., T is a probabilistic ϕ-contraction. Therefore, the conclusion of Corollary 3.3 follows from Theorem 3.1 immediately. 2 Corollary 3.4. Let (X, d) be a complete metric space, and let α : R+ → [0, 1) be a piecewise monotone function. If a mapping T : X → X satisfies the condition d(T x, T y) ≤ α d(x, y) d(x, y) for all x, y ∈ X, (3.14) then T has a unique fixed point x∗ ∈ X, and {T n (x0 )} converges to x∗ for each x0 ∈ X. Proof. Since α : R+ → [0, 1) is a piecewise monotone function, there exists a partition of [0, ∞): 0 = t0 < t1 < · · · < tn−1 < tn = ∞, such that α(t) is monotonic on each subinterval Ii = [ti−1 , ti ) (i = 1, · · · , n). We define a mapping β : R+ → [0, 1) as follows: α(t), if t ∈ Ii = [ti−1 , ti ) (Ii is a monotone increasing interval of α), β(t) = α(tj −1 ), if t ∈ Ij = [tj −1 , tj ) (Ij is a monotone decreasing interval of α). Obviously, β(t) is also a piecewise monotone function and α(t) ≤ β(t) for all t > 0. So, it follows from (3.14) that d(T x, T y) ≤ β d(x, y) d(x, y) for all x, y ∈ X. (3.15) On the other hand, from the proof of Corollary 3.3, we know that (X, F, M ) is a complete Menger space, where F : X × X → D + is defined by (3.12). In addition, as in the proof of Corollary 3.3, we can prove that the condition (3.15) implies FT x,T y β(t)t ≥ Fx,y (t) for all x, y ∈ X and t > 0. This shows that T satisfies the condition (CR). Therefor, the conclusion of Corollary 3.4 follows from Theorem 3.3. 2 Remark 3.7. Corollary 3.4 is a generalization of the Rakotch fixed point theorem [17]. In Rakotch’s theorem, α : (0, ∞) → [0, 1) is a monotone decreasing function.
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4. Fixed point theorems for ϕ-contraction in fuzzy metric spaces In this section, we shall apply the results in Section 3 to obtain the corresponding fixed point theorems for ϕ-contraction in KM-fuzzy metric spaces. Lemma 4.1. Let (X, M, ) be a KM-fuzzy metric space, where the t -norm is continuous at (1, 1). Suppose that there exist x0 , x1 ∈ X such that limt →∞ M(x0 , x1 , t) = 1. Define Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}. Then (Y0 , F, ) is a Menger space, where F is defined by (2.2). If (X, M, ) is complete, then (Y0 , F, ) is a complete Menger space. Proof. It is evident that Y0 is a nonempty subset of X, It is not difficult to show that M satisfies the condition (FM-6) in Y0 , i.e., limt→∞ M(y, z, t) = 1 for all y, z ∈ Y0 . In fact, if y, z ∈ Y0 , then we have limt→∞ M(x0 , y, t) = 1 and limt→∞ M(x0 , z, t) = 1. Note that M(y, z, t) ≥ (M(y, x0 , t/2), M(x0 , z, t/2)). Letting t → ∞, by the continuity of at (1, 1), it follows that limt→∞ M(y, z, t) = 1. Thus, by Lemma 2.2, we conclude that (Y0 , F, ) is a Menger space, where F is defined by (2.2). Next, we show that (Y0 , M, ) is a closed subspace of (X, M, ). Let {yn } be a sequence in Y0 such that limn→∞ yn = y ∈ X. We need to prove that y ∈ Y0 . By the continuity of at (1, 1), for any ε ∈ (0, 1), there exists δ ∈ (0, 1) such that (1 − δ, 1 − δ) > 1 − ε. Taking t0 > 0, because limn→∞ M(yn , y, t0 /2) = 1 there exists n0 ∈ N such that M(yn0 , y, t0 /2) > 1 − δ. Note that yn0 ∈ Y0 , i.e., limt→∞ M(x0 , yn0 , t/2) = 1. Hence there exists t1 ≥ t0 such that M(x0 , yn0 , t/2) > 1 − δ for all t ≥ t1 . Thus, we have M(x0 , y, t) ≥ M(x0 , yn0 , t/2), M(yn0 , y, t/2) ≥ M(x0 , yn0 , t/2), M(yn0 , y, t0 /2) ≥ (1 − δ, 1 − δ) > 1 − ε
for all t ≥ t1 ,
which shows that limt→∞ M(x0 , y, t) = 1, i.e., y ∈ Y0 . Hence (Y0 , M, ) is a closed subspace of (X, M, ). By the completeness of (X, M, ), we infer that (Y0 , M, ) is also complete, and so (Y0 , F, ) is a complete Menger space. 2 Theorem 4.1. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type. Let T : X → X be a mapping such that M T x, T y, ϕ(t) ≥ M(x, y, t) for all x, y ∈ X and t > 0, (4.1) where ϕ ∈ w . Suppose that there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1. Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . Proof. We define a mapping F : Y0 × Y0 → D + by (2.2). Since (X, M, ) is a complete KM-fuzzy metric space and there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1, by Lemma 4.1 we know that (Y0 , F, ) is a complete Menger space. We can prove that (4.1) implies that M(T x, T y, t) ≥ M(x, y, t) for all x, y ∈ X and t > 0.
(4.2)
In fact, since ϕ ∈ w , for each t > 0 there exists r ≥ t such that ϕ(r) < t by Lemma 3.1. So, by (4.1) and the monotony of M(u, v, ·), we have M(T x, T y, t) ≥ M T x, T y, ϕ(r) ≥ M(x, y, r) ≥ M(x, y, t). Eq. (4.2) is proved. It is not difficult to prove that T is a mapping of Y0 into itself. In fact, if y ∈ Y0 , then limt→∞ M(x0 , y, t/2) = 1. By the hypothesis, limt→∞ M(x0 , T x0 , t/2) = 1. In addition, using (FM-4) and (4.2), we get M(x0 , T y, t) ≥ M(x0 , T x0 , t/2), M(T x0 , T y, t/2) ≥ M(x0 , T x0 , t/2), M(x0 , y, t/2) .
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Letting t → ∞ in the above inequality, from the continuity of at (1, 1), we obtain limt→∞ M(x0 , T y, t) = 1, i.e., T y ∈ Y0 . This shows that T is a mapping of Y0 into itself. Clearly, (4.1) implies that FT x,T y (ϕ) ≥ Fx,y (t)
for all x, y ∈ Y0 and t > 0,
where F is defined by (2.2). This shows that T is a probabilistic ϕ-contraction in (Y0 , F, ). Thus, by Theorem 3.1, we conclude that T has a unique fixed point x∗ in Y0 , and {T n (y0 )} converges to x∗ for each y0 ∈ Y . In particular, {T n (x0 )} converges to x∗ . This completes the proof. 2 Remark 4.1. From the above proof, it is easy to see that Theorem 3.1 implies Theorem 4.1. On the other hand, by Lemma 2.2 we know that if (X, F, ) is a complete Menger space, then (X, M, ) is a complete KM-fuzzy metric space with (FM-6), where M is defined by M(x, y, t) = Fx,y (t) for t ≥ 0. So, it is not difficult to prove that Theorem 4.1 implies Theorem 3.1. This shows that Theorem 4.1 is equivalent to Theorem 3.1. That is to say, Theorem 4.1 is an equivalent type of Theorem 3.1 in KM-fuzzy metric spaces. In the same way, from Theorem 3.2, Corollary 3.2 and Theorem 3.3, we can prove the following theorems, respectively. Theorem 4.2. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type, and let T : X → X be a fuzzy ϕ-contraction, i.e., it satisfies (4.1). Suppose that there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1, and ϕ : R+ → R+ is a right-locally monotone function satisfying the condition (CBW). Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . Theorem 4.3. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type, and let T : X → X be a fuzzy ϕ-contraction, i.e., it satisfies (4.1). Suppose that there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1, and ϕ : R+ → R+ is a function satisfying the following conditions: ϕ(t) < t
and
lim sup ϕ(r) < t r→t +
for each t > 0.
(BW)
Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . Theorem 4.4. Let (X, M, ) be a complete KM-fuzzy metric space with a t -norm of H -type, and let α : R+ → [0, 1) be a piecewise monotone function. Suppose that T : X → X is a mapping satisfying the condition M T x, T y, α(t)t ≥ M(x, y, t) for all x, y ∈ X and > 0, (4.3) and there exists some x0 ∈ X such that limt→∞ M(x0 , T x0 , t) = 1. Then T has a unique fixed point x∗ in Y0 = {y ∈ X | limt→∞ M(x0 , y, t) = 1}, and {T n (y0 )} converges to x∗ for each y0 ∈ Y0 . In particular, {T n (x0 )} converges to x∗ . 5. An example The following example shows that Theorem 3.1 is more general than Jachymski’s theorem (i.e. Theorem 1 in [12]). Example 5.1. Let X = [0, ∞) and define F : X × X → D + as follows: t , if |x − y| ≥ t, F (x, y)(t) = Fx,y (t) = t+|x−y| 1, if |x − y| < t. Then (X, F, M ) is a Menger space (see Example in [4]).
(5.1)
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To prove that (X, F, M ) is complete, we first notice that the metric space (X, d) is complete, where d is defined by d(x, y) = |x − y|. Next, we show that if {xn } is a T -Cauchy sequence in (X, F, M ), then {xn } is also a Cauchy sequence in (X, d). In fact, if {xn } is not a Cauchy sequence in (X, d), then there are ε0 > 0 and two sequences {ni } and {mi } such that mi > ni ≥ i and d(xmi , xni ) ≥ ε0 for all i ∈ N. Taking λ0 ∈ (0, 1/2], by (5.1) we have Fxmi ,xni (ε0 ) =
ε0 1 ≤ ≤ 1 − λ0 ε0 + |xmi − xni | 2
for all i ∈ N,
which contradicts that {xn } is a T -Cauchy sequence in (X, F, M ). Hence {xn } is also a Cauchy sequence in (X, d), by the completeness of (X, d), there exists x ∈ X such that limn→∞ xn = x, and so for any ε > 0 there exists n0 ∈ N such that |xn − x| < ε for all n ≥ n0 . Thus, for any ε > 0 and λ ∈ (0, 1], by (5.1) we have Fxn ,x (ε) = 1 > 1 − λ for all n ≥ n0 . This shows that {xn } is T -convergent to x ∈ X. Therefore, (X, F, M ) is complete. x Let T : X → X be defined by T x = 1+x , and let ϕ : R+ → R+ be defined by ⎧ t , 0 ≤ t < 1, ⎪ ⎨ 1+t t 4 (5.2) ϕ(t) = − 3 + 3 , 1 ≤ t ≤ 2, ⎪ ⎩ 4 t − 3, 2 < t < ∞. By Example 3.1, we know that ϕ ∈ w but ϕ ∈ / . We now show that T is a probabilistic ϕ-contraction, i.e., T satisfies (1.2). If |T x − T y| < ϕ(t), then we have FT x,T y (ϕ(t)) = 1 ≥ Fx,y (t), (1.2) holds. Suppose that |T x − T y| ≥ ϕ(t). From (5.2), it is easy to see that ϕ(t) ≥
t 1+t
for all t ≥ 0,
and so |T x − T y| ≥ |T x − T y| =
t 1+t .
Note that
|x − y| |x − y| |x − y| = ≤ . 1 + x + y + xy 1 + |x − y| + 2 min{x, y} + xy 1 + |x − y|
|x−y| t We get 1+t ≤ 1+|x−y| , which implies that |x − y| ≥ t since the function f (u) = t . So, By (5.1), we have Fx,y (t) = t+|x−y|
FT x,T y ϕ(t) =
ϕ(t) ≥ ϕ(t) + |T x − T y|
t 1+t |x−y| t 1+t + 1+|x−y|
≥
u 1+u
is strictly increasing on [0, ∞).
t = Fx,y (t), t + |x − y|
i.e., (1.2) holds. This shows that all the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, we conclude that T has a unique fixed point x∗ in X. Indeed, x = 0 is a unique fixed point of T . However, Theorem 1 in [12] cannot be applied to this example because the ϕ defined by (5.2) does not meet the conditions “ϕ(t) < t and limn→∞ ϕ n (t) = 0 for all t > 0”. 6. Conclusion We have proved a new fixed point theorem for probabilistic ϕ-contraction in Menger spaces with a t -norm of H -type, namely Theorem 3.1. This is a main theorem in this paper. It is worth noting that in the theorem, the contractive gauge function ϕ only needs to meet (1.4), i.e., for each t > 0 there exists r ≥ t such that lim ϕ n (r) = 0, n→∞
which is a quite weak condition. Therefore, this theorem improves and generalizes some important fixed point theo´ c [4], Jachymski [12] and Xiao et al. [25]. As consequences of the theorem, rems, including the recent results of Ciri´ a corrected version of the main theorem in [4], the probabilistic versions of Boyd–Wong’s fixed point theorem [1] and Rakotch’s fixed point theorem [17] and some corresponding fixed point theorems in general metric spaces are
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obtained. Using Theorem 3.1, we also considered the fixed point theorems for ϕ-contraction in fuzzy metric space. An equivalent type of Theorem 3.1 in KM-fuzzy metric spaces is given. Can the condition (1.4) and the reasoning methods and techniques of Theorem 3.1 be applied to generalize some other types of fixed point theorems? This is a question that we want to study in the future. Can we replace the condition (1.4) in Theorem 3.1 by a more weak condition? This question is also worthy of further investigation. References [1] D.W. Boyd, J.S.W. Wong, On nonlinear contractions, Proc. Am. Math. Soc. 20 (1969) 458–464. [2] S.S. Chang, Y.J. Cho, S.M. Kang, Nonlinear Operator Theory in Probabilistic Metric Spaces, Nova Science Publisher, New York, 2001. [3] Y.J. Cho, K.S. Park, S.S. Chang, Fixed point theorems in metric spaces and probabilistic metric spaces, Int. J. Math. Math. Sci. 19 (1996) 243–252. ´ c, Solving the Banach fixed point principle for nonlinear contractions in probabilistic metric spaces, Nonlinear Anal. 72 (2010) [4] L. Ciri´ 2009–2018. [5] M.F. Duh, Y.Y. Huang, Common fixed points for mappings on complete Menger spaces, Far East J. Math. Sci. 6 (2) (1998) 169–193. [6] J.-X. Fang, A note on fixed point theorems of Hadži´c, Fuzzy Sets Syst. 48 (1992) 391–395. [7] J.-X. Fang, Common fixed point theorems of compatible and weakly compatible maps in Menger spaces, Nonlinear Anal. 71 (2009) 1833–1843. [8] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst. 27 (3) (1988) 385–389. [9] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst. 64 (1994) 395–399. [10] O. Hadžíc, Fixed point theorems for multi-valued mappings in probabilistic metric spaces, Mat. Vesn. 3 (1979) 125–133. [11] O. Hadži´c, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, 2001. [12] J. Jachymski, On probabilistic ϕ-contractions on Menger spaces, Nonlinear Anal. 73 (2010) 2199–2203. [13] I. Kramosil, J. Michálek, Fuzzy metrics and statistical metric spaces, Kybernetika 11 (1975) 336–344. [14] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA 28 (1942) 535–537. [15] D. Mihe¸t, A class of contractions in fuzzy metric spaces, Fuzzy Sets Syst. 161 (2010) 1131–1137. [16] D. O’Regan, R. Saadati, Nonlinear contraction theorems in probabilistic spaces, Appl. Math. Comput. 195 (2008) 86–93. [17] E. Rakotch, A note on contractive mappings, Proc. Am. Math. Soc. 13 (1962) 459–465. [18] A. Roldán, J. Martínez-Moreno, C. Roldán, Tripled fixed point theorem in fuzzy metric spaces and applications, Fixed Point Theory Appl. 2013 (2013). Article ID 29. [19] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983. [20] B. Schweizer, A. Sklar, E. Thorp, The metrization of statistical metric spaces, Pac. J. Math. 10 (1960) 673–675. [21] V.M. Sehgal, A.T. Bharucha-Reid, Fixed points of contraction mappings on PM-spaces, Math. Syst. Theory 6 (1972) 97–102. [22] J.Z. Xiao, X.H. Zhu, On linearly topological structure and property of fuzzy normed linear space, Fuzzy Sets Syst. 125 (2002) 153–161. [23] J.Z. Xiao, X.H. Zhu, Y.F. Cao, Common coupled fixed point results for probabilistic φ-contractions in Menger spaces, Nonlinear Anal., Theory Methods Appl. 74 (2011) 4589–4600. [24] J.Z. Xiao, X.H. Zhu, X. Jin, Fixed point theorems for nonlinear contractions in Kaleva–Seikkala’s type fuzzy metric spaces, Fuzzy Sets Syst. 200 (2012) 65–83. [25] J.Z. Xiao, X.H. Zhu, X.Y. Liu, An alternative characterization of probabilistic Menger spaces with H -type triangular norms, Fuzzy Sets Syst. 227 (2013) 107–114.