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Tripled fuzzy metric spaces and fixed point theorem
Tripled fuzzy metric spaces and fixed point theorem
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Tripled fuzzy metric spaces and fixed point theorem Jing-Feng Tian, Ming-Hu Ha, Da-Zeng Tian PII: DOI: Reference:
S0020-0255(20)30006-2 https://doi.org/10.1016/j.ins.2020.01.007 INS 15138
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Information Sciences
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20 March 2019 26 September 2019 6 January 2020
Please cite this article as: Jing-Feng Tian, Ming-Hu Ha, Da-Zeng Tian, Tripled fuzzy metric spaces and fixed point theorem, Information Sciences (2020), doi: https://doi.org/10.1016/j.ins.2020.01.007
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TRIPLED FUZZY METRIC SPACES AND FIXED POINT THEOREM JING-FENG TIAN, MING-HU HA*, AND DA-ZENG TIAN
Abstract. One of the most important topics of research in fuzzy sets is to get an appropriate notion of fuzzy metric space (FMS), in the paper we propose a new FMS–tripled fuzzy metric space (TFMS), which is a new generalization of George and Veeramani’s FMS. Then we present some related examples, topological properties, convergence of sequences, Cauchy sequence (CS) and completeness of the TFMS. Moreover, we introduce two kinds of notions of generalized fuzzy ψ-contractive (Fψ-C) mappings, and derive a fixed point theorem (FPT) on the mappings in the space.
1. Introduction The fuzzy sets [34] plays a very significant role in fuzzy literature, which has been given and studied in many different directions by many researchers[3–6, 19, 20, 24, 26, 31, 35]. One of the most important topics of research in fuzzy sets is to get an appropriate notion of fuzzy metric space (FMS). The topic has been investigated by some researchers in different ways. In 1975, Kramosil and Michalek [17] (KM) initiated the notion of fuzzy metric, a fuzzy set (FS) in the product S × S × [0, ∞[ satisfying some certain conditions. Later, Deng [8] in 1984 presented the notion of fuzzy pseudo metric space, with the metric defined between two fuzzy points, and obtained the fuzzy topological structure and the fuzzy uniform structure on this space. To obtain a Hausdorff topology for FMS introduced by KM, George and Veeramani [11] (GV) in 1994 reintroduced the notion of FMS by modifying the definition of KM’s FMS. Later, the authors in another paper [12] gave a necessary and sufficient condition for the completeness of FMS. Since then, new properties, various fixed point (FP) results for mappings satisfying different contractive conditions and artful applications in both context of FMS above mentioned were established by many researchers[1, 2, 25, 28–30, 32]. Inspired by the above works, the main purpose of the paper is to give a new FMS– tripled fuzzy metric space. The rest of this paper is organized as follows. Section 2 firstly introduces relevant concepts and results applied in developing results of this study, then, it presents a new generalization of FMS, namely, tripled fuzzy metric space. Next, some related examples with particular emphasis on the generalization of FMS are demonstrated in this section. Section 3, based on some topological properties of the studied space, gives the notions of convergence sequence, CS and completeness in TFMS. Section 4 offers two kinds of notions of generalized Fψ-C (2000) Mathematics subject classification: 54A40, 54E35. Key words and phrases: Fuzzy sets; tripled fuzzy metric spaces; Hausdorff topology; fuzzy contractive mapping; fixed point. *Corresponding author: Ming-Hu Ha, e-mail: [email protected]. 1
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
mappings and presents some results such as a FPT on the mappings in TFMS. Finally, a brief summary is given in Section 5. 2. Tripled fuzzy metric spaces The definitions below will be needed in the sequel. Definition 2.1. ([16], [27]) A mapping ∗ : [0, 1]×[0, 1] → [0, 1] is called a continuous t-norm (CTN) if satisfies: (i) ∗(η, ϑ) = ∗(ϑ, η) and ∗(ϑ, ∗(η, κ)) = ∗(∗(ϑ, η), κ), for 0 ≤ ϑ, η, κ ≤ 1, (ii) ∗(ϑ, 1) = ϑ, for 0 ≤ ϑ ≤ 1, (iii) ∗(ϑ, η) ≤ ∗(κ, θ) whenever ϑ ≤ κ and η ≤ θ, for 0 ≤ ϑ, η, κ, θ ≤ 1, (iv) ∗ is continuous. From Definition 2.1 we find that for 0 ≤ ϑ, η ≤ 1, ∗(ϑ, η) ≤ min{ϑ, η} and ∗m (ϑ, η) = min{ϑ, η}, ∗p (ϑ, η) = ϑη are CTNs. Definition 2.2. [27] If ϑ ∗ η > 0 whenever ϑ, η ∈ (0, 1], then we call the t-norm ∗ is positive. It is obviously that for 0 < ϑ, η ≤ 1, ∗m (ϑ, η) = min{ϑ, η} and ∗p (ϑ, η) = ϑη are positive. Definition 2.3. ([17]) Let S be an arbitrary set, let ∗ be a CTN, and let P be a FS on S × S × [0, ∞[ such that for all u, v, w ∈ S, the following conditions are valid: (KM-1) P(u, v, 0) = 0, (KM-2) P(u, v, α) = 1 for α > 0 iff u = v, (KM-3) P(u, v, α) = P(v, u, α) for α > 0, (KM-4) P(u, v, ·) : [0, ∞[→ [0, 1] is left-continuous, (KM-5) P(u, w, α + β) ≥ ∗(P(u, v, α), P(v, w, β)) for α, β ≥ 0. Then the triple (S, P, ∗) is called a FMS (in the sense of KM). In order to obtain a Hausdorff topology (HT) of FMS, GV [11] modified the definition of FMS presented by KM as follows. Definition 2.4. ([11]) Let S be an arbitrary set, let ∗ be a CTN, and let P be a FS on S × S×]0, ∞[ such that for all u, v, w ∈ S and α, β > 0, the following conditions valid: (GV-1) P(u, v, α) > 0, (GV-2) P(u, v, α) = 1 iff u = v, (GV-3) P(u, v, α) = P(v, u, α), (GV-4) P(u, v, ·) :]0, ∞[→]0, 1] is continuous, (GV-5) P(u, w, α + β) ≥ ∗(P(u, v, α), P(v, w, β)). Then the triple (S, P, ∗) is called a FMS (in the sense of GV). Theorem 2.5. ([13]) Let (S, P, ∗) be a FMS. Then for all u, v ∈ S, P(u, v, ·) is nondecreasing. In this paper, the FMS (S, P, ∗) is in the sense of GV. Now, we will give the following notion of TFMSs. Definition 2.6. A triple (S, Q, ∗) is said to be a TFMS if S is an arbitrary set, ∗ is a CTN and Q is a FS on S × S × S×]0, ∞[ such that for all u, v, w, z ∈ S the following conditions valid: (TFM-1) Qu,v,w (α) > 0;
Tripled fuzzy metric spaces and fixed point theorem
(TFM-2) (TFM-3) (TFM-4) (TFM-5) (TFM-6)
3
Qu,v,w (α) = 1 iff u = v = w; Qu,u,v (α) ≥ Qu,v,w (α) for w 6= v; Qu,v,w (α) = Qu,w,v (α) = Qv,u,w (α) = · · · ; Qu,v,w (·) :]0, ∞[→]0, 1] is continuous; Qu,v,w (α + β) ≥ ∗(Qu,z,z (α), Qz,v,w (β)).
Remark 2.7. Let (S, Q, ∗) be a TFMS. Then the mapping P : S×S×]0, ∞[→]0, 1] defined by (1) P(u, v, α) = ∗ Qu,v,v (α), Qv,u,u (α)
is a fuzzy metric (FM). Indeed, it is not difficult to check that P satisfies (GV-1)(GV-4). Next, we prove P satisfies the condition (GV-5). In fact, for all β, α > 0 and u, v, z ∈ S, we find (2) P(u, v, β + α) = ∗ Qu,v,v (β + α), Qv,u,u (β + α) , and
∗ P(u, w, β), P(w, v, α) = ∗ ∗ Qu,w,w (β), Qw,u,u (β) , ∗ Qw,v,v (α), Qv,w,w (α) . From (TFM-6) and (iii) of Definition 2.1, we get ∗ Qu,v,v (β + α), Qv,u,u (β + α) ≥ ∗ ∗ Qu,w,w (β), Qw,v,v (α) , ∗ Qv,w,w (α), Qw,u,u (β) . Since
(3) (4) (5) and (6)
Qu,w,w (β) ≥ min{Qu,w,w (β), Qw,u,u (β)} ≥ ∗ Qu,w,w (β), Qw,u,u (β) , Qw,v,v (α) ≥ min{Qw,v,v (α), Qv,w,w (α)} ≥ ∗ Qw,v,v (α), Qv,w,w (α) ,
Qv,w,w (α) ≥ min{Qw,v,v (α), Qv,w,w (α)} ≥ ∗ Qw,v,v (α), Qv,w,w (α) ,
Qw,u,u (β) ≥ min{Qu,w,w (β), Qw,u,u (β)} ≥ ∗ Qu,w,w (β), Qw,u,u (β) ,
by using (TFM-6), (3), (4), (5), (6) and (1), we have Qu,v,v (β + α) ≥ ∗ Qu,w,w (β), Qw,v,v (α) (7) ≥ ∗ ∗ Qu,w,w (β), Qw,u,u (β) , ∗ Qw,v,v (α), Qv,w,w (α) = ∗ P(u, w, β), P(w, v, α) and (8)
Qv,u,u (β + α) ≥ ∗ Qv,w,w (α), Qw,u,u (β) ≥ ∗ ∗ Qw,v,v (α), Qv,w,w (α) , ∗ Qu,w,w (β), Qw,u,u (β) = ∗ P(w, v, α), P(u, w, β) .
Therefore, from (7), (8) and (2), we get
P(u, v, β + α) = ∗ Qu,v,v (β + α), Qv,u,u (β + α) ≥ ∗ P(u, w, β), P(w, v, α) .
This shows that P satisfies (GV-5), and thus P is a FM.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Example 2.8. Let S = R. Define Q : S × S × S×]0, ∞[→]0, 1] by
Qu,v,w (α) = exp
|u − v| + |v − w| + |w − u| −1 α
for all u, v, w ∈ S and α > 0. If we set ∗ = ∗p , then (S, Q, ∗) is a TFMS. Proof. It is not difficult to prove that Q satisfies (TFM-1)-(TFM-5). Now we prove that for all u, v, w, z ∈ S and all β, α > 0 Qu,v,w (β + α) ≥ ∗(Qu,z,z (β), Qz,v,w (α)). Indeed, since |u − v| + |v − w| + |w − u| |u − z| + |z − v| + |v − w| + |w − z| + |z − u| ≤ β+α β+α 2|u − z| |z − v| + |v − w| + |w − z| + = β+α β+α 2|u − z| |z − v| + |v − w| + |w − z| < + , β α we have |u − v| + |v − w| + |w − u| β+α Qu,v,w (β + α) = e 2|u − z| |z − v| + |v − w| + |w − z| + − β α ≥e −
= Qu,z,z (β)Qz,v,w (α)
= ∗(Qu,z,z (β), Qz,v,w (α)). Hence (S, Q, ∗) is a TFMS.
Remark 2.9. It is obvious that the above (S, Q, ∗) is not a FMS. Example 2.10. Let (S, P, ∗) be a FMS. For u, v, w ∈ S and α > 0, define Q : S × S × S×]0, ∞[→]0, 1] by Qu,v,w (α) = ∗ ∗ P(u, v, α), P(v, w, α) , P(u, w, α) .
Then (S, Q, ∗) is a TFMS.
Proof. It is easy to see that Q satisfies (TFM-1)-(TFM-5). In order to verify that Q satisfies (TFM-6), we need to verify only that (9)
Qu,v,w (β + α) ≥ ∗ Qu,z,z (β), Qz,v,w (α) ,
Tripled fuzzy metric spaces and fixed point theorem
for u, v, w, z ∈ S and β, α > 0, that is, ∗ ∗ P(u, v, β + α), P(v, w, β + α) , P(u, w, β + α) ≥ ∗ ∗ ∗ P(u, z, β), P(z, z, β) , P(z, u, β) , ∗ ∗ P(z, v, α), P(v, w, α) , P(z, w, α) (10) = ∗ ∗ P(u, z, β), P(z, u, β) , ∗ ∗ P(z, v, α), P(v, w, α) , P(z, w, α) . From (GV-5), we find that
P(u, v, β + α) ≥ ∗ P(u, z, β), P(z, v, α) ≥ ∗ min P(u, z, β), P(u, z, β) , (11)
o min min P(z, v, α), P(v, w, α) , P(z, w, α) n
≥ ∗ ∗ P(u, z, β), P(u, z, β) , ∗
∗ P(z, v, α), P(v, w, α) , P(z, w, α) ,
P(u, w, β + α) ≥ ∗ P(u, z, β), P(z, w, α) ≥ ∗ min P(u, z, β), P(u, z, β) , (12)
o min min P(z, v, α), P(v, w, α) , P(z, w, α) n
≥ ∗ ∗ P(u, z, β), P(u, z, β) , ∗
and
∗ P(z, v, α), P(v, w, α) , P(z, w, α)
P(v, w, β + α) ≥ P(v, w, α)
(13)
≥ min{P(z, v, α), P(v, w, α), P(z, w, α)} ≥ ∗ P(u, z, β), min{P(z, v, α), P(z, w, α), P(v, w, α)} = ∗ P(u, z, β), min min{P(z, v, α), P(z, w, α)}, P(v, w, α) ≥ ∗ ∗ P(u, z, β), P(u, z, β) , ∗ ∗ P(z, v, α), P(v, w, α) , P(z, w, α) .
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
By using (11), (12) and (13), we obtain that (10), i.e., (9) holds. Therefore, Q satisfies (TFM-6) and thus (S, Q, ∗) is a TFMS. Letting ∗ = ∗m in Example 2.10, we get the following result. Example 2.11. Let (S, P, ∗m ) be a FMS. For u, v, w ∈ S and α > 0, define Q : S × S × S×]0, ∞[→]0, 1] by Qu,v,w (α) = min{P(u, v, α), P(v, w, α), P(u, w, α)}.
Then (S, Q, ∗m ) is a TFMS.
|u−v|
Remark 2.12. Let S = R, ∗ = ∗m , P(u, v, α) = e− α , where u, v ∈ S, α > 0. Then from [11, Example 2.7 and Remark 2.8] we know that (S, P, ∗) is a FMS. From Example 2.10, we can find that for ∗ = ∗m |u−v| |v−w| |w−u| Qu,v,w (α) = ∗ ∗ e− α , e− α , e− α
is a tripled fuzzy metric and hence (S, Q, ∗) is a TFMS.
Remark 2.13. Let (S, d) be a metric space, let ∗ = ∗p , and let P(u, v, α) =
kαn , kαn + md(u, v)
k, m, n ∈ R+ . Then from [11, Example 2.9] we know that (S, P, ∗) is a FMS. From Example 2.10, we obtain that kαn kαn kαn Qu,v,w (α) = ∗ ∗ , , kαn + md(u, v) kαn + md(v, w) kαn + md(w, u)
is a tripled fuzzy metric and hence (S, Q, ∗) is a TFMS.
Example 2.14. Let (S, P, ∗m ) be a FMS, and let λ1 , λ2 , λ3 > 0 with λ1 +λ2 +λ3 = 1. For u, v, w, ∈ S, α > 0, define a map Q : S × S × S×]0, ∞[→]0, 1] by (14)
Qu,v,z (α) = min{P(u, v, λ1 α), P(v, z, λ2 α), P(u, z, λ3 α)}.
Then (S, Q, ∗m ) is a TFMS. Proof. It is not difficult to obtain that Q satisfies (TFM-1)-(TFM-5). Next, we verify that Q satisfies (TFM-6). From (14), (GV-5) and Theorem 2.5, we find that Qu,v,w (α + β) = min {P (u, v, λ1 (α + β)) , P (v, w, λ2 (α + β)) , P (u, w, λ3 (α + β))} ≥ min {P (u, v, λ1 (α + β)) , P (v, w, λ2 β) , P (u, w, λ3 (α + β))} ≥ min {∗ (P (u, z, λ1 α) , P (z, v, λ1 β)) , P (v, w, λ2 β) , ∗ (P (u, z, λ3 α) , P (z, w, λ3 β))}
= min {min {P (u, z, λ1 α) , P (z, v, λ1 β)} , P (v, w, λ2 β) , min {P (u, z, λ3 α) , P (z, w, λ3 β)}}
= min {min {P (u, z, λ1 α) , P (z, z, λ2 α) , P (u, z, λ3 α)} ,
min {P (z, v, λ1 β) , P (v, w, λ2 β) , P (z, w, λ3 β)}}
= min {Qu,z,z (α), Qz,v,w (β)} = ∗ (Qu,z,z (α), Qz,v,w (β)) .
Thus Q satisfies (TFM-6) and then (S, Q, ∗m ) is a TFMS.
Tripled fuzzy metric spaces and fixed point theorem
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Proposition 2.15. Let (S, Q, ∗) be a TFMS. Then for all u, v ∈ S, Qu,v,v (·) is nondecreasing. Proof. For some β > α > 0, we assume Qu,v,v (α) > Qu,v,v (β). Then Qu,v,v (α) > Qu,v,v (β) = Qu,v,v (α + β − α) ≥ ∗(Qu,v,v (α), Qv,v,v (β − α)). From (TFM-2), we get Qv,v,v (β − α) = 1, and thus
Qu,v,v (α) > Qu,v,v (β) ≥ Qu,v,v (α)
which is a contradiction. Hence the proof of Proposition 2.15 is completed.
3. Topology of TFMS In the section, we first present the notion of neighborhood in the TFMS. Definition 3.1. Let (S, Q, ∗) be a TFMS, let r ∈]0, 1[, α > 0, and let u0 ∈ S. Then the set U (u0 , r, α) = {v ∈ S : Qu0 ,v,v (α) > 1 − r, Qu0 ,u0 ,v (α) > 1 − r} is called the neighborhood with center u0 and radius r with respect to α. Theorem 3.2. Every neighborhood U (u0 , r, α) is an open set. Proof. Let U (u0 , r, α) = {u ∈ S : Qu0 ,u,u (α) > 1 − r, Qu0 ,u0 ,u (α) > 1 − r} be a neighborhood with center u0 and radius r with respect to α. If v ∈ U (u0 , r, α), then we claim that there is a neighborhood of v, Uv , satisfying Uv ⊂ U (u0 , r, α). In fact, since v ∈ U (u0 , r, α), Qu0 ,v,v (α) > 1 − r,
Qu0 ,u0 ,v (α) > 1 − r.
Now, Qu0 ,v,v is continuous at α, so we can obtain that α0 < α and r0 < r such that Qu0 ,v,v (α0 ) > 1 − r0 > 1 − r, Qu0 ,u0 ,v (α0 ) > 1 − r0 > 1 − r. Let Uv = {z ∈ S : Qv,z,z (α1 ) > 1 − r1 , Qv,v,z (α1 ) > 1 − r1 }, where 0 < α1 < α − α0 and r1 is chosen such that ∗(1 − r0 , 1 − r1 ) > 1 − r.
Such a r1 exists since ∗ is continuous, ∗(a, 1) = a for all a ∈ [0, 1] and 1 − r0 > 1 − r. Now, assume that s ∈ Uv , so we have Qv,s,s (α1 ) > 1 − r1 ,
Qv,v,s (α1 ) > 1 − r1 .
By using (TFM-6) and Proposition 2.15, we obtain Qu0 ,s,s (α) ≥ ∗(Qu0 ,v,v (α0 ), Qv,s,s (α − α0 )) ≥ ∗(Qu0 ,v,v (α0 ), Qv,s,s (α1 )) ≥ ∗(1 − r0 , 1 − r1 ) > 1 − r.
Similarly, we also get Qu0 ,u0 ,s (α) > 1 − r. This shows s ∈ U (u0 , r, α) and thus Uv ⊂ U (u0 , r, α). The proof of Theorem 3.2 is completed. Remark 3.3. Let (S, Q, ∗) be a TFMS. Define τ = {A ⊂ S : for any u ∈ A , there exist α > 0 and r ∈ (0, 1) satisfying U (u, r, α) ⊂ A }. Then τ is a topology on S. Moreover, as {U (x, n1 , n1 ) : n = 1, 2, . . . , } is a local base at x, the topology τ is first countable. Theorem 3.4. Every TFMS is Hausdorff.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Proof. Let (S, Q, ∗) be a TFMS, u, v ∈ S and u 6= v. Then there are t > 0 0 00 0 00 0 00 and r , r with 0 < r , r < 1 satisfying Qx,y,y (t) = r and Qu,u,v (α) = r . Since ∗ is continuous, monotone, and ∗(1, 1) = 1, we can find r1 and r2 such that 0 00 0 < r1 , r2 < 1 and ∗(1 − r1 , 1 − r2 ) > r, where r = max{r , r }. Now, we consider the following two neighborhoods Uu and Uv defined by n α α o Uu = z : Qu,z,z > 1 − r1 , Qu,u,z > 1 − r1 , 2 2 and n α α o Uv = z : Qv,z,z > 1 − r2 , Qv,v,z > 1 − r2 . 2 2 T T Clearly Uu Uv = ∅. In fact, if there exists a point s ∈ Uu Uv such that α α α α Qu,s,s > 1−r1 , Qu,u,s > 1−r1 , Qv,s,s > 1−r2 , Qv,v,s > 1−r2 , 2 2 2 2 then by (TFM-6), we have α α 0 0 r = Qu,v,v (α) ≥ ∗ Qu,s,s , Qs,v,v > ∗(1 − r1 , 1 − r2 ) > r ≥ r 2 2 and α α 00 00 , Qs,u,u > ∗(1 − r2 , 1 − r1 ) > r ≥ r , r = Qv,u,u (α) ≥ ∗ Qv,s,s 2 2 which are contradictions. Therefore, Uu and Uv are disjoint and thus (S, Q, ∗) is Hausdorff. Definition 3.5. Let (S, Q, ∗) be a TFMS. A set A ⊂ S is called F-bounded (FBed) if and only if there are α > 0 and 0 < r < 1 satisfying Qu,v,v > 1 − r and Qu,u,v > 1 − r for any u, v ∈ A . Theorem 3.6. Let (S, Q, ∗) be a TFMS. Then every compact subset A of S is FBed. Proof. Let A be a given compact subset of S. Fixing α1 > 0, 0 < r1 < 1, and considering an open cover {U (u, r1 , α1 ) : uS∈ A } of A , as A is compact, we can find u1 , u2 , . . . , un ∈ A such that A ⊆ U (ui , r1 , t1 ). If u, v ∈ A , then for some i, j, u ∈ U (ui , r1 , α1 ) and v ∈ U (uj , r1 , α1 ). Therefore Qui ,u,u (α1 ) > 1 − r1 , Qui ,ui ,u (α1 ) > 1 − r1 , Quj ,v,v (α1 ) > 1 − r1 and Quj ,uj ,v (α1 ) > 1 − r1 . Let and
ζ = min{Qui ,uj ,uj (α1 ) : 1 ≤ i, j ≤ n}
β = min{Qui ,ui ,uj (α1 ) : 1 ≤ i, j ≤ n}. Then ζ, β > 0. Now let γ = min{ζ, β}, then γ > 0. On the one hand
Qu,v,v (3α1 ) ≥ ∗ Qu,ui ,ui (α1 ), ∗(Qui ,uj ,uj (α1 ), Quj ,v,v (α1 )) ≥ ∗ (1 − r1 , ∗(ζ, 1 − r1 ))
≥ ∗ (1 − r1 , ∗(γ, 1 − r1 )) .
On the other hand Qu,u,v (3α1 ) = Qv,u,u (3α1 ) ≥ ∗ Qv,uj ,uj (α1 ), ∗(Quj ,ui ,ui (α1 ), Qui ,u,u (α1 )) ≥ ∗ (1 − r1 , ∗(β, 1 − r1 ))
≥ ∗ (1 − r1 , ∗(γ, 1 − r1 )) .
Putting α = 3α1 and ∗ (1 − r1 , ∗(γ, 1 − r1 )) > 1 − r, 0 < r < 1, we have for all u, v ∈ A , Qu,v,v (α) > 1 − r and Qu,u,v (α) > 1 − r. Hence A is FBed.
Tripled fuzzy metric spaces and fixed point theorem
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Proposition 3.7. Let (S, Q, ∗) be a TFMS and u0 be any point in S. If r1 ≤ r2 and α1 ≤ α2 , then U (u0 , r1 , α1 ) ⊂ U (u0 , r2 , α2 ). Proof. Assume that z ∈ U (u0 , r1 , α1 ), so Qu0 ,z,z (α1 ) > 1 − r1 and Qu0 ,u0 ,z (α1 ) > 1 − r1 . Since Qu,v,v (·) is nondecreasing, we obtain Qu0 ,z,z (α2 ) ≥ Qu0 ,z,z (α1 ) > 1 − r1 ≥ 1 − r2 .
and Qu0 ,u0 ,z (α2 ) ≥ Qu0 ,u0 ,z (α1 ) > 1 − r1 ≥ 1 − r2 .
Therefore, z ∈ U (u0 , r2 , α2 ), and thus U (u0 , r1 , α1 ) ⊂ U (u0 , r2 , α2 ).
Proposition 3.8. Let (S, Q, ∗) be a TFMS and u0 be any point in S. If U1 (u0 , r1 , α1 ) and U2 (u0 , r2 , α2 ) are neighborhoods of point T u0 , then there is a neighborhood of point u0 , U3 (u0 , r3 , α3 ), such that U3 ⊂ U1 U2 . Proof. Let
U1 (u0 , r1 , α1 ) = {v ∈ S : Qu0 ,v,v (α1 ) > 1 − r1 , Qu0 ,u0 ,v (α1 ) > 1 − r1 } and U2 (u0 , r2 , α2 ) = {v ∈ S : Qu0 ,v,v (α2 ) > 1 − r2 , Qu0 ,u0 ,v (α2 ) > 1 − r2 }
be the neighborhoods of u0 . Let r3 = min{r1 , r2 }, α3 = min{α1 , α2 }. Consider U3 (u0 , r3 , α3 ) = {v ∈ S : Qu0 ,v,v (α3 ) > 1 − r3 , Qu0 ,u0 ,v (α3 ) > 1 − r3 }.
Clearly, u0 ∈ U3 (u0 , r3 , α3 ). Since r3 = min{r1 , r2 } ≤ r1 and α3 = min{α1 , α2 } ≤ α1 , and by Proposition 3.7, we have U3 (u0 , r3 , α3 ) ⊂ U1 (u0 , r1 , α1 ). Similarly, U3 (u0 , r3 , α3 ) ⊂ U2 (u0 , r2 , α2 ), and then \ U3 (u0 , r3 , α3 ) ⊂ U1 (u0 , r1 , α1 ) U2 (u0 , r2 , α2 ).
Now, we give the definitions of convergence of sequences, Q-CS and completeness in TFMS. Definition 3.9. Let (S, Q, ∗) be a TFMS. Then (1) A sequence {un } in S is said to be convergent to a point u ∈ S (write un → u), if for any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that un ∈ U (u, r, α) whenever n > Nα,r . (2) A sequence {un } in S is called a Q-CS, or simply CS, if for any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that Qun ,um ,ul (α) > 1 − r whenever m, n, l > Nα,r . (3) A TFMS is said to be Q-complete, or simply complete, if for any Q-CS in S, there exists a point u ∈ S such that the Q-CS is convergent to u. From (1) of Definition3.9 and (TFM-6), it is not difficult to obtain the proposition below. Proposition 3.10. Assume that (S, Q, ∗) is a TFMS and {un } in S is a sequence. Then the statements below are equivalent. (i) The sequence {un } in S is convergent to a point u ∈ S. (ii) As n → ∞, for all α > 0, Qun ,un ,u (α) → 1. (iii) As n → ∞, for all α > 0, Qun ,u,u (α) → 1.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Proposition 3.11. Assume that (S, Q, ∗) is a TFMS. Then the statements below are equivalent. (a) The sequence {un } in S is a Q-CS. 0 (b) For any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that 0 Qun ,um ,um (α) > 1 − r whenever n, m > Nα,r . Proof. That (a) implies (b) follows from (2) of Definition 3.9. Now we prove that (b) implies (a). Since ∗ is continuous, for every 0 1 − r. Let r0 = min r1 , 2r . Then ∗(1 − r0 , 1 − r0 ) > 1 − r. Hence, for any α2 > 0 and 0 < r0 < 1, from (b), there exists a positive integer 0 Nt,r such that Qun ,um ,um α2 > 1 − r0 and Qul ,um ,um α2 > 1 − r0 whenever 0 n, m, l > Nt,r . Then for any α > 0 and 0 < r < 1, we have α α , Qul ,um ,um Qun ,um ,ul (α) ≥ ∗ Qun ,um ,um 2 2 ≥ ∗(1 − r0 , 1 − r0 ) > 1 − r. Then, the sequence {un } is a Q-CS.
Proposition 3.12. Let (S, Q, ∗) be a TFMS and u, v, w ∈ S. Let {un }, {vn } and {wn } be sequences in S. If un → u, vn → v and wn → w as n → ∞, then, for any α > 0, Qun ,vn ,wn (α) → Qu,v,w (α) as n → ∞. Proof. For any α > 0, there exists a ξ > 0 satisfying α > 2ξ. By using (TFM-6), we find Qun ,vn ,wn (α) = Qun ,vn ,wn (ξ + α − ξ) ≥ ∗ Qun ,u,u (ξ), Qu,vn ,wn (α − ξ) 3ξ ξ = ∗ Qun ,u,u (ξ) , Qu,vn ,wn +α− 2 2 (15) ξ 3ξ ≥ ∗ Qun ,u,u (ξ) , ∗ Qvn ,v,v , Qv,u,wn α − 2 2 ξ ξ ≥ ∗ Qun ,u,u (ξ) , ∗ Qvn ,v,v , ∗ Qwn ,w,w , Qw,u,v (α − 2ξ) 2 2 and (16) Qu,v,w (α) = Qu,v,w (ξ + α − ξ) ≥ ∗ Qu,un ,un (ξ), Qun ,v,w (α − ξ) 3ξ ξ +α− = ∗ Qu,un ,un (ξ) , Qun ,v,w 2 2 ξ 3ξ ≥ ∗ Qu,un ,un (ξ) , ∗ Qv,vn ,vn , Qvn ,w,un α − 2 2 ξ ξ ≥ ∗ Qu,un ,un (ξ) , ∗ Qv,vn ,vn , ∗ Qw,wn ,wn , Qun ,vn ,wn (α − 2ξ) . 2 2
Letting n → ∞ in the inequalities (15) and (16), it follows from the continuity of ∗ that (17)
lim Qun ,vn ,wn (α) ≥ Qu,v,w (α − 2ξ)
n→∞
Tripled fuzzy metric spaces and fixed point theorem
11
and (18)
Qu,v,w (α) ≥ lim Qun ,vn ,wn (α − 2ξ). n→∞
Taking the limit as ξ → 0 in (17) and (18), and by the continuity of Q, we have for any α > 0 lim Qun ,vn ,wn (α) = Qu,v,w (α). n→∞
The proof of Theorem 3.12 is completed.
Remark 3.13. Some of the results given in Section 2 and Section 3 are inspired by some well-known conclusions in [11, 12, 36]. 4. FPTs in TFMS It is well known that one of the most important problems in FMS is the FPT, which is playing a very important and basic role in mathematical analysis and topology. For instance, it is used to determine existence and uniqueness of solutions of differential and integral equations. Grabiec in [13] extend firstly the classical Banach FPT to FMS in the sense of KM. Later on, Gregori and Sapena [15] presented another kind of concept on fuzzy contractive mapping (FCM) and studied its applicability to FPT in FMS. In their paper, they successfully extend the Banach FPT to FCM on complete FMS in KM’s sense and GV’s sense. Since then, many researchers have contributed to the study of the FPT in FMS. Recently, Mihet¸ [22] introduced an interesting concept called Fψ-C mapping, which enlarged the category of FCM which was due to Gregori and Sapena [15], and obtained an important FPT for the category of complete KM-fuzzy metrics. In 2016, George and Mi˜ nana [14] gave an important FPT under Fψ-C mapping in FMS in the sense of GV. As the consequence of the authors’ results, they presented a FPT due to Mihet¸ in [22] and generalized a FPT which is presented by Wardowski in [33]. Moreover, Zhou et. al [36] gave some new FPTs in generalized Menger probabilistic metric space. Inspired by the above works, in the section, we first enlarge the concept of Fψ-C mapping in FMS to TFMS which we called generalized Fψ-C mapping, and study its applicability to FPT in TFMS. Then, we obtain some new results including a FPT on GFψ-CM in TFMS. In this section, we suppose that Ψ is the category of all mappings ψ: ]0, 1] →]0, 1] satisfying ψ is continuous, non-decreasing and ψ(α) > α for any α ∈]0, 1[. Definition 4.1. (See Mihet¸ [22]) Let (S, M, ∗) be a FMS, and ψ ∈ Ψ. (I). If a mapping f : S → S such that P(f (u), f (v), α) ≥ ψ(P(u, v, α)), for all u, v ∈ S and α > 0, then f is called a Fψ-C mapping. (II) If a sequence {un } in S such that P(un+1 , un+2 , α) ≥ ψ(P(un , un+1 , α)), for all n ∈ N and α > 0, then {un } is called a Fψ-C sequence. Next, we give the notions of generalized Fψ-C mapping and generalized Fψ-C sequence.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Definition 4.2. Let (S, Q, ∗) be a TFMS, and ψ ∈ Ψ. (I) If a mapping q : S → S such that Qq(u),q(v),q(w) (α) ≥ ψ(Qu,v,w (α)), for all u, v, w ∈ S and α > 0, then q is called a generalized Fψ-C mapping. (II) If a sequence {un } in S such that Qun+1 ,un+2 ,un+3 (α) ≥ ψ(Qun ,un+1 ,un+2 (α)),
(19)
for all n ∈ N and α > 0, then {un } is called a generalized Fψ-C sequence. Remark 4.3. The notion of generalized Fψ-C sequence maybe defined as the following: A sequence {un } in S is called a generalized Fψ-C sequence if it satisfies Qun+1 ,un+2 ,un+2 (α) ≥ ψ(Qun ,un+1 ,un+1 (α)),
for all n ∈ N and α > 0.
Lemma 4.4. [14] If ψ ∈ Ψ, then for each α ∈]0, 1], limn ψ n (α) = 1. Lemma 4.5. Let (S, Q, ∗) V be a TFMS with ∗ positive,Vand let {un } be a generalized Fψ-C sequence in S. If α>0 Qu0 ,u1 ,u1 (α) > 0 and α>0 Qu1 ,u0 ,u0 (α) > 0, then {un } is a CS. V Proof. Assume that {un } is a generalized Fψ-C sequence in V S, and that α>0 Qu0 ,u1 ,u1 (α) = a > 0. We shall show that limn α>0 Qun ,un+1 ,un+1 (α) = 1. To do this, first we shall prove by mathematical induction that for each n ∈ N ^ Qun ,un+1 ,un+1 (α) ≥ ψ n (a). α>0
In fact, as n = 1, for each α > 0, it follows from (19) that
Qu1 ,u2 ,u2 (α) ≥ ψ(Qu0 ,u1 ,u1 (α)) ≥ ψ(a).
V
Then α>0 Qu1 ,u2 ,u2 V (α) ≥ ψ(a). Assume now that α>0 Qun ,un+1 ,un+1 (α) ≥ ψ n (a) holds for some fixed n ∈ N. From (19), the fact that ψ is non-decreasing, and the induction hypothesis we find that Qun+1 ,un+2 ,un+2 (α) ≥ ψ(Qun ,un+1 ,un+1 (α)) ! ^ ≥ψ Qun ,un+1 ,un+1 (α) α>0 n
≥ ψ (ψ (a)) .
V
Therefore α>0 Qun+1 ,un+2 ,un+2 (α) ≥ ψ n+1 (a). Thus, by mathematical induction we conclude that ^ (20) Qun ,un+1 ,un+1 (α) ≥ ψ n (a) α>0
holds for each n ∈ N. Letting n → ∞ on both sides of (20), and by Lemma 4.4 we can find that ^ lim Qun ,un+1 ,un+1 (α) ≥ lim ψ n (a) = 1. n
α>0
n
Tripled fuzzy metric spaces and fixed point theorem
Therefore, (21)
lim n
^
Qun ,un+1 ,un+1 (α)
α>0
!
= 1.
Making similar technique as in the proof of (21) with the condition 0, we have ! ^ (22) lim Qun+1 ,un ,un (α) = 1. n
13
V
α>0
Qu1 ,u0 ,u0 (α) >
α>0
Next we shall prove that {un } is a CS. For it, first we shall show by contradiction, that Qun ,um ,um (α) = 1. lim n,m→∞
Assume that there is a α0 > 0 such that Qun ,um ,um (α0 ) does not converge to 1 as n, m. Hence there exists an ∈]0, 1[ and sequences {n(j)} and {m(j)} such that n(j) > m(j) ≥ j and Qun (j),um (j),um (j) (α0 ) ≤ 1 − , j = 1, 2, . . . .
Fix j ∈ N. Then we can find n(j), m(j) ∈ N such that n(j) > m(j) ≥ j and Qun(j) ,um(j) ,um(j) (α0 ) ≤ 1 − . Given m(j), by choosing nm (j) to be the smallest integer exceeding m(j) and such that Qunm (j) ,um(j) ,um(j) (α0 ) ≤ 1 −
(23) holds. Then
Qunm (j)−1 ,um(j) ,um(j) (α0 ) > 1 − .
(24)
Now we shall show that lim Qunm (j)−1 ,um(j) ,um(j) (α0 ) = 1 − . j
By inequality (23) and (TFM-6), we find that (25)
1 − ≥ Qunm (j) ,um(j) ,um(j) (α0 )
≥ ∗ Qunm (j) ,unm (j)−1 ,unm (j)−1 (δ), Qunm (j)−1 ,um(j) ,um(j) (α0 − δ) ,
for each j ∈ N and each δ ∈]0, α0 [. Therefore, by inequality (25), the continuities of ∗ and Q, we have
1 − ≥ Qunm (j) ,um(j) ,um(j) (α0 ) ≥ lim ∗ Qunm (j) ,unm (j)−1 ,unm (j)−1 (δ), Qunm (j)−1 ,um(j) ,um(j) (α0 − δ) δ→0 = ∗ lim Qunm (j) ,unm (j)−1 ,unm (j)−1 (δ), lim Qunm (j)−1 ,um(j) ,um(j) (α0 − δ) δ→0 δ→0 ! ^ =∗ Qunm (j) ,unm (j)−1 ,unm (j)−1 (α) , Qunm (j)−1 ,um(j) ,um(j) (α0 ) , α>0
for each k ∈ N. Hence, (26)
lim sup Qunm (j) ,um(j) ,um(j) (α0 ) ≤ 1 − , j
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
and (27) lim inf Qunm (j) ,um(j) ,um(j) (α0 ) j
≥ lim ∗ j
= ∗ lim j
^
α>0
^
α>0
!! Qunm (j) ,unm (j)−1 ,unm (j)−1 (α) , Qunm (j)−1 ,um(j) ,um(j) (α0 )
! Qunm (j) ,unm (j)−1 ,unm (j)−1 (α) , lim Qunm (j)−1 ,um(j) ,um(j) (α0 ) j
≥ ∗(1, 1 − ) = 1 − . The second inequality of (27) has been obtained by inequality (24) and equality (22). From inequalities (26) and (27), we have 1 − ≥ lim sup Qunm (j) ,um(j) ,um(j) (α0 ) j
≥ lim inf Qunm (j) ,um(j) ,um(j) (α0 ) ≥ 1 − . j
Thus lim Qunm (j) ,um(j) ,um(j) (α0 ) = 1 − .
(28)
j
On the other hand, it follows by (TFM-6) and equality (28) that Qunm (j) ,um(j) ,um(j) (α0 ) = Qunm (j) ,um(j) ,um(j) (α0 − δ + δ)
≥ ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ), Qunm (j)+1 ,um(j) ,um(j) (α0 − δ)
(29)
= ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ), Qunm (j)+1 ,um(j) ,um(j) (α0 − 2δ + δ) ≥ ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ),
∗ Qunm (j)+1 ,um(j)+1 ,um(j)+1 (α0 − 2δ), Qum(j)+1 ,um(j) ,um(j) (δ)
≥ ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ),
! ∗ ψ Qunm (j) ,um(j) ,um(j) (α0 − 2δ) , Qum(j)+1 ,um(j) ,um(j) (δ) ,
for each j ∈ N and each δ ∈]0, α0 /2[. Letting δ → 0 on both sides of inequality (29), in a similar way that before, we get
(30)
Q(unm (j) ,um(j) ,um(j) (α0 ) ^ ≥∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (α) , α>0
! ^ ∗ ψ Qunm (j) ,um(j) ,um(j) (α0 ) , Qum(j)+1 ,um(j) ,um(j) (α) , α>0
Tripled fuzzy metric spaces and fixed point theorem
15
for each j ∈ N. By using (28), the continuities of ∗ and ψ, equalities (21) and (22), and taking limit as j tends to ∞ on both sides of (30), we have (31) 1 − = lim Qunm (j) ,um(j) ,um(j) (α0 ) j
≥ ∗ lim j
^
α>0
Qunm (j) ,unm (j)+1 ,unm (j)+1 (α) ,
! ^ ∗ ψ lim Qunm (j) ,um(j) ,um(j) (α0 ) , lim Qum(j)+1 ,um(j) ,um(j) (α) j
j
= ∗ 1, ∗ ψ(1 − ), 1 = ψ(1 − ) > 1 − ,
t>0
which is a contradiction (The second equality of (31) has been obtained by equalities (21), (28) and (31)). Hence lim Qun ,um ,um (α) = 1. n,m→∞
for any α > 0. Thus, it follows by Proposition 2.15 that {un } is a CS. The proof of Lemma 4.5 is completed. Theorem 4.6. Assume that (S, Q, ∗) is a complete TFMS with ∗ positive, and q : S → S is a generalized Fψ-C mapping. Then, V V q has a unique FP iff there is a u ∈ S satisfying α>0 Qu,q(u),q(u) (α) > 0 and α>0 Qq(u),u,u (α) > 0.
Proof. Assume that q has a unique FP, then there is a u ∈ S satisfying q(u) = u. Therefore for each α >V0, we have Qu,q(u),q(u) (α) = V Qu,u,u (α) = 1, Qq(u),u,u (α) = Qu,u,u (α) = 1, and so α>0 Qu,q(u),q(u) (α) = 1 and V α>0 Qq(u),u,u (α) = 1. Conversely, assume that there is a u ∈ S satisfying α>0 Qu,q(u),q(u) (α) > 0 and V n Q (α) > 0. Letting u = u and u = q (u) for each n ≥ 1, then we get 0 n q(u),u,u α>0 Qun+1 ,un+2 ,un+3 (α) = Qq(un ),q(un+1 ),q(un+2 ) (α) ≥ ψ(Qun ,un+1 ,un+2 (α)).
Therefore, {un } is a generalized Fψ-C sequence. Moreover ^ ^ Qu0 ,u1 ,u1 (α) = Qu,q(u),q(u) (α) > 0 α>0
and
^
α>0
Qu1 ,u0 ,u0 (α) =
α>0
^
Qq(u),u,u (α) > 0.
α>0
Then by Lemma 4.5, we get that {un } is a CS. Since (S, Q, ∗) is complete, there exists u ∈ S satisfying limn Qun ,u,u (α) = 1 for each α > 0. On the other hand, we find Qq(u),un+1 ,un+1 (α) ≥ ψ(Qu,un ,un (α))
for each n ∈ N and each α > 0. So Qq(u),u,u (α) = lim Qq(u),un+1 ,un+1 (α) n
≥ lim ψ(Qu,un ,un (α)) = 1 n
for each α > 0. Thus, u is a FP of q.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Next we shall verify that u is the unique FP of q. If v is another FP of q, then for any α > 0 Qu,u,v (α) = Qq(u),q(u),q(v) (α) ≥ ψ (Qu,u,v (α)) .
If u 6= v, then for some β > 0
Qu,u,v (β) < 1,
that is, 0 < Qu,u,v (β) < 1. Then Qu,u,v (β) = Qq(u),q(u),q(v) (β) ≥ ψ (Qu,u,v (β)) > Qu,u,v (β), which is a contradiction. Thus, we conclude that u = v. Hence, q has a unique FP in S, which means that Theorem 4.6 holds true. Remark 4.7. The V conditions of Theorem 4.6 involve V the self-mapping (there is a u ∈ S such that α>0 Qu,q(u),q(u) (α) > 0 and α>0 Qq(u),u,u (α) > 0). It is expected that there are not any restrictions on self-mapping in the study of existence of a FP since the expression could be complex. In fact, the following two corollaries can be obtained immediately from Lemma 4.5 and Theorem 4.6, and the demanded conditions do not involve self-mapping. V Corollary V 4.8. Let (S, Q, ∗) be a TFMS with ∗ positive, and let α>0 Qu,v,v (α) > 0 and α>0 Qv,u,u (α) > 0 for each u, v ∈ S. Then, any generalized Fψ-C sequence in S is a CS. Corollary 4.9. Assume V V that (S, Q, ∗) is a complete TFMS with ∗ positive, and α>0 Qu,v,v (α) > 0, α>0 Qv,u,u (α) > 0 for each u, v ∈ S. If q : S → S is a generalized Fψ-C mapping, then q has a unique FP. Remark 4.10. If the notion of generalized Fψ-C sequence is defined as the one in Remark 4.3, then we can prove that Lemma 4.5 and Theorem 4.6 hold in a similar way. However, it should be point out that the Lemma of Gregori and Mi˜ nana [14, Lemma 3.2] and the Theorem of Gregori and Mi˜ nana [14, Theorem 3.3] can be followed from Example 2.11 and the definition in Remark 4.3 by a new proving technique introduced by Zhou et al. in [36]. The corollaries are listed as follows. Corollary 4.11. (Gregori and Mi˜ nana [14, Lemma 3.2]) Assume that (S, P, ∗) is a V FMS with ∗ positive, and {un } is a Fψ-C sequence in S. If α>0 P(u0 , u1 , α) > 0, then {un } is a CS. Proof. Let Qu,v,w (α) = min{P(u, v, α), P(v, w, α), P(u, w, α)} for u, v, w ∈ S, α > 0. From Example 2.11, we know that (S, Q, ∗) is a TFMS. Since Q un+1 ,un+2 ,un+2 (α)
= min{P(un+1 , un+2 , α), P(un+2 , un+2 , α), P(un+2 , un+1 , α)} = P(un+1 , un+2 , α) ≥ ψ P(un , un+1 , α) = ψ min{P(un , un+1 , α), P(un+1 , un+1 , α), P(un+1 , un , α)} = ψ Qun ,un+1 ,un+1 (α) ,
Tripled fuzzy metric spaces and fixed point theorem
that is
17
Qun+1 ,un+2 ,un+2 (α) ≥ ψ Qun ,un+1 ,un+1 (α) ,
for n ∈ N, α > 0. Thus {un } is a generalized Fψ-C sequence. On the other hand,
P(u0 , u1 , α) = min{P(u0 , u1 , α), P(u1 , u1 , α), P(u1 , u0 , α)}
(32)
= Qu0 ,u1 ,u1 (α)
and P(u1 , u0 , α) = min{P(u1 , u0 , α), P(u0 , u0 , α), P(u0 , u1 , α)}
(33)
From (32), (33) and ^
V
α>0
= Qu1 ,u0 ,u0 (α).
P(u0 , u1 , α) > 0, we find that ^ Qu0 ,u1 ,u1 (α) > 0 and Qu1 ,u0 ,u0 (α) > 0. α>0
α>0
Therefore, from Lemma 4.5, we obtain that {un } is a CS in the sense of TFMS (S, Q, ∗). That is, for any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that Qun ,um ,ul (α) > 1 − r for all n, m, l > Nα,r . It follows from the definition of Qu,v,w (α) that min{P(un , um , α), P(um , ul , α), P(un , ul , α)} > 1 − r
for n, m, l > Nα,r . This shows that {un } is a CS in the sense of FMS (S, P, ∗).
By the same method as in Corollary 4.11, we can obtain the following FPT in the sense of FMS (S, P, ∗). Corollary 4.12. (Gregori and Mi˜ nana [14, Theorem 3.3]) Assume that (S, P, ∗) is a complete FMS with ∗ positive, and f : S → V S is a Fψ-C mapping. Then, f has a unique FP iff there is an u ∈ S such that α>0 P(u, f (u), α) > 0. Finally, we give a simple example to illustrate Theorem 4.6.
Example 4.13. Let S =]0, ∞[ and ∗ = ∗p . Define Q : S 3 ×]0, ∞[→]0, 1] by Qu,v,w (α) =
min{u2 , v 2 , w2 } + α max{u2 , v 2 , w2 } + α
for u, v, w ∈ S. It is easy to check that (S, Q, ∗) is a TFMS and min{u2 ,v 2 ,w2 } max{u2 ,v 2 ,w2 } 1 2
> 0. Define a function q : S → S by q(u) = u
1 2
V
α>0
Qu,v,w (α) =
for u ∈ S, and let
ψ(•) = • . Then, from Theorem 4.6, we obtain that q has a unique FP in S. In fact, the FP is u = 1. 5. Conclusions As we know, FM and FPT have been applied in many fields those related information sciences. For example, in dynamical system, fuzzy FPT can be applied in existence, uniqueness and continuity of solution with some vague parameters [10, 23]. In fuzzy game theory, fuzzy FPT or fuzzy common FPT can be used in the proof of the existence of equilibrium solution [7]. In this paper, we present a new FMS–TFMS, which is a new generalization of GV’s FMS, and then give some related examples, topological properties, convergence of sequences, CS and completeness of the TFMS. Moreover, we introduce two kinds of notions of generalized
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Fψ-C mappings, and obtain a FPT on the mappings in the space. We expect that our research results can offer a mathematical basis for the ongoing research on information science. In the future research, we will explore the concrete applications in the field of information science of the obtained results. 6. Acknowledgement The authors would like to express their sincere thanks to the anonymous referees for their making great efforts to improve this paper. References [1] T. Bag, S. K. Samanta, Fixed point theorems on fuzzy normed linear spaces, Inform. Sci., 176 (2006), 2910-2931. [2] T. Bag, S. K. Samanta, Some fixed point theorems in fuzzy normed linear spaces, Inform. Sci., 177 (2007), 3271-3289. [3] S.-M. Chen, S. I. Adam, Weighted fuzzy interpolated reasoning based on ranking values of polygonal fuzzy sets and new scale and move transformation techniques, Inform. Sci., 435 (2018), 184-202. [4] S.-M. Chen, D. Barman, Adaptive weighted fuzzy interpolative reasoning based on representative values and similarity measures of interval type-2 fuzzy sets, Inform. Sci., 478 (2019), 167-185. [5] S.-M. Chen, S.-H. Cheng, C.-H. Chiou, Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology, Inform. Fusion, 478 (2016), 215-227. [6] S.-M. Chen, X.-Y. Zou, D. Barman, Adaptive weighted fuzzy rule interpolation based on ranking values and similarity measures of rough-fuzzy sets, Inform. Sci., 488 (2019), 93-110. [7] H. Chu, J. Hsieh, Fuzzy differential game of guarding a movable territory, Inform. Sci., 176 (1996), 113-131. [8] Z.-K. Deng, Fuzzy pseudo metric spaces, J. Math. Anal. Appl., 86 (1982), 74-95. [9] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69 (1979), 205-230. [10] W. Fei, Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients, Inform. Sci., 177 (2007), 4329-4337. [11] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), 395-399. [12] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst., 90 (1997), 365-368. [13] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 27 (1988), 385-389. [14] V. Gregori, J.-J. Mi˜ nana, On fuzzy ψ-contractive sequences and fixed point theorems, Fuzzy Sets Syst., 300 (2016), 245-252. [15] V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets Syst., 125 (2002), 245-252. [16] O. Had˘zi´c, E. Pap, Fixed Point Theory in Probabilistic Metric Space, Kluwer Academic Publishers, Dordrecht, 2001. [17] I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 15 (1975), 326-334.
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[18] T. Kubiak, D. Zhang, On the L-fuzzy Brouwer fixed point theorem, Fuzzy Sets Syst., 105 (1999), no. 2, 287-292. [19] W. Li, W. Pedrycz, X. Xue, W. Xu, B. Fan, Distance-based double-quantitative rough fuzzy sets with logic operations, Int. J. Approx. Reason., 101 (2018), 206233. [20] Y. Liu and D. Zhang, Lowen spaces, J. Math. Anal. Appl., 241 (2000), 30-38. [21] K. Menger, Statistical metrics, Proc. Natl. Acad. Sci. USA, 28(1942), 535-537. [22] D. Mihet¸, Fuzzy ψ-contractive mappings in non-Archimedean fuzzy metric spaces, Fuzzy Sets Syst., 159 (2008), 739-744. [23] M. T. Mizukoshi, L. C. Barros, Y. Chalco-Canoc, H. Rom´ an-Flores, R. C. Bassanezi, Fuzzy differential equations and the extension principl, Inform. Sci., 177 (2007), 3627-3635. [24] L. T. Ngo, T. H. Dang, W. Pedrycz, Towards interval-valued fuzzy set-based collaborative fuzzy clustering algorithms, Pattern Recogn., 81 (2018), 404-416. [25] D. Qiu, L. Shu, Supremum metric on the space of fuzzy sets and common fixed point theorems for fuzzy mappings, Inform. Sci., 178 (2008), 3595-3604. [26] R. Mesiar, Fuzzy set approach to the utility, preference relations, and aggregation operators, Eur. J. Oper. Res., 176 (2007), 414-422. R. Mesiar [27] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983. [28] L. Spada, An expansion of Basic Logic with fixed points, Soft Comput., 21 (2017), 29-37. [29] P. V. Subrahmanyam, A common fixed point theorem in fuzzy metric spaces, Inform. Sci., 83 (1995), 109-112. [30] J.-F. Tian, X.-M. Hu, H.-S. Zhao, Common tripled fixed point theorem for Wcompatible mappings in fuzzy metric spaces, J. Nonlinear Sci. Appl., 9 (2016), 806-818. [31] E. Trillas, On the use of words and fuzzy sets, Inform. Sci., 176 (2006), 14631487. [32] J.-H. Wang, F.-Y. Meng, L.-P. Pang, X.-H. Hao, An adaptive fixed-point proximity algorithm for solving total variation denoising models, Inform. Sci., 402 (2017), 69-81. [33] D. Wardowski, Fuzzy contractive mappings and fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 222 (2013), 108-114. [34] L. A. Zadeh, Fuzzy sets, Inform. Control, 8 (1965), 338-353. [35] D. Zhang, Sobriety of quantale-valued cotopological spaces, Fuzzy Sets Syst., 350 (2018), 1-19. ´ c, S. M. Alsulami, Generalized probabilistic metric [36] C. Zhou, S. Wang, L. Ciri´ spaces and fixed point theorems, Fixed Point Theory Appl., 2014 (2014): 91.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Declaration of Competing Interest The authors declare that they have no competing interests. Jing-Feng Tian, Department of Mathematics and Physics, North China Electric Power University, Baoding, Hebei Province, 071000, P. R. China E-mail address: [email protected] Ming-Hu Ha, School of Science, Hebei University of Engineering, Handan, Hebei Province, 056038, P. R. China, and College of Mathematics and Information Science, Hebei University, Baoding, Hebei Province, 071000, P. R. China E-mail address: [email protected] Da-Zeng Tian, College of Mathematics and Information Science, Hebei University, Baoding, Hebei Province, 071002, P. R. China E-mail address: [email protected]
Journal Pre-proof
Tripled fuzzy metric spaces and fixed point theorem Jing-Feng Tian, Ming-Hu Ha, Da-Zeng Tian PII: DOI: Reference:
S0020-0255(20)30006-2 https://doi.org/10.1016/j.ins.2020.01.007 INS 15138
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Information Sciences
Received date: Revised date: Accepted date:
20 March 2019 26 September 2019 6 January 2020
Please cite this article as: Jing-Feng Tian, Ming-Hu Ha, Da-Zeng Tian, Tripled fuzzy metric spaces and fixed point theorem, Information Sciences (2020), doi: https://doi.org/10.1016/j.ins.2020.01.007
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TRIPLED FUZZY METRIC SPACES AND FIXED POINT THEOREM JING-FENG TIAN, MING-HU HA*, AND DA-ZENG TIAN
Abstract. One of the most important topics of research in fuzzy sets is to get an appropriate notion of fuzzy metric space (FMS), in the paper we propose a new FMS–tripled fuzzy metric space (TFMS), which is a new generalization of George and Veeramani’s FMS. Then we present some related examples, topological properties, convergence of sequences, Cauchy sequence (CS) and completeness of the TFMS. Moreover, we introduce two kinds of notions of generalized fuzzy ψ-contractive (Fψ-C) mappings, and derive a fixed point theorem (FPT) on the mappings in the space.
1. Introduction The fuzzy sets [34] plays a very significant role in fuzzy literature, which has been given and studied in many different directions by many researchers[3–6, 19, 20, 24, 26, 31, 35]. One of the most important topics of research in fuzzy sets is to get an appropriate notion of fuzzy metric space (FMS). The topic has been investigated by some researchers in different ways. In 1975, Kramosil and Michalek [17] (KM) initiated the notion of fuzzy metric, a fuzzy set (FS) in the product S × S × [0, ∞[ satisfying some certain conditions. Later, Deng [8] in 1984 presented the notion of fuzzy pseudo metric space, with the metric defined between two fuzzy points, and obtained the fuzzy topological structure and the fuzzy uniform structure on this space. To obtain a Hausdorff topology for FMS introduced by KM, George and Veeramani [11] (GV) in 1994 reintroduced the notion of FMS by modifying the definition of KM’s FMS. Later, the authors in another paper [12] gave a necessary and sufficient condition for the completeness of FMS. Since then, new properties, various fixed point (FP) results for mappings satisfying different contractive conditions and artful applications in both context of FMS above mentioned were established by many researchers[1, 2, 25, 28–30, 32]. Inspired by the above works, the main purpose of the paper is to give a new FMS– tripled fuzzy metric space. The rest of this paper is organized as follows. Section 2 firstly introduces relevant concepts and results applied in developing results of this study, then, it presents a new generalization of FMS, namely, tripled fuzzy metric space. Next, some related examples with particular emphasis on the generalization of FMS are demonstrated in this section. Section 3, based on some topological properties of the studied space, gives the notions of convergence sequence, CS and completeness in TFMS. Section 4 offers two kinds of notions of generalized Fψ-C (2000) Mathematics subject classification: 54A40, 54E35. Key words and phrases: Fuzzy sets; tripled fuzzy metric spaces; Hausdorff topology; fuzzy contractive mapping; fixed point. *Corresponding author: Ming-Hu Ha, e-mail: [email protected]. 1
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
mappings and presents some results such as a FPT on the mappings in TFMS. Finally, a brief summary is given in Section 5. 2. Tripled fuzzy metric spaces The definitions below will be needed in the sequel. Definition 2.1. ([16], [27]) A mapping ∗ : [0, 1]×[0, 1] → [0, 1] is called a continuous t-norm (CTN) if satisfies: (i) ∗(η, ϑ) = ∗(ϑ, η) and ∗(ϑ, ∗(η, κ)) = ∗(∗(ϑ, η), κ), for 0 ≤ ϑ, η, κ ≤ 1, (ii) ∗(ϑ, 1) = ϑ, for 0 ≤ ϑ ≤ 1, (iii) ∗(ϑ, η) ≤ ∗(κ, θ) whenever ϑ ≤ κ and η ≤ θ, for 0 ≤ ϑ, η, κ, θ ≤ 1, (iv) ∗ is continuous. From Definition 2.1 we find that for 0 ≤ ϑ, η ≤ 1, ∗(ϑ, η) ≤ min{ϑ, η} and ∗m (ϑ, η) = min{ϑ, η}, ∗p (ϑ, η) = ϑη are CTNs. Definition 2.2. [27] If ϑ ∗ η > 0 whenever ϑ, η ∈ (0, 1], then we call the t-norm ∗ is positive. It is obviously that for 0 < ϑ, η ≤ 1, ∗m (ϑ, η) = min{ϑ, η} and ∗p (ϑ, η) = ϑη are positive. Definition 2.3. ([17]) Let S be an arbitrary set, let ∗ be a CTN, and let P be a FS on S × S × [0, ∞[ such that for all u, v, w ∈ S, the following conditions are valid: (KM-1) P(u, v, 0) = 0, (KM-2) P(u, v, α) = 1 for α > 0 iff u = v, (KM-3) P(u, v, α) = P(v, u, α) for α > 0, (KM-4) P(u, v, ·) : [0, ∞[→ [0, 1] is left-continuous, (KM-5) P(u, w, α + β) ≥ ∗(P(u, v, α), P(v, w, β)) for α, β ≥ 0. Then the triple (S, P, ∗) is called a FMS (in the sense of KM). In order to obtain a Hausdorff topology (HT) of FMS, GV [11] modified the definition of FMS presented by KM as follows. Definition 2.4. ([11]) Let S be an arbitrary set, let ∗ be a CTN, and let P be a FS on S × S×]0, ∞[ such that for all u, v, w ∈ S and α, β > 0, the following conditions valid: (GV-1) P(u, v, α) > 0, (GV-2) P(u, v, α) = 1 iff u = v, (GV-3) P(u, v, α) = P(v, u, α), (GV-4) P(u, v, ·) :]0, ∞[→]0, 1] is continuous, (GV-5) P(u, w, α + β) ≥ ∗(P(u, v, α), P(v, w, β)). Then the triple (S, P, ∗) is called a FMS (in the sense of GV). Theorem 2.5. ([13]) Let (S, P, ∗) be a FMS. Then for all u, v ∈ S, P(u, v, ·) is nondecreasing. In this paper, the FMS (S, P, ∗) is in the sense of GV. Now, we will give the following notion of TFMSs. Definition 2.6. A triple (S, Q, ∗) is said to be a TFMS if S is an arbitrary set, ∗ is a CTN and Q is a FS on S × S × S×]0, ∞[ such that for all u, v, w, z ∈ S the following conditions valid: (TFM-1) Qu,v,w (α) > 0;
Tripled fuzzy metric spaces and fixed point theorem
(TFM-2) (TFM-3) (TFM-4) (TFM-5) (TFM-6)
3
Qu,v,w (α) = 1 iff u = v = w; Qu,u,v (α) ≥ Qu,v,w (α) for w 6= v; Qu,v,w (α) = Qu,w,v (α) = Qv,u,w (α) = · · · ; Qu,v,w (·) :]0, ∞[→]0, 1] is continuous; Qu,v,w (α + β) ≥ ∗(Qu,z,z (α), Qz,v,w (β)).
Remark 2.7. Let (S, Q, ∗) be a TFMS. Then the mapping P : S×S×]0, ∞[→]0, 1] defined by (1) P(u, v, α) = ∗ Qu,v,v (α), Qv,u,u (α)
is a fuzzy metric (FM). Indeed, it is not difficult to check that P satisfies (GV-1)(GV-4). Next, we prove P satisfies the condition (GV-5). In fact, for all β, α > 0 and u, v, z ∈ S, we find (2) P(u, v, β + α) = ∗ Qu,v,v (β + α), Qv,u,u (β + α) , and
∗ P(u, w, β), P(w, v, α) = ∗ ∗ Qu,w,w (β), Qw,u,u (β) , ∗ Qw,v,v (α), Qv,w,w (α) . From (TFM-6) and (iii) of Definition 2.1, we get ∗ Qu,v,v (β + α), Qv,u,u (β + α) ≥ ∗ ∗ Qu,w,w (β), Qw,v,v (α) , ∗ Qv,w,w (α), Qw,u,u (β) . Since
(3) (4) (5) and (6)
Qu,w,w (β) ≥ min{Qu,w,w (β), Qw,u,u (β)} ≥ ∗ Qu,w,w (β), Qw,u,u (β) , Qw,v,v (α) ≥ min{Qw,v,v (α), Qv,w,w (α)} ≥ ∗ Qw,v,v (α), Qv,w,w (α) ,
Qv,w,w (α) ≥ min{Qw,v,v (α), Qv,w,w (α)} ≥ ∗ Qw,v,v (α), Qv,w,w (α) ,
Qw,u,u (β) ≥ min{Qu,w,w (β), Qw,u,u (β)} ≥ ∗ Qu,w,w (β), Qw,u,u (β) ,
by using (TFM-6), (3), (4), (5), (6) and (1), we have Qu,v,v (β + α) ≥ ∗ Qu,w,w (β), Qw,v,v (α) (7) ≥ ∗ ∗ Qu,w,w (β), Qw,u,u (β) , ∗ Qw,v,v (α), Qv,w,w (α) = ∗ P(u, w, β), P(w, v, α) and (8)
Qv,u,u (β + α) ≥ ∗ Qv,w,w (α), Qw,u,u (β) ≥ ∗ ∗ Qw,v,v (α), Qv,w,w (α) , ∗ Qu,w,w (β), Qw,u,u (β) = ∗ P(w, v, α), P(u, w, β) .
Therefore, from (7), (8) and (2), we get
P(u, v, β + α) = ∗ Qu,v,v (β + α), Qv,u,u (β + α) ≥ ∗ P(u, w, β), P(w, v, α) .
This shows that P satisfies (GV-5), and thus P is a FM.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Example 2.8. Let S = R. Define Q : S × S × S×]0, ∞[→]0, 1] by
Qu,v,w (α) = exp
|u − v| + |v − w| + |w − u| −1 α
for all u, v, w ∈ S and α > 0. If we set ∗ = ∗p , then (S, Q, ∗) is a TFMS. Proof. It is not difficult to prove that Q satisfies (TFM-1)-(TFM-5). Now we prove that for all u, v, w, z ∈ S and all β, α > 0 Qu,v,w (β + α) ≥ ∗(Qu,z,z (β), Qz,v,w (α)). Indeed, since |u − v| + |v − w| + |w − u| |u − z| + |z − v| + |v − w| + |w − z| + |z − u| ≤ β+α β+α 2|u − z| |z − v| + |v − w| + |w − z| + = β+α β+α 2|u − z| |z − v| + |v − w| + |w − z| < + , β α we have |u − v| + |v − w| + |w − u| β+α Qu,v,w (β + α) = e 2|u − z| |z − v| + |v − w| + |w − z| + − β α ≥e −
= Qu,z,z (β)Qz,v,w (α)
= ∗(Qu,z,z (β), Qz,v,w (α)). Hence (S, Q, ∗) is a TFMS.
Remark 2.9. It is obvious that the above (S, Q, ∗) is not a FMS. Example 2.10. Let (S, P, ∗) be a FMS. For u, v, w ∈ S and α > 0, define Q : S × S × S×]0, ∞[→]0, 1] by Qu,v,w (α) = ∗ ∗ P(u, v, α), P(v, w, α) , P(u, w, α) .
Then (S, Q, ∗) is a TFMS.
Proof. It is easy to see that Q satisfies (TFM-1)-(TFM-5). In order to verify that Q satisfies (TFM-6), we need to verify only that (9)
Qu,v,w (β + α) ≥ ∗ Qu,z,z (β), Qz,v,w (α) ,
Tripled fuzzy metric spaces and fixed point theorem
for u, v, w, z ∈ S and β, α > 0, that is, ∗ ∗ P(u, v, β + α), P(v, w, β + α) , P(u, w, β + α) ≥ ∗ ∗ ∗ P(u, z, β), P(z, z, β) , P(z, u, β) , ∗ ∗ P(z, v, α), P(v, w, α) , P(z, w, α) (10) = ∗ ∗ P(u, z, β), P(z, u, β) , ∗ ∗ P(z, v, α), P(v, w, α) , P(z, w, α) . From (GV-5), we find that
P(u, v, β + α) ≥ ∗ P(u, z, β), P(z, v, α) ≥ ∗ min P(u, z, β), P(u, z, β) , (11)
o min min P(z, v, α), P(v, w, α) , P(z, w, α) n
≥ ∗ ∗ P(u, z, β), P(u, z, β) , ∗
∗ P(z, v, α), P(v, w, α) , P(z, w, α) ,
P(u, w, β + α) ≥ ∗ P(u, z, β), P(z, w, α) ≥ ∗ min P(u, z, β), P(u, z, β) , (12)
o min min P(z, v, α), P(v, w, α) , P(z, w, α) n
≥ ∗ ∗ P(u, z, β), P(u, z, β) , ∗
and
∗ P(z, v, α), P(v, w, α) , P(z, w, α)
P(v, w, β + α) ≥ P(v, w, α)
(13)
≥ min{P(z, v, α), P(v, w, α), P(z, w, α)} ≥ ∗ P(u, z, β), min{P(z, v, α), P(z, w, α), P(v, w, α)} = ∗ P(u, z, β), min min{P(z, v, α), P(z, w, α)}, P(v, w, α) ≥ ∗ ∗ P(u, z, β), P(u, z, β) , ∗ ∗ P(z, v, α), P(v, w, α) , P(z, w, α) .
5
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
By using (11), (12) and (13), we obtain that (10), i.e., (9) holds. Therefore, Q satisfies (TFM-6) and thus (S, Q, ∗) is a TFMS. Letting ∗ = ∗m in Example 2.10, we get the following result. Example 2.11. Let (S, P, ∗m ) be a FMS. For u, v, w ∈ S and α > 0, define Q : S × S × S×]0, ∞[→]0, 1] by Qu,v,w (α) = min{P(u, v, α), P(v, w, α), P(u, w, α)}.
Then (S, Q, ∗m ) is a TFMS.
|u−v|
Remark 2.12. Let S = R, ∗ = ∗m , P(u, v, α) = e− α , where u, v ∈ S, α > 0. Then from [11, Example 2.7 and Remark 2.8] we know that (S, P, ∗) is a FMS. From Example 2.10, we can find that for ∗ = ∗m |u−v| |v−w| |w−u| Qu,v,w (α) = ∗ ∗ e− α , e− α , e− α
is a tripled fuzzy metric and hence (S, Q, ∗) is a TFMS.
Remark 2.13. Let (S, d) be a metric space, let ∗ = ∗p , and let P(u, v, α) =
kαn , kαn + md(u, v)
k, m, n ∈ R+ . Then from [11, Example 2.9] we know that (S, P, ∗) is a FMS. From Example 2.10, we obtain that kαn kαn kαn Qu,v,w (α) = ∗ ∗ , , kαn + md(u, v) kαn + md(v, w) kαn + md(w, u)
is a tripled fuzzy metric and hence (S, Q, ∗) is a TFMS.
Example 2.14. Let (S, P, ∗m ) be a FMS, and let λ1 , λ2 , λ3 > 0 with λ1 +λ2 +λ3 = 1. For u, v, w, ∈ S, α > 0, define a map Q : S × S × S×]0, ∞[→]0, 1] by (14)
Qu,v,z (α) = min{P(u, v, λ1 α), P(v, z, λ2 α), P(u, z, λ3 α)}.
Then (S, Q, ∗m ) is a TFMS. Proof. It is not difficult to obtain that Q satisfies (TFM-1)-(TFM-5). Next, we verify that Q satisfies (TFM-6). From (14), (GV-5) and Theorem 2.5, we find that Qu,v,w (α + β) = min {P (u, v, λ1 (α + β)) , P (v, w, λ2 (α + β)) , P (u, w, λ3 (α + β))} ≥ min {P (u, v, λ1 (α + β)) , P (v, w, λ2 β) , P (u, w, λ3 (α + β))} ≥ min {∗ (P (u, z, λ1 α) , P (z, v, λ1 β)) , P (v, w, λ2 β) , ∗ (P (u, z, λ3 α) , P (z, w, λ3 β))}
= min {min {P (u, z, λ1 α) , P (z, v, λ1 β)} , P (v, w, λ2 β) , min {P (u, z, λ3 α) , P (z, w, λ3 β)}}
= min {min {P (u, z, λ1 α) , P (z, z, λ2 α) , P (u, z, λ3 α)} ,
min {P (z, v, λ1 β) , P (v, w, λ2 β) , P (z, w, λ3 β)}}
= min {Qu,z,z (α), Qz,v,w (β)} = ∗ (Qu,z,z (α), Qz,v,w (β)) .
Thus Q satisfies (TFM-6) and then (S, Q, ∗m ) is a TFMS.
Tripled fuzzy metric spaces and fixed point theorem
7
Proposition 2.15. Let (S, Q, ∗) be a TFMS. Then for all u, v ∈ S, Qu,v,v (·) is nondecreasing. Proof. For some β > α > 0, we assume Qu,v,v (α) > Qu,v,v (β). Then Qu,v,v (α) > Qu,v,v (β) = Qu,v,v (α + β − α) ≥ ∗(Qu,v,v (α), Qv,v,v (β − α)). From (TFM-2), we get Qv,v,v (β − α) = 1, and thus
Qu,v,v (α) > Qu,v,v (β) ≥ Qu,v,v (α)
which is a contradiction. Hence the proof of Proposition 2.15 is completed.
3. Topology of TFMS In the section, we first present the notion of neighborhood in the TFMS. Definition 3.1. Let (S, Q, ∗) be a TFMS, let r ∈]0, 1[, α > 0, and let u0 ∈ S. Then the set U (u0 , r, α) = {v ∈ S : Qu0 ,v,v (α) > 1 − r, Qu0 ,u0 ,v (α) > 1 − r} is called the neighborhood with center u0 and radius r with respect to α. Theorem 3.2. Every neighborhood U (u0 , r, α) is an open set. Proof. Let U (u0 , r, α) = {u ∈ S : Qu0 ,u,u (α) > 1 − r, Qu0 ,u0 ,u (α) > 1 − r} be a neighborhood with center u0 and radius r with respect to α. If v ∈ U (u0 , r, α), then we claim that there is a neighborhood of v, Uv , satisfying Uv ⊂ U (u0 , r, α). In fact, since v ∈ U (u0 , r, α), Qu0 ,v,v (α) > 1 − r,
Qu0 ,u0 ,v (α) > 1 − r.
Now, Qu0 ,v,v is continuous at α, so we can obtain that α0 < α and r0 < r such that Qu0 ,v,v (α0 ) > 1 − r0 > 1 − r, Qu0 ,u0 ,v (α0 ) > 1 − r0 > 1 − r. Let Uv = {z ∈ S : Qv,z,z (α1 ) > 1 − r1 , Qv,v,z (α1 ) > 1 − r1 }, where 0 < α1 < α − α0 and r1 is chosen such that ∗(1 − r0 , 1 − r1 ) > 1 − r.
Such a r1 exists since ∗ is continuous, ∗(a, 1) = a for all a ∈ [0, 1] and 1 − r0 > 1 − r. Now, assume that s ∈ Uv , so we have Qv,s,s (α1 ) > 1 − r1 ,
Qv,v,s (α1 ) > 1 − r1 .
By using (TFM-6) and Proposition 2.15, we obtain Qu0 ,s,s (α) ≥ ∗(Qu0 ,v,v (α0 ), Qv,s,s (α − α0 )) ≥ ∗(Qu0 ,v,v (α0 ), Qv,s,s (α1 )) ≥ ∗(1 − r0 , 1 − r1 ) > 1 − r.
Similarly, we also get Qu0 ,u0 ,s (α) > 1 − r. This shows s ∈ U (u0 , r, α) and thus Uv ⊂ U (u0 , r, α). The proof of Theorem 3.2 is completed. Remark 3.3. Let (S, Q, ∗) be a TFMS. Define τ = {A ⊂ S : for any u ∈ A , there exist α > 0 and r ∈ (0, 1) satisfying U (u, r, α) ⊂ A }. Then τ is a topology on S. Moreover, as {U (x, n1 , n1 ) : n = 1, 2, . . . , } is a local base at x, the topology τ is first countable. Theorem 3.4. Every TFMS is Hausdorff.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Proof. Let (S, Q, ∗) be a TFMS, u, v ∈ S and u 6= v. Then there are t > 0 0 00 0 00 0 00 and r , r with 0 < r , r < 1 satisfying Qx,y,y (t) = r and Qu,u,v (α) = r . Since ∗ is continuous, monotone, and ∗(1, 1) = 1, we can find r1 and r2 such that 0 00 0 < r1 , r2 < 1 and ∗(1 − r1 , 1 − r2 ) > r, where r = max{r , r }. Now, we consider the following two neighborhoods Uu and Uv defined by n α α o Uu = z : Qu,z,z > 1 − r1 , Qu,u,z > 1 − r1 , 2 2 and n α α o Uv = z : Qv,z,z > 1 − r2 , Qv,v,z > 1 − r2 . 2 2 T T Clearly Uu Uv = ∅. In fact, if there exists a point s ∈ Uu Uv such that α α α α Qu,s,s > 1−r1 , Qu,u,s > 1−r1 , Qv,s,s > 1−r2 , Qv,v,s > 1−r2 , 2 2 2 2 then by (TFM-6), we have α α 0 0 r = Qu,v,v (α) ≥ ∗ Qu,s,s , Qs,v,v > ∗(1 − r1 , 1 − r2 ) > r ≥ r 2 2 and α α 00 00 , Qs,u,u > ∗(1 − r2 , 1 − r1 ) > r ≥ r , r = Qv,u,u (α) ≥ ∗ Qv,s,s 2 2 which are contradictions. Therefore, Uu and Uv are disjoint and thus (S, Q, ∗) is Hausdorff. Definition 3.5. Let (S, Q, ∗) be a TFMS. A set A ⊂ S is called F-bounded (FBed) if and only if there are α > 0 and 0 < r < 1 satisfying Qu,v,v > 1 − r and Qu,u,v > 1 − r for any u, v ∈ A . Theorem 3.6. Let (S, Q, ∗) be a TFMS. Then every compact subset A of S is FBed. Proof. Let A be a given compact subset of S. Fixing α1 > 0, 0 < r1 < 1, and considering an open cover {U (u, r1 , α1 ) : uS∈ A } of A , as A is compact, we can find u1 , u2 , . . . , un ∈ A such that A ⊆ U (ui , r1 , t1 ). If u, v ∈ A , then for some i, j, u ∈ U (ui , r1 , α1 ) and v ∈ U (uj , r1 , α1 ). Therefore Qui ,u,u (α1 ) > 1 − r1 , Qui ,ui ,u (α1 ) > 1 − r1 , Quj ,v,v (α1 ) > 1 − r1 and Quj ,uj ,v (α1 ) > 1 − r1 . Let and
ζ = min{Qui ,uj ,uj (α1 ) : 1 ≤ i, j ≤ n}
β = min{Qui ,ui ,uj (α1 ) : 1 ≤ i, j ≤ n}. Then ζ, β > 0. Now let γ = min{ζ, β}, then γ > 0. On the one hand
Qu,v,v (3α1 ) ≥ ∗ Qu,ui ,ui (α1 ), ∗(Qui ,uj ,uj (α1 ), Quj ,v,v (α1 )) ≥ ∗ (1 − r1 , ∗(ζ, 1 − r1 ))
≥ ∗ (1 − r1 , ∗(γ, 1 − r1 )) .
On the other hand Qu,u,v (3α1 ) = Qv,u,u (3α1 ) ≥ ∗ Qv,uj ,uj (α1 ), ∗(Quj ,ui ,ui (α1 ), Qui ,u,u (α1 )) ≥ ∗ (1 − r1 , ∗(β, 1 − r1 ))
≥ ∗ (1 − r1 , ∗(γ, 1 − r1 )) .
Putting α = 3α1 and ∗ (1 − r1 , ∗(γ, 1 − r1 )) > 1 − r, 0 < r < 1, we have for all u, v ∈ A , Qu,v,v (α) > 1 − r and Qu,u,v (α) > 1 − r. Hence A is FBed.
Tripled fuzzy metric spaces and fixed point theorem
9
Proposition 3.7. Let (S, Q, ∗) be a TFMS and u0 be any point in S. If r1 ≤ r2 and α1 ≤ α2 , then U (u0 , r1 , α1 ) ⊂ U (u0 , r2 , α2 ). Proof. Assume that z ∈ U (u0 , r1 , α1 ), so Qu0 ,z,z (α1 ) > 1 − r1 and Qu0 ,u0 ,z (α1 ) > 1 − r1 . Since Qu,v,v (·) is nondecreasing, we obtain Qu0 ,z,z (α2 ) ≥ Qu0 ,z,z (α1 ) > 1 − r1 ≥ 1 − r2 .
and Qu0 ,u0 ,z (α2 ) ≥ Qu0 ,u0 ,z (α1 ) > 1 − r1 ≥ 1 − r2 .
Therefore, z ∈ U (u0 , r2 , α2 ), and thus U (u0 , r1 , α1 ) ⊂ U (u0 , r2 , α2 ).
Proposition 3.8. Let (S, Q, ∗) be a TFMS and u0 be any point in S. If U1 (u0 , r1 , α1 ) and U2 (u0 , r2 , α2 ) are neighborhoods of point T u0 , then there is a neighborhood of point u0 , U3 (u0 , r3 , α3 ), such that U3 ⊂ U1 U2 . Proof. Let
U1 (u0 , r1 , α1 ) = {v ∈ S : Qu0 ,v,v (α1 ) > 1 − r1 , Qu0 ,u0 ,v (α1 ) > 1 − r1 } and U2 (u0 , r2 , α2 ) = {v ∈ S : Qu0 ,v,v (α2 ) > 1 − r2 , Qu0 ,u0 ,v (α2 ) > 1 − r2 }
be the neighborhoods of u0 . Let r3 = min{r1 , r2 }, α3 = min{α1 , α2 }. Consider U3 (u0 , r3 , α3 ) = {v ∈ S : Qu0 ,v,v (α3 ) > 1 − r3 , Qu0 ,u0 ,v (α3 ) > 1 − r3 }.
Clearly, u0 ∈ U3 (u0 , r3 , α3 ). Since r3 = min{r1 , r2 } ≤ r1 and α3 = min{α1 , α2 } ≤ α1 , and by Proposition 3.7, we have U3 (u0 , r3 , α3 ) ⊂ U1 (u0 , r1 , α1 ). Similarly, U3 (u0 , r3 , α3 ) ⊂ U2 (u0 , r2 , α2 ), and then \ U3 (u0 , r3 , α3 ) ⊂ U1 (u0 , r1 , α1 ) U2 (u0 , r2 , α2 ).
Now, we give the definitions of convergence of sequences, Q-CS and completeness in TFMS. Definition 3.9. Let (S, Q, ∗) be a TFMS. Then (1) A sequence {un } in S is said to be convergent to a point u ∈ S (write un → u), if for any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that un ∈ U (u, r, α) whenever n > Nα,r . (2) A sequence {un } in S is called a Q-CS, or simply CS, if for any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that Qun ,um ,ul (α) > 1 − r whenever m, n, l > Nα,r . (3) A TFMS is said to be Q-complete, or simply complete, if for any Q-CS in S, there exists a point u ∈ S such that the Q-CS is convergent to u. From (1) of Definition3.9 and (TFM-6), it is not difficult to obtain the proposition below. Proposition 3.10. Assume that (S, Q, ∗) is a TFMS and {un } in S is a sequence. Then the statements below are equivalent. (i) The sequence {un } in S is convergent to a point u ∈ S. (ii) As n → ∞, for all α > 0, Qun ,un ,u (α) → 1. (iii) As n → ∞, for all α > 0, Qun ,u,u (α) → 1.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Proposition 3.11. Assume that (S, Q, ∗) is a TFMS. Then the statements below are equivalent. (a) The sequence {un } in S is a Q-CS. 0 (b) For any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that 0 Qun ,um ,um (α) > 1 − r whenever n, m > Nα,r . Proof. That (a) implies (b) follows from (2) of Definition 3.9. Now we prove that (b) implies (a). Since ∗ is continuous, for every 0
Proposition 3.12. Let (S, Q, ∗) be a TFMS and u, v, w ∈ S. Let {un }, {vn } and {wn } be sequences in S. If un → u, vn → v and wn → w as n → ∞, then, for any α > 0, Qun ,vn ,wn (α) → Qu,v,w (α) as n → ∞. Proof. For any α > 0, there exists a ξ > 0 satisfying α > 2ξ. By using (TFM-6), we find Qun ,vn ,wn (α) = Qun ,vn ,wn (ξ + α − ξ) ≥ ∗ Qun ,u,u (ξ), Qu,vn ,wn (α − ξ) 3ξ ξ = ∗ Qun ,u,u (ξ) , Qu,vn ,wn +α− 2 2 (15) ξ 3ξ ≥ ∗ Qun ,u,u (ξ) , ∗ Qvn ,v,v , Qv,u,wn α − 2 2 ξ ξ ≥ ∗ Qun ,u,u (ξ) , ∗ Qvn ,v,v , ∗ Qwn ,w,w , Qw,u,v (α − 2ξ) 2 2 and (16) Qu,v,w (α) = Qu,v,w (ξ + α − ξ) ≥ ∗ Qu,un ,un (ξ), Qun ,v,w (α − ξ) 3ξ ξ +α− = ∗ Qu,un ,un (ξ) , Qun ,v,w 2 2 ξ 3ξ ≥ ∗ Qu,un ,un (ξ) , ∗ Qv,vn ,vn , Qvn ,w,un α − 2 2 ξ ξ ≥ ∗ Qu,un ,un (ξ) , ∗ Qv,vn ,vn , ∗ Qw,wn ,wn , Qun ,vn ,wn (α − 2ξ) . 2 2
Letting n → ∞ in the inequalities (15) and (16), it follows from the continuity of ∗ that (17)
lim Qun ,vn ,wn (α) ≥ Qu,v,w (α − 2ξ)
n→∞
Tripled fuzzy metric spaces and fixed point theorem
11
and (18)
Qu,v,w (α) ≥ lim Qun ,vn ,wn (α − 2ξ). n→∞
Taking the limit as ξ → 0 in (17) and (18), and by the continuity of Q, we have for any α > 0 lim Qun ,vn ,wn (α) = Qu,v,w (α). n→∞
The proof of Theorem 3.12 is completed.
Remark 3.13. Some of the results given in Section 2 and Section 3 are inspired by some well-known conclusions in [11, 12, 36]. 4. FPTs in TFMS It is well known that one of the most important problems in FMS is the FPT, which is playing a very important and basic role in mathematical analysis and topology. For instance, it is used to determine existence and uniqueness of solutions of differential and integral equations. Grabiec in [13] extend firstly the classical Banach FPT to FMS in the sense of KM. Later on, Gregori and Sapena [15] presented another kind of concept on fuzzy contractive mapping (FCM) and studied its applicability to FPT in FMS. In their paper, they successfully extend the Banach FPT to FCM on complete FMS in KM’s sense and GV’s sense. Since then, many researchers have contributed to the study of the FPT in FMS. Recently, Mihet¸ [22] introduced an interesting concept called Fψ-C mapping, which enlarged the category of FCM which was due to Gregori and Sapena [15], and obtained an important FPT for the category of complete KM-fuzzy metrics. In 2016, George and Mi˜ nana [14] gave an important FPT under Fψ-C mapping in FMS in the sense of GV. As the consequence of the authors’ results, they presented a FPT due to Mihet¸ in [22] and generalized a FPT which is presented by Wardowski in [33]. Moreover, Zhou et. al [36] gave some new FPTs in generalized Menger probabilistic metric space. Inspired by the above works, in the section, we first enlarge the concept of Fψ-C mapping in FMS to TFMS which we called generalized Fψ-C mapping, and study its applicability to FPT in TFMS. Then, we obtain some new results including a FPT on GFψ-CM in TFMS. In this section, we suppose that Ψ is the category of all mappings ψ: ]0, 1] →]0, 1] satisfying ψ is continuous, non-decreasing and ψ(α) > α for any α ∈]0, 1[. Definition 4.1. (See Mihet¸ [22]) Let (S, M, ∗) be a FMS, and ψ ∈ Ψ. (I). If a mapping f : S → S such that P(f (u), f (v), α) ≥ ψ(P(u, v, α)), for all u, v ∈ S and α > 0, then f is called a Fψ-C mapping. (II) If a sequence {un } in S such that P(un+1 , un+2 , α) ≥ ψ(P(un , un+1 , α)), for all n ∈ N and α > 0, then {un } is called a Fψ-C sequence. Next, we give the notions of generalized Fψ-C mapping and generalized Fψ-C sequence.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Definition 4.2. Let (S, Q, ∗) be a TFMS, and ψ ∈ Ψ. (I) If a mapping q : S → S such that Qq(u),q(v),q(w) (α) ≥ ψ(Qu,v,w (α)), for all u, v, w ∈ S and α > 0, then q is called a generalized Fψ-C mapping. (II) If a sequence {un } in S such that Qun+1 ,un+2 ,un+3 (α) ≥ ψ(Qun ,un+1 ,un+2 (α)),
(19)
for all n ∈ N and α > 0, then {un } is called a generalized Fψ-C sequence. Remark 4.3. The notion of generalized Fψ-C sequence maybe defined as the following: A sequence {un } in S is called a generalized Fψ-C sequence if it satisfies Qun+1 ,un+2 ,un+2 (α) ≥ ψ(Qun ,un+1 ,un+1 (α)),
for all n ∈ N and α > 0.
Lemma 4.4. [14] If ψ ∈ Ψ, then for each α ∈]0, 1], limn ψ n (α) = 1. Lemma 4.5. Let (S, Q, ∗) V be a TFMS with ∗ positive,Vand let {un } be a generalized Fψ-C sequence in S. If α>0 Qu0 ,u1 ,u1 (α) > 0 and α>0 Qu1 ,u0 ,u0 (α) > 0, then {un } is a CS. V Proof. Assume that {un } is a generalized Fψ-C sequence in V S, and that α>0 Qu0 ,u1 ,u1 (α) = a > 0. We shall show that limn α>0 Qun ,un+1 ,un+1 (α) = 1. To do this, first we shall prove by mathematical induction that for each n ∈ N ^ Qun ,un+1 ,un+1 (α) ≥ ψ n (a). α>0
In fact, as n = 1, for each α > 0, it follows from (19) that
Qu1 ,u2 ,u2 (α) ≥ ψ(Qu0 ,u1 ,u1 (α)) ≥ ψ(a).
V
Then α>0 Qu1 ,u2 ,u2 V (α) ≥ ψ(a). Assume now that α>0 Qun ,un+1 ,un+1 (α) ≥ ψ n (a) holds for some fixed n ∈ N. From (19), the fact that ψ is non-decreasing, and the induction hypothesis we find that Qun+1 ,un+2 ,un+2 (α) ≥ ψ(Qun ,un+1 ,un+1 (α)) ! ^ ≥ψ Qun ,un+1 ,un+1 (α) α>0 n
≥ ψ (ψ (a)) .
V
Therefore α>0 Qun+1 ,un+2 ,un+2 (α) ≥ ψ n+1 (a). Thus, by mathematical induction we conclude that ^ (20) Qun ,un+1 ,un+1 (α) ≥ ψ n (a) α>0
holds for each n ∈ N. Letting n → ∞ on both sides of (20), and by Lemma 4.4 we can find that ^ lim Qun ,un+1 ,un+1 (α) ≥ lim ψ n (a) = 1. n
α>0
n
Tripled fuzzy metric spaces and fixed point theorem
Therefore, (21)
lim n
^
Qun ,un+1 ,un+1 (α)
α>0
!
= 1.
Making similar technique as in the proof of (21) with the condition 0, we have ! ^ (22) lim Qun+1 ,un ,un (α) = 1. n
13
V
α>0
Qu1 ,u0 ,u0 (α) >
α>0
Next we shall prove that {un } is a CS. For it, first we shall show by contradiction, that Qun ,um ,um (α) = 1. lim n,m→∞
Assume that there is a α0 > 0 such that Qun ,um ,um (α0 ) does not converge to 1 as n, m. Hence there exists an ∈]0, 1[ and sequences {n(j)} and {m(j)} such that n(j) > m(j) ≥ j and Qun (j),um (j),um (j) (α0 ) ≤ 1 − , j = 1, 2, . . . .
Fix j ∈ N. Then we can find n(j), m(j) ∈ N such that n(j) > m(j) ≥ j and Qun(j) ,um(j) ,um(j) (α0 ) ≤ 1 − . Given m(j), by choosing nm (j) to be the smallest integer exceeding m(j) and such that Qunm (j) ,um(j) ,um(j) (α0 ) ≤ 1 −
(23) holds. Then
Qunm (j)−1 ,um(j) ,um(j) (α0 ) > 1 − .
(24)
Now we shall show that lim Qunm (j)−1 ,um(j) ,um(j) (α0 ) = 1 − . j
By inequality (23) and (TFM-6), we find that (25)
1 − ≥ Qunm (j) ,um(j) ,um(j) (α0 )
≥ ∗ Qunm (j) ,unm (j)−1 ,unm (j)−1 (δ), Qunm (j)−1 ,um(j) ,um(j) (α0 − δ) ,
for each j ∈ N and each δ ∈]0, α0 [. Therefore, by inequality (25), the continuities of ∗ and Q, we have
1 − ≥ Qunm (j) ,um(j) ,um(j) (α0 ) ≥ lim ∗ Qunm (j) ,unm (j)−1 ,unm (j)−1 (δ), Qunm (j)−1 ,um(j) ,um(j) (α0 − δ) δ→0 = ∗ lim Qunm (j) ,unm (j)−1 ,unm (j)−1 (δ), lim Qunm (j)−1 ,um(j) ,um(j) (α0 − δ) δ→0 δ→0 ! ^ =∗ Qunm (j) ,unm (j)−1 ,unm (j)−1 (α) , Qunm (j)−1 ,um(j) ,um(j) (α0 ) , α>0
for each k ∈ N. Hence, (26)
lim sup Qunm (j) ,um(j) ,um(j) (α0 ) ≤ 1 − , j
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
and (27) lim inf Qunm (j) ,um(j) ,um(j) (α0 ) j
≥ lim ∗ j
= ∗ lim j
^
α>0
^
α>0
!! Qunm (j) ,unm (j)−1 ,unm (j)−1 (α) , Qunm (j)−1 ,um(j) ,um(j) (α0 )
! Qunm (j) ,unm (j)−1 ,unm (j)−1 (α) , lim Qunm (j)−1 ,um(j) ,um(j) (α0 ) j
≥ ∗(1, 1 − ) = 1 − . The second inequality of (27) has been obtained by inequality (24) and equality (22). From inequalities (26) and (27), we have 1 − ≥ lim sup Qunm (j) ,um(j) ,um(j) (α0 ) j
≥ lim inf Qunm (j) ,um(j) ,um(j) (α0 ) ≥ 1 − . j
Thus lim Qunm (j) ,um(j) ,um(j) (α0 ) = 1 − .
(28)
j
On the other hand, it follows by (TFM-6) and equality (28) that Qunm (j) ,um(j) ,um(j) (α0 ) = Qunm (j) ,um(j) ,um(j) (α0 − δ + δ)
≥ ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ), Qunm (j)+1 ,um(j) ,um(j) (α0 − δ)
(29)
= ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ), Qunm (j)+1 ,um(j) ,um(j) (α0 − 2δ + δ) ≥ ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ),
∗ Qunm (j)+1 ,um(j)+1 ,um(j)+1 (α0 − 2δ), Qum(j)+1 ,um(j) ,um(j) (δ)
≥ ∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (δ),
! ∗ ψ Qunm (j) ,um(j) ,um(j) (α0 − 2δ) , Qum(j)+1 ,um(j) ,um(j) (δ) ,
for each j ∈ N and each δ ∈]0, α0 /2[. Letting δ → 0 on both sides of inequality (29), in a similar way that before, we get
(30)
Q(unm (j) ,um(j) ,um(j) (α0 ) ^ ≥∗ Qunm (j) ,unm (j)+1 ,unm (j)+1 (α) , α>0
! ^ ∗ ψ Qunm (j) ,um(j) ,um(j) (α0 ) , Qum(j)+1 ,um(j) ,um(j) (α) , α>0
Tripled fuzzy metric spaces and fixed point theorem
15
for each j ∈ N. By using (28), the continuities of ∗ and ψ, equalities (21) and (22), and taking limit as j tends to ∞ on both sides of (30), we have (31) 1 − = lim Qunm (j) ,um(j) ,um(j) (α0 ) j
≥ ∗ lim j
^
α>0
Qunm (j) ,unm (j)+1 ,unm (j)+1 (α) ,
! ^ ∗ ψ lim Qunm (j) ,um(j) ,um(j) (α0 ) , lim Qum(j)+1 ,um(j) ,um(j) (α) j
j
= ∗ 1, ∗ ψ(1 − ), 1 = ψ(1 − ) > 1 − ,
t>0
which is a contradiction (The second equality of (31) has been obtained by equalities (21), (28) and (31)). Hence lim Qun ,um ,um (α) = 1. n,m→∞
for any α > 0. Thus, it follows by Proposition 2.15 that {un } is a CS. The proof of Lemma 4.5 is completed. Theorem 4.6. Assume that (S, Q, ∗) is a complete TFMS with ∗ positive, and q : S → S is a generalized Fψ-C mapping. Then, V V q has a unique FP iff there is a u ∈ S satisfying α>0 Qu,q(u),q(u) (α) > 0 and α>0 Qq(u),u,u (α) > 0.
Proof. Assume that q has a unique FP, then there is a u ∈ S satisfying q(u) = u. Therefore for each α >V0, we have Qu,q(u),q(u) (α) = V Qu,u,u (α) = 1, Qq(u),u,u (α) = Qu,u,u (α) = 1, and so α>0 Qu,q(u),q(u) (α) = 1 and V α>0 Qq(u),u,u (α) = 1. Conversely, assume that there is a u ∈ S satisfying α>0 Qu,q(u),q(u) (α) > 0 and V n Q (α) > 0. Letting u = u and u = q (u) for each n ≥ 1, then we get 0 n q(u),u,u α>0 Qun+1 ,un+2 ,un+3 (α) = Qq(un ),q(un+1 ),q(un+2 ) (α) ≥ ψ(Qun ,un+1 ,un+2 (α)).
Therefore, {un } is a generalized Fψ-C sequence. Moreover ^ ^ Qu0 ,u1 ,u1 (α) = Qu,q(u),q(u) (α) > 0 α>0
and
^
α>0
Qu1 ,u0 ,u0 (α) =
α>0
^
Qq(u),u,u (α) > 0.
α>0
Then by Lemma 4.5, we get that {un } is a CS. Since (S, Q, ∗) is complete, there exists u ∈ S satisfying limn Qun ,u,u (α) = 1 for each α > 0. On the other hand, we find Qq(u),un+1 ,un+1 (α) ≥ ψ(Qu,un ,un (α))
for each n ∈ N and each α > 0. So Qq(u),u,u (α) = lim Qq(u),un+1 ,un+1 (α) n
≥ lim ψ(Qu,un ,un (α)) = 1 n
for each α > 0. Thus, u is a FP of q.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Next we shall verify that u is the unique FP of q. If v is another FP of q, then for any α > 0 Qu,u,v (α) = Qq(u),q(u),q(v) (α) ≥ ψ (Qu,u,v (α)) .
If u 6= v, then for some β > 0
Qu,u,v (β) < 1,
that is, 0 < Qu,u,v (β) < 1. Then Qu,u,v (β) = Qq(u),q(u),q(v) (β) ≥ ψ (Qu,u,v (β)) > Qu,u,v (β), which is a contradiction. Thus, we conclude that u = v. Hence, q has a unique FP in S, which means that Theorem 4.6 holds true. Remark 4.7. The V conditions of Theorem 4.6 involve V the self-mapping (there is a u ∈ S such that α>0 Qu,q(u),q(u) (α) > 0 and α>0 Qq(u),u,u (α) > 0). It is expected that there are not any restrictions on self-mapping in the study of existence of a FP since the expression could be complex. In fact, the following two corollaries can be obtained immediately from Lemma 4.5 and Theorem 4.6, and the demanded conditions do not involve self-mapping. V Corollary V 4.8. Let (S, Q, ∗) be a TFMS with ∗ positive, and let α>0 Qu,v,v (α) > 0 and α>0 Qv,u,u (α) > 0 for each u, v ∈ S. Then, any generalized Fψ-C sequence in S is a CS. Corollary 4.9. Assume V V that (S, Q, ∗) is a complete TFMS with ∗ positive, and α>0 Qu,v,v (α) > 0, α>0 Qv,u,u (α) > 0 for each u, v ∈ S. If q : S → S is a generalized Fψ-C mapping, then q has a unique FP. Remark 4.10. If the notion of generalized Fψ-C sequence is defined as the one in Remark 4.3, then we can prove that Lemma 4.5 and Theorem 4.6 hold in a similar way. However, it should be point out that the Lemma of Gregori and Mi˜ nana [14, Lemma 3.2] and the Theorem of Gregori and Mi˜ nana [14, Theorem 3.3] can be followed from Example 2.11 and the definition in Remark 4.3 by a new proving technique introduced by Zhou et al. in [36]. The corollaries are listed as follows. Corollary 4.11. (Gregori and Mi˜ nana [14, Lemma 3.2]) Assume that (S, P, ∗) is a V FMS with ∗ positive, and {un } is a Fψ-C sequence in S. If α>0 P(u0 , u1 , α) > 0, then {un } is a CS. Proof. Let Qu,v,w (α) = min{P(u, v, α), P(v, w, α), P(u, w, α)} for u, v, w ∈ S, α > 0. From Example 2.11, we know that (S, Q, ∗) is a TFMS. Since Q un+1 ,un+2 ,un+2 (α)
= min{P(un+1 , un+2 , α), P(un+2 , un+2 , α), P(un+2 , un+1 , α)} = P(un+1 , un+2 , α) ≥ ψ P(un , un+1 , α) = ψ min{P(un , un+1 , α), P(un+1 , un+1 , α), P(un+1 , un , α)} = ψ Qun ,un+1 ,un+1 (α) ,
Tripled fuzzy metric spaces and fixed point theorem
that is
17
Qun+1 ,un+2 ,un+2 (α) ≥ ψ Qun ,un+1 ,un+1 (α) ,
for n ∈ N, α > 0. Thus {un } is a generalized Fψ-C sequence. On the other hand,
P(u0 , u1 , α) = min{P(u0 , u1 , α), P(u1 , u1 , α), P(u1 , u0 , α)}
(32)
= Qu0 ,u1 ,u1 (α)
and P(u1 , u0 , α) = min{P(u1 , u0 , α), P(u0 , u0 , α), P(u0 , u1 , α)}
(33)
From (32), (33) and ^
V
α>0
= Qu1 ,u0 ,u0 (α).
P(u0 , u1 , α) > 0, we find that ^ Qu0 ,u1 ,u1 (α) > 0 and Qu1 ,u0 ,u0 (α) > 0. α>0
α>0
Therefore, from Lemma 4.5, we obtain that {un } is a CS in the sense of TFMS (S, Q, ∗). That is, for any α > 0 and 0 < r < 1, there is an integer Nα,r > 0 such that Qun ,um ,ul (α) > 1 − r for all n, m, l > Nα,r . It follows from the definition of Qu,v,w (α) that min{P(un , um , α), P(um , ul , α), P(un , ul , α)} > 1 − r
for n, m, l > Nα,r . This shows that {un } is a CS in the sense of FMS (S, P, ∗).
By the same method as in Corollary 4.11, we can obtain the following FPT in the sense of FMS (S, P, ∗). Corollary 4.12. (Gregori and Mi˜ nana [14, Theorem 3.3]) Assume that (S, P, ∗) is a complete FMS with ∗ positive, and f : S → V S is a Fψ-C mapping. Then, f has a unique FP iff there is an u ∈ S such that α>0 P(u, f (u), α) > 0. Finally, we give a simple example to illustrate Theorem 4.6.
Example 4.13. Let S =]0, ∞[ and ∗ = ∗p . Define Q : S 3 ×]0, ∞[→]0, 1] by Qu,v,w (α) =
min{u2 , v 2 , w2 } + α max{u2 , v 2 , w2 } + α
for u, v, w ∈ S. It is easy to check that (S, Q, ∗) is a TFMS and min{u2 ,v 2 ,w2 } max{u2 ,v 2 ,w2 } 1 2
> 0. Define a function q : S → S by q(u) = u
1 2
V
α>0
Qu,v,w (α) =
for u ∈ S, and let
ψ(•) = • . Then, from Theorem 4.6, we obtain that q has a unique FP in S. In fact, the FP is u = 1. 5. Conclusions As we know, FM and FPT have been applied in many fields those related information sciences. For example, in dynamical system, fuzzy FPT can be applied in existence, uniqueness and continuity of solution with some vague parameters [10, 23]. In fuzzy game theory, fuzzy FPT or fuzzy common FPT can be used in the proof of the existence of equilibrium solution [7]. In this paper, we present a new FMS–TFMS, which is a new generalization of GV’s FMS, and then give some related examples, topological properties, convergence of sequences, CS and completeness of the TFMS. Moreover, we introduce two kinds of notions of generalized
18
Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Fψ-C mappings, and obtain a FPT on the mappings in the space. We expect that our research results can offer a mathematical basis for the ongoing research on information science. In the future research, we will explore the concrete applications in the field of information science of the obtained results. 6. Acknowledgement The authors would like to express their sincere thanks to the anonymous referees for their making great efforts to improve this paper. References [1] T. Bag, S. K. Samanta, Fixed point theorems on fuzzy normed linear spaces, Inform. Sci., 176 (2006), 2910-2931. [2] T. Bag, S. K. Samanta, Some fixed point theorems in fuzzy normed linear spaces, Inform. Sci., 177 (2007), 3271-3289. [3] S.-M. Chen, S. I. Adam, Weighted fuzzy interpolated reasoning based on ranking values of polygonal fuzzy sets and new scale and move transformation techniques, Inform. Sci., 435 (2018), 184-202. [4] S.-M. Chen, D. Barman, Adaptive weighted fuzzy interpolative reasoning based on representative values and similarity measures of interval type-2 fuzzy sets, Inform. Sci., 478 (2019), 167-185. [5] S.-M. Chen, S.-H. Cheng, C.-H. Chiou, Fuzzy multiattribute group decision making based on intuitionistic fuzzy sets and evidential reasoning methodology, Inform. Fusion, 478 (2016), 215-227. [6] S.-M. Chen, X.-Y. Zou, D. Barman, Adaptive weighted fuzzy rule interpolation based on ranking values and similarity measures of rough-fuzzy sets, Inform. Sci., 488 (2019), 93-110. [7] H. Chu, J. Hsieh, Fuzzy differential game of guarding a movable territory, Inform. Sci., 176 (1996), 113-131. [8] Z.-K. Deng, Fuzzy pseudo metric spaces, J. Math. Anal. Appl., 86 (1982), 74-95. [9] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl., 69 (1979), 205-230. [10] W. Fei, Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients, Inform. Sci., 177 (2007), 4329-4337. [11] A. George, P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst., 64 (1994), 395-399. [12] A. George, P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets Syst., 90 (1997), 365-368. [13] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Syst., 27 (1988), 385-389. [14] V. Gregori, J.-J. Mi˜ nana, On fuzzy ψ-contractive sequences and fixed point theorems, Fuzzy Sets Syst., 300 (2016), 245-252. [15] V. Gregori, A. Sapena, On fixed-point theorems in fuzzy metric spaces, Fuzzy Sets Syst., 125 (2002), 245-252. [16] O. Had˘zi´c, E. Pap, Fixed Point Theory in Probabilistic Metric Space, Kluwer Academic Publishers, Dordrecht, 2001. [17] I. Kramosil, J. Michalek, Fuzzy metrics and statistical metric spaces, Kybernetika, 15 (1975), 326-334.
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Jing-Feng Tian, Ming-Hu Ha*, and Da-Zeng Tian
Declaration of Competing Interest The authors declare that they have no competing interests. Jing-Feng Tian, Department of Mathematics and Physics, North China Electric Power University, Baoding, Hebei Province, 071000, P. R. China E-mail address: [email protected] Ming-Hu Ha, School of Science, Hebei University of Engineering, Handan, Hebei Province, 056038, P. R. China, and College of Mathematics and Information Science, Hebei University, Baoding, Hebei Province, 071000, P. R. China E-mail address: [email protected] Da-Zeng Tian, College of Mathematics and Information Science, Hebei University, Baoding, Hebei Province, 071002, P. R. China E-mail address: [email protected]