@ Appl. Gen. Topol. 21, no. 2 (2020), 285-294 doi:10.4995/agt.2020.13126 c© AGT, UPV, 2020 The depth and the attracting centre for a continuous map on a fuzzy metric interval Taixiang Sun a, Lue Li a, Guangwang Su a, Caihong Han a,∗ and Guoen Xia b a College of Information and Statistics, Guangxi University of Finance and Economics, Nanning, 530003, China. (stx1963@163.com,li1982lue@163.com,s1g6w3@163.com,h198204c@163.com) b College of of Business Administration, Guangxi University of Finance and Economics, Nanning, 530003, China. (x3009h@163.com) Communicated by D. Georgiou Abstract Let I be a fuzzy metric interval and f be a continuous map from I to I. Denote by R(f), Ω(f) and ω(x, f) the set of recurrent points of f, the set of non-wandering points of f and the set of ω- limit points of x under f, respectively. Write ω(f) = ∪x∈Iω(x, f), ωn+1(f) = ∪x∈ωn(f)ω(x, f) and Ωn+1(f) = Ω(f|Ωn(f)) for any positive integer n. In this paper, we show that Ω2(f) = R(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2. 2010 MSC: 54E35; 54H25. Keywords: fuzzy metric interval; attracting centre; depth. 1. Introduction By extending the notion of Menger space to the fuzzy setting, Kramosil and Michalek [7] obtained the notion of fuzzy metric space with the help of continuous t-norms. In order to obtain a Hausdorff topology in fuzzy metric ∗Corresponding author. Project supported by NNSF of China (11761011, 71862003) and NSF of Guangxi (2018GXNS- FAA294010) and SF of Guangxi University of Finance and Economics (2019QNB10). Received 13 February 2020 – Accepted 11 May 2020 http://dx.doi.org/10.4995/agt.2020.13126 T. Sun, L. Li, G. Su, C. Han and G. Xia spaces, George and Veeramani [1] modified the notion given by Kramosil and Michalek in a slight but appealing way. Recently many authors studied several properties of the Hausdorff fuzzy metric spaces (see [6, 8, 11]) and introduced and investigated the different types of fuzzy contractive maps and obtained a lot of fixed point theorems (see [3, 4, 9, 10, 13, 14, 15, 16]). Until now, there are little of works that investigates some properties of discrete dynamical systems on fuzzy metric spaces. In this paper, we introduce the notion of fuzzy metric interval and study the depth and the attracting centre for a continuous map on a fuzzy metric interval. The rest of this paper is organized as follows. In Section 2 we give some definitions and notations. In Section 3 we study the depth for a continuous map on a fuzzy metric interval. In Section 4 we study the depth of the attracting centre for a continuous map on a fuzzy metric interval. 2. Preliminaries Throughout the paper, let N be the set of all positive integers and N! = N ∪ {0}. Firstly, we recall the basic definitions and the properties about fuzzy metric spaces. Definition 2.1 (see [12]). We say that a continuous map ξ : [0, 1]2 −→ [0, 1] is a continuous t-norm if for any a, b, c, d ∈ [0, 1], the following conditions hold: (1) ξ(a, b) = ξ(b, a). (2) ξ(a, b) ≤ ξ(c, d) for a ≤ c and b ≤ d. (3) ξ(ξ(a, b), c) = ξ(a, ξ(b, c)). (4) ξ(a, 0) = 0 and ξ(a, 1) = a. For a, b ∈ [0, 1], we will use the notation a∗b instead of ξ(a, b). For example, ξ(a, b) = min{a, b}, ξ(a, b) = ab and ξ(a, b) = max{a + b − 1, 0} are the most commonly used t-norms. In the present paper, we also use the following definition of the fuzzy metric space. Definition 2.2 (see [1]). We say that a triple (X, M, ∗) is a fuzzy metric space if X is a nonempty set, ∗ is a continuous t-norm and M is a map defined on X2×(0, +∞) into [0, 1] and for any x, y, z ∈ X and s, t ∈ (0, +∞), the following conditions hold: (1) M(x, y, t) > 0. (2) M(x, y, t) = 1 (for any t > 0) ⇐⇒ x = y. (3) M(x, y, t) = M(y, x, t). (4) M(x, z, t + s) ≥ M(x, y, t) ∗ M(y, z, s). (5) Mxy : (0, +∞) −→ [0, 1] is a continuous mapping ( where Mxy(t) = M(x, y, t)). Remark 2.3. (1) Mxy is a non-decreasing function on (0, ∞) for all x, y ∈ X (see [2]). (2) M is a continuous function on X × X × (0, +∞) (see [11]). If (X, M, ∗) is a fuzzy metric space, then we will say that (M, ∗), or simply M, is a fuzzy metric on X. In [1], George and Veeramani showed that every c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 286 The depth and the attracting centre for map fuzzy metric M on X generates a topology τM on X which has as a base the family of open sets of the form {BM(x, ε, t) : x ∈ X, 0 < ε < 1, t > 0} , where BM(x, ε, t) = {y ∈ X : M(x, y, t) > 1 − ε} for all x ∈ X, ε ∈ (0, 1) and t > 0, and (X, τM) is a Hausdorff space. Definition 2.4 (see [5]). Let (X, M, ∗) be a fuzzy metric space. We say that a sequence of points xn ∈ X converges to x ( denoted by xn −→ x) ⇐⇒ limn−→+∞ M(xn, x, t) = 1 (for any t > 0), i.e. for each δ ∈ (0, 1) and t > 0, there exists N ∈ N such that M(xn, x, t) > 1 − δ for all n ≥ N. Definition 2.5. (1) We say that a fuzzy metric space (X, M, ∗) is compact if (X, τM) is compact. We say that a subset A of X is compact if A as a fuzzy metric subspace is compact. (2) We say that a fuzzy metric space (X, M, ∗) is connected if (X, τM) is connected. We say that a subset A of X is connected if A as a fuzzy metric subspace is connected. By [6] we know that: (1) (X, M, ∗) is a compact fuzzy metric space ⇐⇒ each sequence of points in X has a convergent subsequence. (2) If (X, M, ∗) is a compact fuzzy metric space and A is a subset of X, then A is compact ⇐⇒ A is closed. (3) If (X, M, ∗) is a compact fuzzy metric space, then for any x, y ∈ X with x ∕= y, there exist B(x, ε1, t1) and B(y, ε2, t2) with ε1, ε1 ∈ (0, 1) and t1, t2 ∈ (0, +∞) such that B(x, ε1, t1) ∩ B(y, ε2, t2) = ∅. Definition 2.6. Let (X, M, ∗) be a compact fuzzy metric space and a, b ∈ X. We say that X is a fuzzy metric interval with ends a and b if the following conditions hold: (1) M(a, x, t) ≥ M(a, b, t) for any x ∈ X and t > 0. (2) For any x, y ∈ X with x ∕= y, we have M(a, x, t) < M(a, y, t) for any t > 0, which is denoted by x > y, or M(a, x, t) > M(a, y, t) for any t > 0, which is denoted by x < y . (3) For any x, y ∈ X with M(a, x, t) ≥ M(a, y, t) for any t > 0, set {z ∈ X : M(a, x, t) ≥ M(a, z, t) ≥ M(a, y, t) for any t > 0}, which is denoted by [x; y], is a connected subset of X. (4) If y ∈ B(x, ε, t) for some x ∈ X and some ε ∈ (0, 1) and some t > 0, then [y; x] ⊂ B(x, ε, t) if y ≤ x or [x; y] ⊂ B(x, ε, t) if y ≥ x. Write [x; y) = [x; y] − {y} ≡ {z ∈ X : M(a, x, t) ≥ M(a, z, t) > M(a, y, t) and (x; y) = [x; y) − {x} ≡ {z ∈ X : M(a, x, t) > M(a, z, t) ≥ M(a, y, t). Remark 2.7. (1) [a; b] = X. (2) Let x, y ∈ [a; b]. If M(a, x, t) = M(a, y, t) for some t > 0, then x = y. Example 2.8. Let I = [a, b] be a compact interval of R= (−∞, +∞). Define s ∗ t = st for any s; t ∈ [0, 1], and let Md : I × I × (0, ∞) −→ [0, 1] such that for any x, y ∈ I and t > 0, Md(x, y, t) = t t + |x − y| . c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 287 T. Sun, L. Li, G. Su, C. Han and G. Xia Then (I, Md, ∗) is a fuzzy metric interval. Further, it was proven [1] that the topologies induced by (I, d) with d(x, y) = |x−y| for any x, y ∈ I and (I, Md, ∗) are the same. Let (I, M, ∗) be a fuzzy metric interval and A ⊂ I. Denote by A the closure of A in (I, τM). Let C 0(I) denote the set of all continuous maps on I. For any f ∈ C0(I) and x ∈ I, write f0(x) = x and fn = f ◦ fn−1 for any n ∈ N. We also introduce a number of notations: O(x, f) = {fn(x) : n ∈ N!}. Λ(x, f) = ∪∞n=1f −n(x). Fix(f) = {x : f(x) = x}. P(f) = {x : there exists some n ∈ N such that fn(x) = x}. ω(x, f) = {y : there exists a sequence of positive integers k1 < k2 < · · · such that lim n−→∞ M(fkn(x), y, t) = 1 (for any t > 0)}. R(f) = {x : x ∈ ω(x, f)}. UΓ(f) = {y : there exist a connected component J of I − {y}, a point x ∈ J, a sequence of points x1, x2, · · · ∈ Λ(x, f) ∩ J and a sequence of positive integers k1 < k2 < · · · such that fkn(x) ∈ J for any n ∈ N and lim n−→∞ M(xn, y, t) = lim n−→∞ M(fkn(x), y, t) = 1(for any t > 0)}. Ω(f) = {y : there exist a sequence of points x1, x2, · · · ∈ I and a sequence of positive integers k1 ≤ k2 ≤ · · · such that lim n−→∞ M(xn, y, t) = lim n−→∞ M(fkn(xn), y, t) = 1 (for any t > 0)}. P(f), R(f), UΓ(f) and Ω(f) are called the set of periodic points, the set of recurrent points, the set of unilateral γ- limit points and the set of non- wandering points of f, respectively. O(x, f), Λ(x, f) and ω(x, f) are called the orbit of x under f, the reverse orbit of x under f and the set of ω-limit points of x under f, respectively. Remark 2.9. Let (I, M, ∗) be a fuzzy metric interval and f ∈ C0(I). Then the following statements hold: (1) f(Ω(f)) ⊂ Ω(f) and Ω(f) is closed. (2) Fix(f) ⊂ P(f) ⊂ R(f) ⊂ ω(f) ⊂ Ω(f). (3) AΓ(f) ⊂ ω(f) and R(f) ⊂ Ωn(f) for any ∈ N. Definition 2.10. Let (I, M, ∗) be a fuzzy metric interval and f ∈ C0(I). (1) For any A ⊂ I, write ω(A) = ∪x∈Aω(x, f) and ω1(f) = ω(f) = ω(I) and ωn+1(f) = ∪x∈ωn(f)ω(x, f) for any n ∈ N. The minimal m ∈ N ∪ {∞} such that ωm(f) = ωm+1(f) is called the depth of the attracting centre of f. (2) Write Ω1(f) = Ω(f) and Ωn+1(f) = Ω(f|Ωn(f)) for any n ∈ N. The minimal m ∈ N ∪ {∞} such that Ωm(f) = Ωm+1(f) is called the depth of f. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 288 The depth and the attracting centre for map In this paper, we show that if f is a continuous map on a fuzzy metric interval, then Ω2(f) = R(f) and the depth of f is at most 2, and ω 3(f) = ω2(f) and the depth of the attracting centre of f is at most 2. 3. The depth for a continuous map on a fuzzy metric interval In this section, we study the depth of a continuous map on a fuzzy metric interval I = [a, b] with a ∕= b. Proposition 3.1. The following two statements hold: (1) Let xn ∈ I for any n ∈ N. Then xn −→ x ⇐⇒ M(a, xn, t) −→ M(a, x, t) for any t > 0. (2) For any x, y ∈ I with x ≤ y, [x; y] is closed. (3) For any x, y, z ∈ X with x < y < z, [x; z] − {y} is not connected. Proof. (1) =⇒ is obvious since M is continuous. ⇐= Assume on the contrary that xn ∕−→ x. Then there exist an open neighbourhood U of x and a sequence of positive integers k1 < k2 < · · · such that fkn(x) ∈ I − U for any n ∈ N. By taking subsequence we let xkn −→ u ∈ I − U since I − U is closed. Then by the above (=⇒) we have M(a, xkn, t) −→ M(a, u, t) for any t > 0. Thus M(a, x, t) = M(a, u, t). By Remark 2.7 we have x = u. This is a contradiction. (2) Let xn ∈ [x; y] and xn −→ u ∈ I since I is compact. Then M(a, x, t) ≥ M(a, xn, t) ≥ M(a, y, t) with M(a, xn, t) −→ M(a, u, t) ∈ [M(a, y, t), M(a, x, t)], which implies u ∈ [x; y]. (3) We claim that [x; y) is an open subset of [x; z]. Indeed, for any w ∈ [x; y), there exists an open neighbourhood U = B(w, ε, t0) of w such that y ∕∈ U. By Definition 2.6 (4) we see that U ∩ [x; z] ⊂ [x; y), which implies that [x; y) is an open subset of [x; z]. In a similar fashion we can also show that (y; z] is an open subset of [x; z]. Since [x; y) ∩ (y; z] = ∅ and [x; z] − {y} = [x; y) ∩ (y; z], from which it follows that [x; z] − {y} is not connected. The proof is completed. □ Lemma 3.2. Let f ∈ C(I). If there exist x, y ∈ I with x ≤ y such that f(x) ≤ x ≤ y ≤ f(y) or x ≤ f(x) and f(y) ≤ y, then [x; y] ∩ Fix(f) ∕= ∅. Proof. We can assume that f(x) ∕= x and f(y) ∕= y. Define F : [x; y] −→ R such that for any z ∈ [x; y], F(z) = M(a, z, t) − M(a, f(z), t). By Remark 2.3 we see that F is continuous. Since F(x)F(y) < 0 and [x; y] is connected, we have 0 ∈ f([x; y]) and there exists p ∈ [x; y] such that F(p) = 0. Thus M(a, p, t) = M(a, f(p), t) for t > 0. By Remark 2.7 we see f(p) = p. The proof is completed. □ Corollary 3.3. Let f ∈ C(I). If there exist x ∈ I and n ∈ N such that fn(x) < x and [fn(x); x] ∩ P(f) = ∅, Then fkn(x) < fn(x) for any k ≥ 2. If there exist x ∈ I and n ∈ N such that fn(x) > x and [x; fn(x)] ∩ P(f) = ∅, Then fkn(x) > fn(x) for any k ≥ 2. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 289 T. Sun, L. Li, G. Su, C. Han and G. Xia Proof. Let fn(x) < x. Assume on the contrary that fn(x) ≤ fkn(x) for some k ≥ 2. Then it follows from Lemma 3.2 that [fn(x); x] ∩ Fix(f(k−1)n) ∕= ∅. This contradicts [fn(x); x] ∩ P(f) = ∅. For the another case, the proof is similar. The proof is completed. □ By Lemma 3.2 and Corollary 3.3 we obtain the following corollary. Corollary 3.4. Let f ∈ C(I) and x, y ∈ I with x < y and [x; y] ∩ P(f) = ∅. If u, v, fm(u), fn(v) ∈ [x; y] for some m, n ∈ N, then u ∈ [x; fm(u)) and v ∈ [x; fn(v)), or u ∈ (fm(u); y] and v ∈ (fn(v); y]. Theorem 3.5. Let f ∈ C(I). Then P(f) = R(f). Proof. By Remark 2.9 it is suffice to show that P(f) ⊃ R(f). Let x ∈ R(f) − P(f). Take an open neighbourhood U = B(x, ε, t0) of x. Then there exists a sequence of positive integers k1 < k2 < · · · such that fkn(x) −→ x and fkn(x) ∈ U. Choose fkm(x) ∈ U. Without loss of generality we may assume that fkm(x) < x. Then by Remark 2.7 we see that there exists r > m such that fkm(x) < fkr (x). By Corollary 3.4 we obtain [fkm(x); fkr (x)]∩P(f) ∕= ∅ if fkr (x) ≥ x or [fkm(x); x] ∩ P(f) ∕= ∅ if fkr (x) ≤ x. Thus U ∩ P(f) ∕= ∅ (Definition 2.6 (4)), which implies x ∈ P(f) and R(f) ⊂ P(f). The proof is completed. □ Theorem 3.6. Let f ∈ C(I). Then Ω(f|Ω(f)) = R(f) and the depth of f is at most 2. Proof. By Remark 2.9 it is suffice to show that Ω(f|Ω(f)) − R(f) ⊂ R(f). Let x ∈ Ω(f|Ω(f)) − R(f). Take an open neighbourhood U = B(x, ε, t0) of x. Then there exist a sequence of positive integer k1 ≤ k2 ≤ · · · and a sequence of points xn ∈ Ω(f) such that fkn(xn) −→ x, xn −→ x and fkn(xn), xn ∈ U for any n ∈ N. Without loss of generality we may assume that xn ∕∈ P(f) for any n ∈ N. Choose xm, fkm(xm) ∈ U. Without loss of generality we may assume that xm < f km(xm). We can choose an open neighbourhood W = B(xn, δ, t1) of xm such that W, f km(W) ⊂ U and W ∩ fkm(W) = ∅ since I is a compact Hausdorff space. Note that xm ∈ Ω(f) and xm ∕∈ P(f). Then there exist a sequence of positive integers r1 < r2 < · · · and a sequence of points yn ∈ I such that frn(yn) −→ xm, yn −→ xm and frn(yn), yn ∈ W for any n ∈ N. By Proposition 3.1 we see that there exists rn > km such that M(a, frn(yn), t) > M(a, f km(yn), t) and M(a, yn, t) > M(a, f km(yn), t). Thus frn(yn) < f km(yn) and yn < f km(yn). By Corollary 3.4 we obtain [yn; f km(yn)] ∩ P(f) ∕= ∅ if yn ≤ frn(yn), or [frn(yn); fkm(yn)] ∩ P(f) ∕= ∅ if yn ≥ frn(yn), which implies U ∩ P(f) ∕= ∅ (Definition 2.6 (4)). Thus x ∈ P(f) = R(f). The proof is completed. □ 4. The attracting centre for a continuous map on a fuzzy metric interval In this section, we study the attracting centre of a continuous map on a fuzzy metric interval I = [a; b] with a ∕= b. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 290 The depth and the attracting centre for map Lemma 4.1. Let f ∈ C0(I) and x, w, y ∈ I with w ∈ [x; y] and [x; y] ∩ (R(f) ∪ Λ(w, f)) = ∅. If there exist x1, x2, · · · ∈ Λ(w, f) with xn < x and y1, y2, · · · ∈ Λ(w, f) with y < yn for any n ∈ N such that limn−→∞ M(xn, x, t) = limn−→∞ M(yn, y, t) = 1 (for any t > 0), then {x, y} ∩ AΓ(f) ∕= ∅. Proof. We claim that fn([x; y]) ∩ [x; y] = ∅ for any n ∈ N. Indeed, if fm([x; y]) ∩ [x; y] ∕= ∅ for some m ∈ N, then fm([x; y]) ∕⊂ [x; y] since [x; y] ∩ P(f) = ∅. Thus there exists N ∈ N such that xn ∈ fm([x; y]) or yn ∈ fm([x; y]) for any n ≥ N (Proposition 3.1), which implies that [x; y]∩Λ(w, f) ∕= ∅. This is a contradiction. Since x, y ∕∈ R(f), we see that there exist δ0 ∈ (0, 1), t1 > 0 and t2 > 0 such that M(fn(x), x, t1) ≤ 1−δ0 and M(fn(y), y, t2) ≤ 1−δ0 for any n ∈ N. Thus by taking subsequence we can assume that fn([x; y]) ∩ [x1; y1] = ∅ for any n ∈ N and choose two sequences of positive integers µ1, µ2, · · · and λ1, λ2, · · · such that (1) fµn(xn) = f λn(yn) = w for all n ∈ N. (2) (i) fµn([xn; x]) ⊃ [x1; x] for any n ∈ N or (ii) fµn([xn; x]) ⊃ [y; y1] for any n ∈ N. (3) (i) fλn([y; yn]) ⊃ [x1; x] for any n ∈ N or (ii) fλn([y; yn]) ⊃ [y; y1] for any n ∈ N. If (2(i)) holds, then write E1 = [x1; x] and for n ≥ 2, write En = En−1 ∩ f−µ1−···−µn−1([xn; x]). Thus En+1 ⊂ En and En ∕= ∅ for all n ∈ N, which implies that ∩∞n=1En ∕= ∅. Let u ∈ ∩∞n=1En. Then one has fµ1+···+µn−1(u) ∈ [xn; x] and u ∈ fµn([xn; x]) for all n ∈ N, from which we see that x ∈ AΓ(f). If (3(i)) holds, then using arguments similar to the ones developed in the proof of above case, also we can show that y ∈ AΓ(f). If (2(ii)) and (3(ii)) hold, then fλn+µn([yn; y]) ⊃ [y1; y] and fλn+µn([x; xn]) ⊃ [x; x1] for any n ∈ N. In a similar fashion, also we can show that x, y ∈ AΓ(f). The proof is completed. □ Lemma 4.2. Let f ∈ C0(I). Then R(f) ∪ AΓ(f) ⊂ ω(R(f) ∪ AΓ(f)). Proof. If x ∈ R(f), then x ∈ ω(x, f) ⊂ ω(R(f) ∪ AΓ(f)). In the following we show that x ∈ ω(R(f) ∪ AΓ(f)) if x ∈ AΓ(f) − R(f). Since x ∈ AΓ(f) − R(f), there exist δ0 ∈ (0, 1) and t1 > 0 such that M(fn(x), x, t1) ≤ 1 − δ0 for any n ∈ N and without loss of generality we may assume that there exist a sequence of points x0, x1, · · · , xn, · · · ∈ (x; b], and two sequences of positive integers λ1 ≤ λ2 ≤ · · · and µ1 < µ2 < · · · such that: (1) M(x, xn, t1) > 1 − δ0/2 for every n ∈ N!. (2) fλn(xn) = x0 and xn ∈ (x; xn−1) for every n ∈ N and limn−→∞ M(xn, x, t) = 1 (for any t > 0). (3) fµn(x0) ∈ (x; x0) and fµn+1(x0) ∈ (x; fµn(x0)) for every n ∈ N and limn−→∞ M(f µn(x0), x, t) = 1 (for any t > 0). c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 291 T. Sun, L. Li, G. Su, C. Han and G. Xia By Corollary 3.4 we see that there exists a sequence of periodic points p1, p2, · · · such that pn+1 ∈ (x; pn) for any n ∈ N and limn−→∞ M(pn, x, t) = 1 (for any t > 0). Without loss of generality we may assume that x < · · · < x2n < x2n−1 < fµn(x0) < · · · < fµ3(x0) < x4 < x3 < f µ2(x0) < x2 < x1 < p2 < f µ1(x0) < p1 < x0. Let En be the connected component of I − Λ(fµ1(x0), f) containing fn+µ1(x0) and E0 = [u; v]. Then f n([u; v]) ⊂ En. We claim that [u; v] ∩ (AΓ(f) ∪ R(f)) ∕= ∅. Without loss of generality we may assume that [u; v]∩R(f) = ∅. Then [u; v] ⊂ [p2, p1] and fn([u; v]) ∕⊂ [u; v] for all n ∈ N and the following two statements hold: (4) (i) u ∈ Λ(fµ1(x0), f), or (ii) there exist u1, u2, · · · ∈ Λ(fµ1(x0), f) with p2 < u1 < u2 < · · · < u such that limn−→∞ M(un, u, t) = 1 (for any t > 0). (5) (i) v ∈ Λ(fµ1(x0), f), or (ii) there exist v1, v2, · · · ∈ Λ(fµ1(x0), f) with v < · · · < v2 < v1 < p1 such that limn−→∞ M(vn, v, t) = 1 (for any t > 0). Now we show that u ∕∈ Λ(fµ1(x0), f). Otherwise, if u ∈ Λ(fµ1(x0), f), then there exists some n ∈ N such that fn(u) = fµ1(x0) ∕= u. By Lemma 3.2 and Corollary 3.4 we have fn(v) > v and v ∕∈ Λ(fµ1(x0), f) since [u; v] ∩ P(f) = ∅. Thus by (5(ii)) we see that there exists N ∈ N such that vn ∈ fn((u; v)) for any n ≥ N, which contradicts the definition of E0. In a similar fashion we can also show that v ∕∈ Λ(fµ1(x0), f). Since u, v ∕∈ Λ(fµ1(x0), f), we know that (4(ii)) and (5(ii)) hold. By Lemma 4.1 we see that {u, v} ∩ AΓ(f) ∕= ∅. The claim is proven It follows from above claim that x ∈ ω(u, f) ∩ ω(v, f). The proof is com- pleted. □ Lemma 4.3. Let f ∈ C0(I). Then ω(Ω(f)) ⊂ R(f) ∪ AΓ(f). Proof. We may assume that y ∈ ω(Ω(f)) − R(f). Then there exist δ0 ∈ (0, 1) and t1 > 0 such that M(f n(y), y, t1) ≤ 1 − δ0 for any n ∈ N and without loss of generality we may assume that there exists some z ∈ Ω(f) such that y ∈ ω(z, f) and there exist two sequences of positive integers µ1 < µ2 < · · · and λ1 < λ2 < · · · , a sequence of points z1, z2, · · · such that (1) limn−→∞ M(f µn(z), y, t) = limn−→∞ M(f λn(zn), z, t) = limn−→∞ M(zn, z, t) = 1 (for any t > 0). (2) y < · · · < fµn(z) < · · · < fµ2(z) < fµ1(z) with M(fµn(z), y, t1) > 1 − δ0/2. For every n ≥ 2, there exists a point zi with sn = λi +µn −µn+4 > 0 such that fµn+5(z) < un+4 = f µn+4(zi) < f µn+3(z) < fµn+2(z) < fµn+1(z) < vn = f λi+µn(zi) = f sn(un+4) < f µn−1(z). If there exist k1 < k2 < · · · such that fskn (y) > y, then fskn (y) > fµ1(z) and fskn ([y; ukn+4]) ⊃ [vkn; fµ1(z)], then there exists wn ∈ (y; ukn+4] such that fskn (wn) = f µ1(z) for any n ≥ 2 with wn −→ y, which implies that y ∈ AΓ(f). In the following we may assume that fsn(y) < y for any n ≥ 2. c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 292 The depth and the attracting centre for map Since fsn([y; un+4]) ⊃ [y; vn] for any n ≥ 2, there exists wn ∈ [y; un+4] such that fsn(wn) = wn−1 for any n ≥ 2, w1 = fµ3(z) and wn −→ y, which implies that y ∈ AΓ(f). The proof is completed. □ Theorem 4.4. Let f ∈ C0(I). Then for any n ∈ N, ωn+2(f) = ω2(f) = ω(Ω(f)) = ω(R(f) ∪ AΓ(f)) and the depth of the attracting centre of f is at most 2. Proof. It follows from Lemma 4.2 and Lemma 4.3 that ω(R(f) ∪ (AΓ(f))) ⊂ ω2(f) ⊂ ω(Ω(f)) ⊂ R(f) ∪ (AΓ(f) ⊂ ω(R(f) ∪ AΓ(f)). The last implies that ω(R(f) ∪ (AΓ(f))) = ω2(f) = ω(Ω(f)) = R(f) ∪ (AΓ(f) = ω(R(f) ∪ AΓ(f)). Thus we know that for any n ∈ N, ωn+2(f) = ω2(f) = ω(Ω(f)) = ω(R(f) ∪ AΓ(f)) = ω(R(f) ∪ AΓ(f)). The proof is completed. □ 5. Conclusion In this paper, we introduce the notion of fuzzy metric interval, and study the depth and the attracting centre for a continuous map f on a fuzzy metric interval, and show that Ω2(f) = R(f) and the depth of f is at most 2, and ω3(f) = ω2(f) and the depth of the attracting centre of f is at most 2. Acknowledgements. The authors thank the referee for his/her valuable sug- gestions which improved the paper. References [1] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Sys. 64 (1994), 395–399. [2] M. Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets Sys. 27 (1989), 385–389. [3] V. Gregori and J. J. 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