Home The fuzzy metric space based on fuzzy measure
Article Open Access

The fuzzy metric space based on fuzzy measure

  • Jialiang Xie , Qingguo Li EMAIL logo , Shuili Chen and Huan Huang
Published/Copyright: September 20, 2016
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 14 Issue 1

Abstract

In this paper, we study the relation between a fuzzy measure and a fuzzy metric which is induced by the fuzzy measure. We also discuss some basic properties of the constructed fuzzy metric space. In particular, we show that the nonatom of fuzzy measure space can be characterized in the constructed fuzzy metric space.

Keywords: Fuzzy measure; Fuzzy metric space; t-Norm, Nonatom; Convex
MSC 2010: 54E70; 54A40; 60B05

1 Introduction

The theory of metric space is an important topic in topology. The methods of constructing a fuzzy metric have been extensively studied [1–4]. It is worth noting that George and Veeramani [5] introduced the concept of a fuzzy metric with the help of continuous t-norms. Despite being restrictive, this kind of fuzzy metric provides a more natural and intuitive way to connect with the metrizable topological spaces. This concept has been widely used in various papers devoted to fuzzy topology [5–11]. It also has been applied to color image filtering to improve image quality (see [9] and the references therein).

On the other hand, measure theory is one of the most important theories in mathematics and it has been extensively studied. The concept of fuzzy measure was first introduced by Sugeno [12]. It can be regarded as an extension of classical measure in which the additivity is replaced by a weaker condition, monotonicity. Klement et. al establish the axiomatic theory of fuzzy σ-algebras and develop a measure theory of fuzzy sets [13–15]. So far, there are many different classes of fuzzy measures, such as possibility measure [16, 17], decomposable measure [18–20], pseudo-additive measure [21, 22], and T-measure [23–26] etc. A systematic study of fuzzy measure theory can be found in [27–30].

Recently, the study of constructing a fuzzy metric using a fuzzy measure technique has been actively pursued. In particular, a fuzzy Prokhorov metric and ultrametric defined on the set of all probability measures in a compact fuzzy metric space have been developed in [31, 32]. Cao et. al [33] introduce fuzzy analogue of the Kantorovich metric among the set of possibility distributions. In [34, 35], the authors discuss the relations between the decomposable measure and the fuzzy metric. More specifically, it has been proven that, with a Hausdorff fuzzy pseudo-metric constructed on its power set, a stationary fuzzy ultrametric space can induce a σ-⊥-superdecomposable measure. The authors of [35] further use a topological approach to extend the t-conorm-based decomposable measures by introducing a fuzzy pseudometric structure on an algebra of sets. In [36] we constructed a pseudo-metric (in the sense of Pap) on the measurable sets of a given σ-⊥-decomposable measure, and then analyzed the connection between the induced pseudo-metric and the σ-⊥-decomposable measure.

In this paper we focus on the following problems: how to construct a fuzzy metric by using a fuzzy measure developed in [14, 15] and what is the relation between these two? And what is the relations between these two? Specifically, by introducing the concept of an equivalence relation on fuzzy measurable sets, we construct a fuzzy metric on the associated quotient sets from a given fuzzy measure. Furthermore, we study some basic properties of the constructed fuzzy metric space such as completeness and continuity. To gain better insight into our proposed method of constructing a fuzzy metric, we study the properties of the constructed fuzzy metric which can precisely reflect those of fuzzy measure. As an illustration we obtain that the nonatom of fuzzy measure space can be characterized in the constructed fuzzy metric space.

The rest of the paper is organized as follows. In Section 2, some basic notions and results are given. Sections 3 and 4 are devoted to constructing a fuzzy metric and discussing its properties. In Section 5, we discuss the relationships between the constructed fuzzy metric and the fuzzy measure. Finally, some concluding remarks are given in Section 6.

2 Preliminaries

We start this section by recalling the concept of triangular norms from [20, 37]. They are an important tool in extending the classical metric space to fuzzy metric space.

Definition 2.1 (Klement et al.)

A function ⊤ : [0, 1]2 → [0, 1] is called triangular norm (t-norm for short) if it satisfies the following conditions for all a, b, c, d ∈ [0, 1]:

  1. a⊤1 = a. (boundary condition)

  2. a⊤b ≤ c⊤dwhenevera ≤ c and b ≤ d.(monotonicity)

  3. a⊤b = b⊤b. (commutativity)

  4. a⊤(b⊤c) = (a⊤b)⊤c. (associativity)

A t-Norm ⊤ is said to be continuous if it is a continuous function in [0, 1]2. Typical examples of continuous t-Norms are the minimum TM, the product TP and the Łukasiewicz t-norm TL, which are given by, respectively:

TM(x, y)=min(x, y), TP(x, y)=x⋅y, TL(x, y)=max(0, x+y−1).

Because of the associative property, the t-Norm ⊤ can be extended by induction to n-ary operation by setting

⊤i=1nxi=(⊤i=1n−1xi)⊤xn

Due to monotonicity, for each sequence (xi)i∈ℕ of elements of [0, 1], the following limit can be considered:

⊤i=1∞xi=limn→∞⊤i=1nxi.

Next we recall the concept of a fuzzy metric with the help of the continuous t-norm, which is a generalization of the concept of Menger probabilistic metric to the fuzzy setting.

Definition 2.2 (George and Veeramani [5])

The 3-tuple (X, M, ⊤) is said to be a fuzzy metric space if X is an arbitrary nonempty set, ⊤ is a continuous t-norm and M is a fuzzy set on X2 × (0, ∞) satisfying the following conditions, for all x, y, z ∈ X, t, s > 0:

  1. M(x, y, t) > 0,

  2. M(x, y, t) = 1 iff x = y,

  3. M(x, y, t) = M(x, y, t)

  4. M(x, z, t + s) ≥ M(x, y, t)⊤M(y, z, t)

  5. M(x, y, ·) : (0, ∞) → (0, 1] is continuous.

If the condition (GV2) is replaced by the condition (GVp): M(x, x, t) = 1, then (X, M, ⊤) is said to be a fuzzy pseudometric space.

It was proved in [5] that in a fuzzy metric space X, the function M(x, y, ·) is nondecreasing for all x, y ∈ X. A sequence (xi)i∈ℕ in a fuzzy metric space (X, M, ⊤) is said to converge [6] to x if limi→∞M(xi, x, t) = 1 for all t > 0; a sequence (xi)i∈ℕ in a fuzzy metric space (X, M, ⊤) is said to be Cauchy [6] if limi, j →∞M(xi, x, t) = 1 for all t > 0; (X, M, ⊤) is said to be complete [8] if every Cauchy sequence is convergent. A mapping f from a fuzzy metric space (X, M, ⊤1) to a fuzzy metric space (X, M, ⊤2) is called uniformly continuous [7] if for each ε ∈ (0, 1) and each t > 0, their exist r ∈ (0, 1) and s > 0 such that N(f(x), f(y), t) > 1 – ε whenever M(x, y, s) > 1 – r.

In this follow-up, we recall several concepts from the measure theory of fuzzy sets (see e.g. [13–15]).

Definition 2.3

Let X be a nonempty set, I the unit interval [0, 1]. A subsetAof IX is a fuzzy σ-algebra iff

  1. 0, 1 ∈ A,

  2. A ∈ Aimplies1 — A ∈ A,

  3. if{Ai}i=1∞is a sequence inA, then∨i=1∞Ai = sup Ai ∈ A.

Definition 2.4

A finite fuzzy measure (or F-measure) on a fuzzyσ-algebraAis a function μ : A → [0, ∞) satisfying:

  1. μ(0) = 0,

  2. for A ∈ A, μ(1 – A) = μ(1) – μ(A),

  3. for A, B ∈ A, μ(A ∨ B) + μ (A ∧ B) = μ(A) + μ(B),

  4. if{Ai}i=1∞is a sequence inAsuch thatAi ↗ A, A ∈ A, thenμ(A) sup = μ(Ai).

We call (X, A, μ) an F-measure space, elements of A are referred as fuzzy measurable sets.

3 Constructing fuzzy metric based on F-measure

In this follow-up, ⊤ stands for the minimum t-norm TM. The following result is the natural fuzzy metric structure on fuzzy measurable sets.

Theorem 3.1

Let (X, A, μ) be an F-measure space. If we define the fuzzy set M on A2 × (0, ∞) by

M(A, B, t)=tt+μ(A∨B)−μ(A∧B),

where A, B ∈ A. Then M is a fuzzy pseudometric on A.

Proof

From the definition of M, it is obvious that for any A, B ∈ A, t > 0, we have (i) M(A, B, t) > 0; (ii) M(A, A, t) = 1 and M(A, B, t) = M(B, A, t); (iii) M(x, y, ·) : (0, ∞) → [0, 1] is continuous. The only thing that we need to prove is the triangular inequality. For any A, B, C ∈ A and t, s > 0, we have M(A, B, t)=tt+μ(A∨B)−μ(A∧B), M(B, C, s)=ss+μ(B∨C)−μ(B∧C) and M(A, C, t+s)=t+st+s+μ(A∨C)−μ(A∧C). We are going to verify M(A, B, C, t + s) ≥ M(A, B, t) ⊤ M(B, C, s). For each A, B, C, ∈ A, we have

μ(A∨B)+μ(B∨C)+μ(A∧C)=μ(A∨B∨C)+μ(A∨B)∧(B∨C)+μ(A∧C)=μ(A∨B∨C)+μB∨(A∧C)+μ(A∧C)≥μ(A∨C)+μB∧(A∧B)∨C+μ(A∧B∧C)=μ(A∨C)+μ(A∧B)+μ(B∧C)

and so

μ(A∨C)−μ(A∧C)≤μ(A∨B)−μ(A∨B)+μ(B∨C)−μ(B∨C)

we can distinguish two cases: M(B, C, s) ≥ M(A, B, s) or M(A, B, t) ≥ M(B, C, s).

Case (i). If M(B, C, s) ≥ M(A, B, t), or equivlently,

ss+μ(B∨C)−μ(B∧C)≥tt+μ(A∨B)−μ(A∧B)

and hence

s(μ(A∨B)−μ(A∧B))≥t(μ(B∨C)−μ(B∧C))

Consequently,

t(μ(A∨C)−μ(A∧C))≤t(μ(A∨B)−μ(A∧B))+t(μ(B∨C)−μ(B∧C))≤t(μ(A∨B)−μ(A∧B))+s(μ(A∨B)−μ(A∧B)).

This implies that

tt+μ(A∨B)−μ(A∧B)≤t+st+s+μ(A∨C)−μ(A∧C),

and hence

M(A, C, t+s)≥M(A, B, t)=M(A, B, t)⊤M(B, C, s).

Case (ii). Similar to (i). Thus, M is a fuzzy pesudometric on A.

Remark 3.2

Based on Theorem 3.1, M is a fuzzy pseudometric onAunder any left-continuous t-norms since TM is the biggest left-continuous t-norm.

Generally speaking, the fuzzy pseudometric space ( A, M, ⊤) is usually not a fuzzy metric space. But we can construct a fuzzy metric space from a fuzzy pseudometric metric space ( A, M, ⊤) and, at the same time, keep the general characteristics of the fuzzy pseudometric metric space.

Lemma 3.3

Let (X, A, μ) be an F-measure space. For each A, B ∈, we define the relation “~” on A : A ~ B if and only if μ(A ∨ B) = μ(A ∧ B). Then ~ is an equivalence relation on A.

Proof

The identity μ, (A ∨ C) - μ, (A ∧ C) ≥ μ(A ∨ B) - μ(A ∧ B) + μ(B ∨ C) μ(B ∧ C) - which was proved in Theorem 3.1, and the monotonicity of μ imply that ∼ is transitive, and it is clearly reflexive and symmetric.

Theorem 3.4

Let (X, A, μ) be an F-measure space. LetA/μ be the set of all equivalence classes for the relation “∼”. The fuzzy pseudometric M has a natural extension to

M([A], [B], t)=M(A, B, t),

where [A] ([B]) denote the equivalence class of A (B). Then M is a fuzzy metric onA/μ.

Proof

We first prove that M([A], [B], t) is well defined on A/μ. If A1 ∈ [A] and B1 ∈ [B], we conclude that μ(A ∨ B1) - μ(A ∧ B1) = μ(A ∨ B) - μ(A ∧ B). In fact

μ(A1∨B1)−μ(A1∧B1)≤μ(A1∨A)−μ(A1∧A)+μ(A∨B)−μ(A∧B)+μ(B∨B1)−μ(B∧B1).

Since A1 ∈ [A] and B1 ∈ [B], we have μ(A1 ∨ A) - μ(A1 ∧ A) = μ(B ∨ B1) - μ(B ∧ B1) = 0, then μ(A1 ∨ B1) - μ(A1 ∧ B1) ≤ μ(A ∨ B) - μ(A ∧ B). In a similar way that μ(A ∨ B) - μ(A ∧ B) ≤ μ(A1 ∨ B1) - μ(A1 ∧ B1) holds. We obtain M(A1, B1, t) = M(A, B, t), which shows that M([A] [B], t) does not depend on our choice of representatives in the equivalence classes and is well-defined on A/μ. Secondly, we need to prove that M is a fuzzy metric on A/μ. It is obvious that M([A] [B], t) = 1 ⇔ μ(A ∨ B) - μ(A ∧ B) = 0 ⇔ [A] = [B]. Thus, M is a fuzzy metric on A/μ.

The following Lemma shows that the collection of equivalence class A/μ forms a fuzzy σ-algebra.

Lemma 3.5

Let (XAμ) be an F -measure space. Then forAi, Bi ∈ A, Ai ∼ Bi, i ∈ ℕ, we have

  1. 1 - Ai ∼ 1 - Bi

  2. A1 ∨ A2 ~ B1 ∨ B2andA1 ∨ A2 ~ B1 ∨ B2

  3. ∧i=0∞Ai∼∧i=1∞Bi.

Proof

  1. For Ai, Bi ∈ A, Ai ∼ Bi, then μ(Ai ∨ Bi) - μ(Ai ∧ Bi) = 0. Hence, 1 - Ai ∼ 1 - Bi holds by μ((1−Ai)∨(1−Bi))−μ((1−Ai)∧(1−Bi))=μ(Ai∨Bi)−μ(Ai∧Bi)=0

  2. We first show that for each P ∈ A, A1 ∨ P ∼ B1 ∨ P. Since A1 ∼ B1, we have

    μ(A1∨!P)≤μ(A1∨B1∨P)=μ(A1∨B1)+μ(P)−μ(A1∨B1)∧P≤μ(B1)+μ(P)−μ(B1∧P)=μ(B1∨P).

    Similarly,

    μ(B1∨P)≤μ(A1∨B1∨P)≤μ(A1∨P).

    Consequently,

    μ(A1∨P)=μ(A1∨B1∨P)=μ(B1∨P).

    Thus we obtain

    μ(A1∨A2)=μ(B1∨B2)=μ(A1∨A2∨B1∨B2).

    This implies that A1 ∨ A2 ∼ B1 ∨ B2. Similarly, we have A1 ∧ A2 ∼ B1 ∧ B2.

  3. For i ∈ ℕ, we have μ(Ai ∨ Bi) = μ(Ai ∧ Bi), then μ(Ai) = μ(Bi) = μ(Ai ∧ Bi). Hence

    μ(∧i=1∞Ai)=limk→∞μ(∧i=1kAi)=limk→∞μ(∧i=1kBi)=μ(∧i=1∞Bi).

    Also,

    μ(∧i=1∞Ai)=limk→∞⁡μ(∧i=1kAi)=limk→∞⁡μ(∧i=1k⁡(Ai∧Bi))=μ(∧i=1∞⁡(Ai∧Bi))=μ((∧i=1∞Ai)∧(∧i=1∞Bi)).

    Thus, ∧i=1∞Ai∼∧i=1∞Bi.

Lemma 3.6

Let (X, Aμ) be an F-measure space. If, forAi, Bi ∈ A, Ai ∼ Bi, i ∈ ℕ, Ai↗A, Bi↗BandthenA ∈ B.

Proof

The proof is straightforward.

According to Lemma 3.5 and Lemma 3.6, by means of representatives of classes, we can introduce the operations of union, intersection and complementation on A/μ:∨i=1∞[Ai]=[∨i=1∞Ai];∧i=1∞[Ai][∧i=1∞Ai];1−[Ai]=[1−Ai], [Ai] ∈ A/μ denote the equivalence class of Ai in A. Hence, A/μ is a fuzzy σ algbra. We therefore properly define μ on A/μ by setting

μ([A])=μ(A),forallA∈A.

The pair (A/μ, μ) is a said to be an F-measure algebra.

For convenience and simplicity, we denote members [A] of A/μ by A, and functions μ: A/μ → [0, ∞) by μ: A/μ → [0, ∞)

4 Properties of the fuzzy metric space( A, M, ⊤)

In this section, we study some properties of the fuzzy metric space ( A, M, ⊤) based on F -measure µ.

Theorem 4.1

Let (X, A, µ) be an F -measure space and M be the fuzzy metric on A defined in Theorem 3.4. Then the maps (i) (A, B) ↦ A ∨ B and (ii) (A, B) ↦ A ∧ B are uniformly continuous fromA × AtoA.

Proof

  1. For any A1, B1, A2, B2 ∈ , t, s > 0, we first prove the relation M(A1, A2, t) ⊤ M(B1, B2, s) ≤ M(A1 ∨ B1, A2 ∨ B2, t + s). For each A1, B1, A2, B2 ∈ A, we have

    μ(A1∨B1)∨(A2∨B2)−μ(A1∨B1)∧(A2∨B2)≤μ(A1∨A2)∨(B1∨B2)−μ(A1∧A2)∨(B1∧B2)=μ(A1∨A2)+μ(B1∨B2)−μ(A1∨A2)∧(B1∨B2)−μ(A1∧A2)−μ(B1∧B2)+μ(A1∧A2)∧(B1∧B2)≤μ(A1∨A2)−μ(A1∧A2)+μ(B1∨B2)−μ(B1∧B2).

    we can distinguish two cases: M(A2, B2, s) ≥ M(A1, B1, t) or M(A1, B1, t) ≥ M(A2, B2, s).

    Case one. If M(B1, B2, s) ≥ M(A1, A2, t), or equivlently,

    ss+μ(B1∨B2)−μ(B1∧B2)≥tt+μ(A1∨A2)−μ(A1∧A2)

    and hence

    s(μ(A1∨A2)−μ(A1∧A2))≥t(μ(B1∨B2)−μ(B1∧B2))

    In consequence,

    tμ(A1∨B1)∨(A2∨B2)−μ(A1∨B1)∧(A2∨B2)≤tμ(A1∨A2)−μ(A1∧A2)+tμ(B1∨B2)−μ(B1∧B2)≤tμ(A1∨A2)−μ(A1∧A2)+sμ(A1∨A2)−μ(A1∧A2)

    This implies that

    tt+μ(A1∨A2)−μ(A1∧A2)≤t+st+s+μ((A1∨B1)∨(A2∨B2))−μ((A1∨B1)∧(A2∨B2))

    and hence

    M(A1∨B1, A2∨B2, t+s)≥M(A1, A2, t)=M(A1, A2, t)⊤M(B1, B2, s).

    Case two. Similar to case one.

    Thus, M(A1 ∨ B1, A2 ∨ B2, t + s) ≥ M(A1, A2, t)⊤M(B1, B2, s).

    If we fix ε ∈ (0, 1), there exists δ ∈ (0, 1) such that (1 -δ)⊤(1-δ) > 1 - ε, by the continuity of ⊤. Thus, for each ε ∈ (0, 1) there exist δ ∈ (0, 1) such that M(A1 ∨ B1, A2 ∨ B2, t + s) > 1-ε whenever M(A1, A2, t) > 1-δ and M(B1, B2, s) > 1-δ. We conclude that (A, B) ↦ A ∨ B is uniformly continuous.

  2. Proceeding as in the proof of (i), we prove the relation M(A1 ∧ B1, A2 ∧ B2, t + s) ≥ M(A1, A2, t)⊤M(B1, B2, s), for any A1, B1, A2, B2 ∈ A, t, s > 0. Moreover, we conclude that (A, B) ↦ A ∧ B is uniformly continuous by using similar technique in (i).

Now, we shall consider the completeness of the fuzzy metric space A, M, ⊤.

Lemma 4.2

Let (X, A, µ) be an F -measure space and Ai (i = 1, 2, ...) be elements ofA. Then the fuzzy metric M as defined in Theorem 3.4 satisfies the following properties:

  1. M(A1, A2, t)≤M(A1, A1∨A2, t)for each t > 0;

  2. M(Ak, Ak+1, tk)≤M(∨i=1kAi, ∨i=1k+1Ai, ∑i=1kti), for all k ∈ ℕ, ti > 0.

Proof

  1. For any A1, A2 ∈ A, we get

    μA1∨(A1∨A2)−μA1∧(A1∨A2)=μ(A1∨A2)−μ(A1)≤μ(A1∨A2)−μ(A1∧A2)

    and so M(A1, A2, t) ≤ M(A1, A1 ∨ A2, t), for each t > 0.

  2. By (i) and Theorem 4.1 (i), for all k ∈ ℕ, ti > 0, we have

    M(∨ki=1Ai,∨k+1i=1Ai,∑i=1kti)=M((∨k−1i=1Ai)∨Ak,(∨k−1i=1Ai)∨(Ak∨Ak+1),∑i=1kti)≥M(∨k−1i=1Ai,∨k−1i=1Ai,∑i=1k−1ti)⊤M(Ak,Ak∨Ak+1,tk)≥1⊤M(Ak,Ak+1,tk)=M(Ak,Ak+1,tk)

Lemma 4.3

Let (xn)n∈ℕbe a Cauchy sequence in fuzzy metric space (x, M, ⊤). If there is a subsequence (xk(n)n∈ℕof (xn)n∈ℕthat converges to x in X, then the Cauchy sequence (xn)n∈ℕconverges to x.

Proof

Let (xk(n))n∈ℕ be a subsequence of (xn)n∈ℕ Then, given r with 0 < r < 1 and t > 0, there is an n0 ∈ ℕ such that for each n≥n0, M(x, xk(n), t2)>1−s, where s > 0 satisfies (1-s)⊤(1-s) > 1-r. Since (xn)n∈ℕ be a Cauchy sequence, there is n1 ≥ k(n0) such that M(xn, xm, t2)>1−s for each n, m ≥ n1. Therefore, for each n ≥ n1, we have

M(x, xnt)≥M(x, xk(n), t2)⊤M(x, xk(n), t2)≥(1−s)⊤(1−s)>1−r.

We conclude that the Cauchy sequence (xn)n∈ℕ converges to x.

Theorem 4.4

The fuzzy metric space (A, M, ⊤) based on F -measure µ is complete.

Proof

Let (An)n∈ℕ be a Cauchy sequence in A. We need to show that there exists a set A ∈ A such that limn→∞M(An, A, t)=1 for each t > 0. By Lemma 4.3, it suffices to show that there is a convergent subsequence in (An)n∈ℕ. Hence, passing to subsequence, there exists n0 ∈ ℕ such that M(An, An+1, tn) > 1-2−n for all n ≥ n0, tn > 0.

For n ∈ ℕ, let Bp=∨i=nn+pAi. Then (Bp) is a monotonic increasing sequence and limp→∞Bp=∨i=n∞Ai≜Dn so, μ(Dn)=limp→∞μ(Bp). By Lemma 4.2, we have

    M(An, Bp, ptn+(p−1)tn+1+⋯+tn+p−1)≥M(An, B1, tn)⊤M(B1, B2, tn+tn+1)⊤⋯⊤M(Bp−1, Bp, tn+tn+1+⋯+tn+p−1)≥M(An, An+1, tn)⊤M(An+1, An+2+tn+1)⊤⋯⊤M(An+p−1, +An+p+tn+p−1)>(1−2−n)⊤(1−2−n−1)⊤⋯⊤(1−2−n−p+1)

Hence, limn→∞limp→∞M(An, Dn, ptn+(p−1)tn+1+⋯+tn+p−1)=1. Moreover, since the sequence (Dn)n∈ℕ is monotonic decresing, we can set A=∩n=1∞Dn, t=limn→∞limp→∞(ptn+(p−1)tn+1+⋯+tn+p−1), A ∈ A in the light of A is a fuzzy σ-algebra. Thus, limn→∞M(An, A, t)=1 and complete the proof.

5 The correspondence between fuzzy metric space and F-measure space

In the following, we give the characteristics of the nonatom of the F-measure algebra (A, µ).

Definition 5.1

If, for two distinct elements A, B ⊂ X, there exists t, s > 0 and an element P ⊂ X, different from both A and B such that M(A, B, t + s) = M(A, P, t)⊤M(P, B, s), then fuzzy metric space (X, M, ⊤) is called convex..

Definition 5.2

Let (A, µ) be an F-measure algebra. Then the measure µ is called nonatom if for A, B ∈ A, A ≤ B and µ(A) < µ(B), there exists P ∈ A, A ≤ P ≤ B such that µ(A) < µ(P) <(B).

Theorem 5.3

The F-measure algebra (A, µ) is nonatom if and only if fuzzy metric space (A, M, ⊤) is convex.

Proof

(Sufficiency) Suppose that A is convex. Let A, C ∈ A with A ≤ C and µ(A) < µ(C). Since A ≠ C, there exists t, s > 0 and B ∈ A, which is different from both A and C such that M(A, C, t + s) = M(A, B, t)⊤M(B, C, s). We can distinguish two cases: M(A, B, t) ≥ M(B, C, s) or M(B, C, s) ≥ M(A, B, t). Since M is nondecreasing with respect to t, it is easy to verify that in the two cases the inequalities

μ(A∨C)−μ(A∧C)≥μ(A∨B)−μ(A∧B),  μ(A∨C)−μ(A∧C)≥μ(B∨C)−μ(B∧C)(∗)

hold. Set P = A ∨ (B ∧ C) = (A ∨ B) ∧ C, then A ≤ P ≤ C.

Suppose that µ(A ∧ B) = µ(B ∧ C). Then

μB∨C−μB∧C=μB∨C−μA∧B≥μC−μA=μA∨C−μA∧C

which contradicts the condition (*). Hence, we have μ(B ∧ C ) > μ(A ∧ B). Consequently, μ(P)- μ(A) = μ(B ∧ C)-μ(A ∧ B) > 0, i.e, μ(P) > μ(A).

On the other hand, suppose that μ(A ∨ B) = μ(B ∨ C). Then

μA∨B−μA∧B=μB∨C−μA∧B≥μC−μA=μA∨C−μA∧C

which contradicts the condition (∗). Hence, we obtain μ(B ∨ C) > μ(A ∨ B). Consequently, μ(C)-μ(P) = μ(B ∨ C)-μ(A ∨ B) > 0, i.e, μ(C) > μ(P). Thus the F-measure μ is nonatom.

(Necessity) Suppose that μ is nonatom, A and C ∈ A with A ≠ C.

Case (i). If A ∧ C ≠ A and A ∧ C ≠ C, put B = A ∧ C. Then μ(A ∨ B)-μ(A ∧ B)+μ(B ∨ C)-μ(B ∧ C) = μ(A)-μ(A ∨ C)+μ(C)-μ(A ∧ C) = μ(A ∨ C)-μ(A ∧ C).

Case (ii). Suppose A ∧ C = A. Then μ(A ∧ C) = μ(A) and also μ(A ∨ C) = μ(C). We deduce that μ(A) < μ(C) in the light of (A, µ) is F-measure algebra. By the assumption, there exists B ∈ A, A ≤ B ≤ C such that μ(A) < μ(B) < (C). Hence μ(A ∨ B)-μ(A ∧ B)+μ(B ∨ C)-μ(B ∧ C) = μ(B)-μ(A)+μ(C)-μ(B) = μ(A ∨ C)-μ(A ∧ C). So, both in the case (i) and (ii) we deduce that

μ(A∨B)−μ(A∧B)+μ(B∨C)−μ(B∧C)=μ(A∨C)−μ(A∧C). (∗∗)

Next we need to prove that there exists t, s > 0, such that M(A, C, t + s) = M(A, B, t)⊤M(B, C, s). If there exists t, s > 0, such that

tt+μ(A∨B)−μ(A∧B)=t+st+s+μ(A∨C)−μ(A∧C),

so

μ(A∨B)−μ(A∧B)=t(μ(A∨C)−μ(A∧C))t+s,

according to the equality (∗∗), we get

μ(B∨C)−μ(B∧C)=s(μ(A∨C)−μ(A∧C))t+s,

and hence

ss+μ(B∨C)−μ(B∧C)=t+st+s+μ(A∨C)−μ(A∧C).

In consequence, M(A, C, t + s) = M(A, B, t)⊤M(B, C, s). Also, if there exists t, s > 0, such that

ss+μ(B∨C)−μ(B∧C)=t+st+s+μ(A∨C)−μ(A∧C),

we can get M(A, C, t + s) = M(A, B, t)⊤M(B, C, s) by the same method as employed in the above. This completes the proof of theorem.

6 Conclusion

In this paper, by constructing a fuzzy metric on the fuzzy measurable sets, we studied the relations between these two. In particular, several satisfactory properties of the constructed fuzzy metric have been obtained. In addition, we investigated the charaterization of the nonatom of the fuzzy measure and the corresponding properties of constructed fuzzy metric space. The main results and methods presented in this paper generalize some well known results in [38, 39].

Acknowledgement

The authors are enormously grateful to the editors and the anonymous reviewers for their professional comments and valuable suggestions. This work is supported in part by the National Natural Science Foundation of China (No. 11371130, 61103052, 11401195), the Natural Science Foundation of Fujian Province (No. 2014H0034, 2016J01022), the projects of Education Department of Fujian Province (No. JA15280) and Li Shangda Discipline Construction Fund of Jimei University.

References

[1] Deng Z. K., Fuzzy pseudometric spaces, Journal of Mathematical Analysis and Applications, 1982, 86, 74–9510.1016/0022-247X(82)90255-4Search in Google Scholar

[2] Kaleva O., Seikkala S., On fuzzy metric spaces, Fuzzy Sets and Systems, 1984, 12, 215–22910.1016/0165-0114(84)90069-1Search in Google Scholar

[3] Kramosil I., Michálek J., Fuzzy metric and statistical metric spaces, Kybernetika, 1975, 11, 336–334Search in Google Scholar

[4] Yue Y. L., Shi F. G., On fuzzy pseudo-metric spaces, Fuzzy Sets and Systems, 2010, 161, 1105–111610.1016/j.fss.2009.10.001Search in Google Scholar

[5] George A., Veeramani P., On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 1994, 64, 395–39910.1016/0165-0114(94)90162-7Search in Google Scholar

[6] George A., Veeramani P., On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 1997, 90, 365–36810.1016/S0165-0114(96)00207-2Search in Google Scholar

[7] Gregori V., Romaguera S., Sapena A., Uniform continuity in fuzzy metric spaces, Rendiconti dell'Istituto di Matematica dell'University di Trieste Suppl., 2001, 2, 81–88Search in Google Scholar

[8] Gregori V., Sapena A., On fixed point theorems in fuzzy metric spaces, Fuzzy Sets and Systems, 2002, 125, 245–25310.1016/S0165-0114(00)00088-9Search in Google Scholar

[9] Gregori V., Morillas S., Sapena A., Examples of fuzzy metrics and applications, Fuzzy Sets and Systems, 2011, 170, 95–11110.1016/j.fss.2010.10.019Search in Google Scholar

[10] Shen Y. H., Qiu D., Chen W., Fixed point theorems in fuzzy metric spaces, Applied Mathematics Letters, 2012, 25, 138–14110.1016/j.aml.2011.08.002Search in Google Scholar

[11] Shen Y. H., Qiu D., Chen W., Fixed point theory for cyclic ϕ-contractions in fuzzy metric spaces, Iranian Journal of Fuzzy Systems, 2013, 10, 125–13310.1186/1687-1812-2013-133Search in Google Scholar

[12] Sugeno M., Theory of fuzzy integrals and its applications, Ph.D. Thesis, Tokyo Institute of Technology, Tokyo, 1974.Search in Google Scholar

[13] Klement E. P., Fuzzy σ-algebras and fuzzy measurable functions, Fuzzy Sets and Systems, 1980, 4, 83–9310.1016/0165-0114(80)90066-4Search in Google Scholar

[14] Klement E. P., Characterization of finite fuzzy measures using markoff-kernels, Journal of mathematical analysis and applications, 1980, 75, 330–33910.1016/0022-247X(80)90083-9Search in Google Scholar

[15] Srivastava Y. K., Fuzzy probability measures and dynamical systems, Ph.D. Thesis, Allahabad University, 1993.Search in Google Scholar

[16] Klir G. J., Generalized information theory, Fuzzy Sets and Systems, 1991, 40, 127–14210.1016/0165-0114(91)90049-VSearch in Google Scholar

[17] Dubois D., Prade M., Sabbadin R., Decision-theoretic foundations of possibility theory, European Journal of Operational, 2001, 128, 459–47810.1016/S0377-2217(99)00473-7Search in Google Scholar

[18] Dubois D., Prade M., A class of fuzzy measures based on triangular norms, International Journal of General System, 1982, 41, 59–95Search in Google Scholar

[19] Dubois D., Prade M., Aggregation of decomposable measures with application to utility theory, Theory and Decision, 1996, 41, 59–9510.1007/BF00134116Search in Google Scholar

[20] Weber S., ┴-decomposable measures and integrals for archimedean t-conorm, Journal of Mathematical Analysis and Applications, 1984, 101, 114–13810.1016/0022-247X(84)90061-1Search in Google Scholar

[21] Sugeno M., Murofushi T., Pseudo-additive measures and integrals, Journal of Mathematical Analysis and Applications, 1987, 122, 197–22210.1016/0022-247X(87)90354-4Search in Google Scholar

[22] Pap E., Pseudo-analysis, Computational Cybernetics (ICCC), 2013 IEEE 9th International Conference, IEEE, 2013, 213–21810.1109/ICCCyb.2013.6617590Search in Google Scholar

[23] Butnariu D., Klement E. P., Triangular Norm-Based Measures and Games with Fuzzy Coalitions, Springer Netherlands, 1993.10.1007/978-94-017-3602-2Search in Google Scholar

[24] Klement E. P., construction of fuzzy σ-algebras using triangular norms, Journal of mathematical analysis and applications, 1982, 85, 543–56510.1016/0022-247X(82)90015-4Search in Google Scholar

[25] Klement E. P., Characterization of fuzzy measures constructed by means of triangular norms, Journal of mathematical analysis and applications, 1982, 86, 345–35810.1016/0022-247X(82)90227-XSearch in Google Scholar

[26] Klement E. P., Weber S., Generalized measures, Fuzzy Sets and Systems, 1991, 40, 375–39410.1016/0165-0114(91)90166-NSearch in Google Scholar

[27] Denneberg D., Non-additive measure and integral, Kluwer Academic Publishers, 199410.1007/978-94-017-2434-0Search in Google Scholar

[28] Pap E., Null-additive set functions, Ister Science, 1995Search in Google Scholar

[29] Pap E., Handbook of measure Theory: In two volumes, volume 2, Elsevier Science, Amsterdam, 2002Search in Google Scholar

[30] Wang Z. Y., Klir G. J., Generalized Measure Theory, Spalgebraer Press, New York, 200910.1007/978-0-387-76852-6Search in Google Scholar

[31] Repovš D., Savchenko A., Zarichnyi M., Fuzzy prokhorov metric on the set of probability measures, Fuzzy Sets and Systems, 2011, 175, 96–10410.1016/j.fss.2011.02.014Search in Google Scholar

[32] Savchenko A., Zarichnyi M., Fuzzy ultrametrics on the set of probability measures, Topology, 2009, 48, 130–13610.1016/j.top.2009.11.011Search in Google Scholar

[33] Cao Y. Z., Sun S. X., Wang H., Chen G., A behavioral distance for fuzzy-transition systems, IEEE Transactions on Fuzzy Systems, 2013, 21, 735–74710.1109/TFUZZ.2012.2230177Search in Google Scholar

[34] Qiu D., Zhang W. Q., Li C., On decomposable measures constructed by using stationary fuzzy pseudo-ultrametrics, International Journal of General Systems, 2013, 42, 395–40410.1080/03081079.2012.758879Search in Google Scholar

[35] Qiu D., Zhang W. Q., Li C., Extension of a class of decomposable measures using fuzzy pseudometrics, International Journal of General Systems, 2013, 222, 33–4410.1016/j.fss.2012.09.010Search in Google Scholar

[36] Xie J. L., Li Q. G., Chen S. L., Gao Y., On pseudo-metric spaces induced by σ┴-decomposable measures, Fuzzy Sets and Systems, 2016, 289, 33-4210.1016/j.fss.2015.04.018Search in Google Scholar

[37] Klement E. P., Mesiar R., Pap E., Triangular norms, Kluwer Academic Publishers, Dordrecht, 200010.1007/978-94-015-9540-7Search in Google Scholar

[38] Halmos P. R., Measure Theory, Spalgebraer-Verlag, New York, 1974Search in Google Scholar

[39] Royden H. L., Real Analysis, Macmillan, NewYork, 1989Search in Google Scholar

Received: 2016-5-14
Accepted: 2016-7-5
Published Online: 2016-9-20
Published in Print: 2016-1-1

© 2016 Xie etal., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Articles in the same Issue

  1. Regular Article
  2. A metric graph satisfying w41=1 that cannot be lifted to a curve satisfying dim⁡ (W41)=1
  3. Regular Article
  4. On the Riemann-Hilbert problem in multiply connected domains
  5. Regular Article
  6. Hamilton cycles in almost distance-hereditary graphs
  7. Regular Article
  8. Locally adequate semigroup algebras
  9. Regular Article
  10. Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  11. Corrigendum
  12. Corrigendum to: parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  13. Regular Article
  14. Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix
  15. Regular Article
  16. Integral inequalities involving generalized Erdélyi-Kober fractional integral operators
  17. Regular Article
  18. Results on the deficiencies of some differential-difference polynomials of meromorphic functions
  19. Regular Article
  20. General numerical radius inequalities for matrices of operators
  21. Regular Article
  22. The best uniform quadratic approximation of circular arcs with high accuracy
  23. Regular Article
  24. Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions
  25. Regular Article
  26. A note on the rate of convergence for Chebyshev-Lobatto and Radau systems
  27. Regular Article
  28. On the weakly(α, ψ, ξ)-contractive condition for multi-valued operators in metric spaces and related fixed point results
  29. Regular Article
  30. Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces
  31. Regular Article
  32. Bounds for the Z-eigenpair of general nonnegative tensors
  33. Regular Article
  34. Subsymmetry and asymmetry models for multiway square contingency tables with ordered categories
  35. Regular Article
  36. End-regular and End-orthodox generalized lexicographic products of bipartite graphs
  37. Regular Article
  38. Refinement of the Jensen integral inequality
  39. Regular Article
  40. New iterative codes for 𝓗-tensors and an application
  41. Regular Article
  42. A result for O2-convergence to be topological in posets
  43. Regular Article
  44. A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation
  45. Regular Article
  46. Uncertainty orders on the sublinear expectation space
  47. Regular Article
  48. Generalized derivations of Lie triple systems
  49. Regular Article
  50. The BV solution of the parabolic equation with degeneracy on the boundary
  51. Regular Article
  52. Malliavin method for optimal investment in financial markets with memory
  53. Regular Article
  54. Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces
  55. Regular Article
  56. On annihilators in BL-algebras
  57. Regular Article
  58. On derivations of quantales
  59. Regular Article
  60. On the closed subfields of Q¯~p
  61. Regular Article
  62. A class of tridiagonal operators associated to some subshifts
  63. Regular Article
  64. Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations
  65. Regular Article
  66. Weighted fractional differential equations with infinite delay in Banach spaces
  67. Regular Article
  68. Laplace-Stieltjes transform of the system mean lifetime via geometric process model
  69. Regular Article
  70. Various limit theorems for ratios from the uniform distribution
  71. Regular Article
  72. On α-almost Artinian modules
  73. Regular Article
  74. Limit theorems for the weights and the degrees in anN-interactions random graph model
  75. Regular Article
  76. An analysis on the stability of a state dependent delay differential equation
  77. Regular Article
  78. The hybrid mean value of Dedekind sums and two-term exponential sums
  79. Regular Article
  80. New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations
  81. Regular Article
  82. On the concept of general solution for impulsive differential equations of fractional-order q ∈ (2,3)
  83. Regular Article
  84. A Riesz representation theory for completely regular Hausdorff spaces and its applications
  85. Regular Article
  86. Oscillation of impulsive conformable fractional differential equations
  87. Regular Article
  88. Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex
  89. Regular Article
  90. Homoclinic solutions of 2nth-order difference equations containing both advance and retardation
  91. Regular Article
  92. When do L-fuzzy ideals of a ring generate a distributive lattice?
  93. Regular Article
  94. Fully degenerate poly-Bernoulli numbers and polynomials
  95. Commentary
  96. Commentary to: Generalized derivations of Lie triple systems
  97. Regular Article
  98. Simple sufficient conditions for starlikeness and convexity for meromorphic functions
  99. Regular Article
  100. Global stability analysis and control of leptospirosis
  101. Regular Article
  102. Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise
  103. Regular Article
  104. The fuzzy metric space based on fuzzy measure
  105. Regular Article
  106. A classification of low dimensional multiplicative Hom-Lie superalgebras
  107. Regular Article
  108. Structures of W(2.2) Lie conformal algebra
  109. Regular Article
  110. On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
  111. Regular Article
  112. Parabolic Marcinkiewicz integrals on product spaces and extrapolation
  113. Regular Article
  114. Prime, weakly prime and almost prime elements in multiplication lattice modules
  115. Regular Article
  116. Pochhammer symbol with negative indices. A new rule for the method of brackets
  117. Regular Article
  118. Outcome space range reduction method for global optimization of sum of affine ratios problem
  119. Regular Article
  120. Factorization theorems for strong maps between matroids of arbitrary cardinality
  121. Regular Article
  122. A convergence analysis of SOR iterative methods for linear systems with weak H-matrices
  123. Regular Article
  124. Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions
  125. Regular Article
  126. Some congruences for 3-component multipartitions
  127. Regular Article
  128. Bound for the largest singular value of nonnegative rectangular tensors
  129. Regular Article
  130. Convolutions of harmonic right half-plane mappings
  131. Regular Article
  132. On homological classification of pomonoids by GP-po-flatness of S-posets
  133. Regular Article
  134. On CSQ-normal subgroups of finite groups
  135. Regular Article
  136. The homogeneous balance of undetermined coefficients method and its application
  137. Regular Article
  138. On the saturated numerical semigroups
  139. Regular Article
  140. The Bruhat rank of a binary symmetric staircase pattern
  141. Regular Article
  142. Fixed point theorems for cyclic contractive mappings via altering distance functions in metric-like spaces
  143. Regular Article
  144. Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space
  145. Regular Article
  146. An S-type upper bound for the largest singular value of nonnegative rectangular tensors
  147. Regular Article
  148. Fuzzy ideals of ordered semigroups with fuzzy orderings
  149. Regular Article
  150. On meromorphic functions for sharing two sets and three sets in m-punctured complex plane
  151. Regular Article
  152. An incremental approach to obtaining attribute reduction for dynamic decision systems
  153. Regular Article
  154. Very true operators on MTL-algebras
  155. Regular Article
  156. Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations
  157. Regular Article
  158. A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors
  159. Regular Article
  160. Robust dynamic output feedback fault-tolerant control for Takagi-Sugeno fuzzy systems with interval time-varying delay via improved delay partitioning approach
  161. Regular Article
  162. New bounds for the minimum eigenvalue of M-matrices
  163. Regular Article
  164. Semi-quotient mappings and spaces
  165. Regular Article
  166. Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces
  167. Regular Article
  168. A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic
  169. Regular Article
  170. Solution to Fredholm integral inclusions via (F, δb)-contractions
  171. Regular Article
  172. An Ulam stability result on quasi-b-metric-like spaces
  173. Regular Article
  174. On the arrowhead-Fibonacci numbers
  175. Regular Article
  176. Rough semigroups and rough fuzzy semigroups based on fuzzy ideals
  177. Regular Article
  178. The general solution of impulsive systems with Riemann-Liouville fractional derivatives
  179. Regular Article
  180. A remark on local fractional calculus and ordinary derivatives
  181. Regular Article
  182. Elastic Sturmian spirals in the Lorentz-Minkowski plane
  183. Topical Issue: Metaheuristics: Methods and Applications
  184. Bias-variance decomposition in Genetic Programming
  185. Topical Issue: Metaheuristics: Methods and Applications
  186. A novel generalized oppositional biogeography-based optimization algorithm: application to peak to average power ratio reduction in OFDM systems
  187. Special Issue on Recent Developments in Differential Equations
  188. Modeling of vibration for functionally graded beams
  189. Special Issue on Recent Developments in Differential Equations
  190. Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs
  191. Special Issue on Recent Developments in Differential Equations
  192. Differential equations associated with generalized Bell polynomials and their zeros
  193. Special Issue on Recent Developments in Differential Equations
  194. Differential equations for p, q-Touchard polynomials
  195. Special Issue on Recent Developments in Differential Equations
  196. A new approach to nonlinear singular integral operators depending on three parameters
  197. Special Issue on Recent Developments in Differential Equations
  198. Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor
  199. Special Issue on Recent Developments in Differential Equations
  200. On new characterization of inextensible flows of space-like curves in de Sitter space
  201. Special Issue on Recent Developments in Differential Equations
  202. Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces
  203. Special Issue on Recent Developments in Differential Equations
  204. Fractional virus epidemic model on financial networks
  205. Special Issue on Recent Developments in Differential Equations
  206. Reductions and conservation laws for BBM and modified BBM equations
  207. Special Issue on Recent Developments in Differential Equations
  208. Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances
https://doi.org/10.1515/math-2016-0051
Keywords for this article
Fuzzy measure; Fuzzy metric space; t-Norm, Nonatom; Convex
Creative Commons
BY-NC-ND 3.0

Articles in the same Issue

  1. Regular Article
  2. A metric graph satisfying w41=1 that cannot be lifted to a curve satisfying dim⁡ (W41)=1
  3. Regular Article
  4. On the Riemann-Hilbert problem in multiply connected domains
  5. Regular Article
  6. Hamilton cycles in almost distance-hereditary graphs
  7. Regular Article
  8. Locally adequate semigroup algebras
  9. Regular Article
  10. Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  11. Corrigendum
  12. Corrigendum to: parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces
  13. Regular Article
  14. Some new bounds of the minimum eigenvalue for the Hadamard product of an M-matrix and an inverse M-matrix
  15. Regular Article
  16. Integral inequalities involving generalized Erdélyi-Kober fractional integral operators
  17. Regular Article
  18. Results on the deficiencies of some differential-difference polynomials of meromorphic functions
  19. Regular Article
  20. General numerical radius inequalities for matrices of operators
  21. Regular Article
  22. The best uniform quadratic approximation of circular arcs with high accuracy
  23. Regular Article
  24. Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions
  25. Regular Article
  26. A note on the rate of convergence for Chebyshev-Lobatto and Radau systems
  27. Regular Article
  28. On the weakly(α, ψ, ξ)-contractive condition for multi-valued operators in metric spaces and related fixed point results
  29. Regular Article
  30. Existence of a common solution for a system of nonlinear integral equations via fixed point methods in b-metric spaces
  31. Regular Article
  32. Bounds for the Z-eigenpair of general nonnegative tensors
  33. Regular Article
  34. Subsymmetry and asymmetry models for multiway square contingency tables with ordered categories
  35. Regular Article
  36. End-regular and End-orthodox generalized lexicographic products of bipartite graphs
  37. Regular Article
  38. Refinement of the Jensen integral inequality
  39. Regular Article
  40. New iterative codes for 𝓗-tensors and an application
  41. Regular Article
  42. A result for O2-convergence to be topological in posets
  43. Regular Article
  44. A fixed point approach to the Mittag-Leffler-Hyers-Ulam stability of a fractional integral equation
  45. Regular Article
  46. Uncertainty orders on the sublinear expectation space
  47. Regular Article
  48. Generalized derivations of Lie triple systems
  49. Regular Article
  50. The BV solution of the parabolic equation with degeneracy on the boundary
  51. Regular Article
  52. Malliavin method for optimal investment in financial markets with memory
  53. Regular Article
  54. Parabolic sublinear operators with rough kernel generated by parabolic calderön-zygmund operators and parabolic local campanato space estimates for their commutators on the parabolic generalized local morrey spaces
  55. Regular Article
  56. On annihilators in BL-algebras
  57. Regular Article
  58. On derivations of quantales
  59. Regular Article
  60. On the closed subfields of Q¯~p
  61. Regular Article
  62. A class of tridiagonal operators associated to some subshifts
  63. Regular Article
  64. Some notes to existence and stability of the positive periodic solutions for a delayed nonlinear differential equations
  65. Regular Article
  66. Weighted fractional differential equations with infinite delay in Banach spaces
  67. Regular Article
  68. Laplace-Stieltjes transform of the system mean lifetime via geometric process model
  69. Regular Article
  70. Various limit theorems for ratios from the uniform distribution
  71. Regular Article
  72. On α-almost Artinian modules
  73. Regular Article
  74. Limit theorems for the weights and the degrees in anN-interactions random graph model
  75. Regular Article
  76. An analysis on the stability of a state dependent delay differential equation
  77. Regular Article
  78. The hybrid mean value of Dedekind sums and two-term exponential sums
  79. Regular Article
  80. New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations
  81. Regular Article
  82. On the concept of general solution for impulsive differential equations of fractional-order q ∈ (2,3)
  83. Regular Article
  84. A Riesz representation theory for completely regular Hausdorff spaces and its applications
  85. Regular Article
  86. Oscillation of impulsive conformable fractional differential equations
  87. Regular Article
  88. Dynamics of doubly stochastic quadratic operators on a finite-dimensional simplex
  89. Regular Article
  90. Homoclinic solutions of 2nth-order difference equations containing both advance and retardation
  91. Regular Article
  92. When do L-fuzzy ideals of a ring generate a distributive lattice?
  93. Regular Article
  94. Fully degenerate poly-Bernoulli numbers and polynomials
  95. Commentary
  96. Commentary to: Generalized derivations of Lie triple systems
  97. Regular Article
  98. Simple sufficient conditions for starlikeness and convexity for meromorphic functions
  99. Regular Article
  100. Global stability analysis and control of leptospirosis
  101. Regular Article
  102. Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise
  103. Regular Article
  104. The fuzzy metric space based on fuzzy measure
  105. Regular Article
  106. A classification of low dimensional multiplicative Hom-Lie superalgebras
  107. Regular Article
  108. Structures of W(2.2) Lie conformal algebra
  109. Regular Article
  110. On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
  111. Regular Article
  112. Parabolic Marcinkiewicz integrals on product spaces and extrapolation
  113. Regular Article
  114. Prime, weakly prime and almost prime elements in multiplication lattice modules
  115. Regular Article
  116. Pochhammer symbol with negative indices. A new rule for the method of brackets
  117. Regular Article
  118. Outcome space range reduction method for global optimization of sum of affine ratios problem
  119. Regular Article
  120. Factorization theorems for strong maps between matroids of arbitrary cardinality
  121. Regular Article
  122. A convergence analysis of SOR iterative methods for linear systems with weak H-matrices
  123. Regular Article
  124. Existence theory for sequential fractional differential equations with anti-periodic type boundary conditions
  125. Regular Article
  126. Some congruences for 3-component multipartitions
  127. Regular Article
  128. Bound for the largest singular value of nonnegative rectangular tensors
  129. Regular Article
  130. Convolutions of harmonic right half-plane mappings
  131. Regular Article
  132. On homological classification of pomonoids by GP-po-flatness of S-posets
  133. Regular Article
  134. On CSQ-normal subgroups of finite groups
  135. Regular Article
  136. The homogeneous balance of undetermined coefficients method and its application
  137. Regular Article
  138. On the saturated numerical semigroups
  139. Regular Article
  140. The Bruhat rank of a binary symmetric staircase pattern
  141. Regular Article
  142. Fixed point theorems for cyclic contractive mappings via altering distance functions in metric-like spaces
  143. Regular Article
  144. Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space
  145. Regular Article
  146. An S-type upper bound for the largest singular value of nonnegative rectangular tensors
  147. Regular Article
  148. Fuzzy ideals of ordered semigroups with fuzzy orderings
  149. Regular Article
  150. On meromorphic functions for sharing two sets and three sets in m-punctured complex plane
  151. Regular Article
  152. An incremental approach to obtaining attribute reduction for dynamic decision systems
  153. Regular Article
  154. Very true operators on MTL-algebras
  155. Regular Article
  156. Value distribution of meromorphic solutions of homogeneous and non-homogeneous complex linear differential-difference equations
  157. Regular Article
  158. A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors
  159. Regular Article
  160. Robust dynamic output feedback fault-tolerant control for Takagi-Sugeno fuzzy systems with interval time-varying delay via improved delay partitioning approach
  161. Regular Article
  162. New bounds for the minimum eigenvalue of M-matrices
  163. Regular Article
  164. Semi-quotient mappings and spaces
  165. Regular Article
  166. Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces
  167. Regular Article
  168. A family of singular functions and its relation to harmonic fractal analysis and fuzzy logic
  169. Regular Article
  170. Solution to Fredholm integral inclusions via (F, δb)-contractions
  171. Regular Article
  172. An Ulam stability result on quasi-b-metric-like spaces
  173. Regular Article
  174. On the arrowhead-Fibonacci numbers
  175. Regular Article
  176. Rough semigroups and rough fuzzy semigroups based on fuzzy ideals
  177. Regular Article
  178. The general solution of impulsive systems with Riemann-Liouville fractional derivatives
  179. Regular Article
  180. A remark on local fractional calculus and ordinary derivatives
  181. Regular Article
  182. Elastic Sturmian spirals in the Lorentz-Minkowski plane
  183. Topical Issue: Metaheuristics: Methods and Applications
  184. Bias-variance decomposition in Genetic Programming
  185. Topical Issue: Metaheuristics: Methods and Applications
  186. A novel generalized oppositional biogeography-based optimization algorithm: application to peak to average power ratio reduction in OFDM systems
  187. Special Issue on Recent Developments in Differential Equations
  188. Modeling of vibration for functionally graded beams
  189. Special Issue on Recent Developments in Differential Equations
  190. Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs
  191. Special Issue on Recent Developments in Differential Equations
  192. Differential equations associated with generalized Bell polynomials and their zeros
  193. Special Issue on Recent Developments in Differential Equations
  194. Differential equations for p, q-Touchard polynomials
  195. Special Issue on Recent Developments in Differential Equations
  196. A new approach to nonlinear singular integral operators depending on three parameters
  197. Special Issue on Recent Developments in Differential Equations
  198. Performance and stochastic stability of the adaptive fading extended Kalman filter with the matrix forgetting factor
  199. Special Issue on Recent Developments in Differential Equations
  200. On new characterization of inextensible flows of space-like curves in de Sitter space
  201. Special Issue on Recent Developments in Differential Equations
  202. Convergence theorems for a family of multivalued nonexpansive mappings in hyperbolic spaces
  203. Special Issue on Recent Developments in Differential Equations
  204. Fractional virus epidemic model on financial networks
  205. Special Issue on Recent Developments in Differential Equations
  206. Reductions and conservation laws for BBM and modified BBM equations
  207. Special Issue on Recent Developments in Differential Equations
  208. Extinction of a two species non-autonomous competitive system with Beddington-DeAngelis functional response and the effect of toxic substances
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 26.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2016-0051/html?lang=en
Scroll to top button