Startseite Mathematik A result for O2-convergence to be topological in posets
Artikel Open Access

A result for O2-convergence to be topological in posets

  • Qingguo Li EMAIL logo und Zhiwei Zou
Veröffentlicht/Copyright: 23. April 2016
Artikel downloaden (PDF)
Open Mathematics
Aus der Zeitschrift Open Mathematics Band 14 Heft 1

Abstract

In this paper, the α waybelow relation, which is determined by O2-convergence, is characterized by the order on a poset, and a sufficient and necessary condition for O2-convergence to be topological is obtained.

Keywords: O2-convergence; O2-doubly continuous poset; Topological, α-doubly continuous poset; α-doubly continuous poset
MSC 2010: 54A20; 06B35

1. Introduction

The O2-convergence on a poset P, which is a generalization of O-convergence, is defined as follows [1–4]: Let P be a poset, a net (xi)i ∈ I is said to O2-converges to x ∈ P if there exist subsets M and N of P such that

  1. sup M = inf N = x;

  2. For each a ∈ M and b ∈ N, there exists k ∈ I such that a ≤ xi ≤ b holds for all i ≥ k.

By saying the O2-convergence on a poset P is topological, we mean that there exists a topology 𝒯 on P such that a net (xi)i ∈ I in PO2-converges to x ∈ P if and only if it converges to x with respect to the topology 𝒯. Otherwise, we say that O2-convergence is not topological. As is pointed in [5], in general, O2-convergence is not topological. The same problem exists for O-convergence and Lim-inf-convergence, Zhao and others have done some excellent work in [6–8].

In [5], Zhao and Li proved that if the O2-convergence on a poset P is topological, then P is an α-doubly continuous poset. Also they give a sufficient condition: If P is an α*-doubly continuous poset, then O2-convergence is topological. Moreover, if P is a poset satisfying condition (*), O2-convergence is topological if and only if P is an α-doubly continuous poset.

But as pointed in this paper, there exists poset which is α-doubly continuous but not α*-doubly continuous. Then for such a poset, is the O2-convergence topological? In other words, can we find a sufficient and necessary condition for O2-convergence to be topological for any poset?

In this paper, the α waybelow relation is characterized just by the order on a poset. The concept of O2-doubly continuous poset is given, which is strictly stronger than α-doubly continuous and strictly weaker than α*-doubly continuous. Most important, we prove that for a poset P, the O2-convergence on it is topological if and only if P is an O2-doubly continuous poset.

2. Preliminaries

This section will briefly mention some of the important results that Zhang and Li presented in [5]. For terms which are not defined here, please refer to [9, 10] and related references.

For a poset P, a, b ∈ P, we denote [a, b] = {x ∈ P : a ≤ x ≤ b}. We use M0 ⊆finM to indicate M0 is a finite subset of M.

Definition 2.1

([5]).Let P be a poset. For x, y, z ∈ P, define x ≪αy if for every net (xi)i ∈ Iin P which O2-converges to y, x ≤ xi holds eventually; z ⊳α y if for every net (xi)i ∈ Iin P which O2-converges to y, xi ≤ z holds eventually.

It follows from Definition 2.1 that x ≪α y ⟹ x ≤ y and z ⊳α y ⟹ z ≥ y. If P has a bottom element ⊥ (top element ⊤), then ⊥ ≪α x (⊤ ⊳α x) for each x ∈ P

Definition 2.2

([5]). A poset P is called an α-doubly continuous poset if a = sup {x ∈ P : x ≪α a} = inf {y ∈ P: z ⊳α a} for every a ∈ P.

Remark 2.3

  1. Every finite lattice is α-doubly continuous, every chain or antichain is α-doubly continuous.

  2. If P is α-doubly continuous, then for every a ∈ P, the set {x ∈ P : x ≪α a} and {y ∈ P : z ⊳α a} are both nonempty.

Definition 2.4

([5]). Let P be a poset. For x, y, z ∈ P, define x ≪α*y if for every net (xi)i ∈ Iin P which O2-converges to some ω ∈ P with y ≤ ω, x ≤ xi holds eventually. We define the order dual relation z ⊳α*y if z ≪α*y in Pop, where Pop denotes P endowed with the reverse order.

Definition 2.5

([5]). A poset P is called an α*-doubly continuous poset if a= sup {x ∈ P : x ≪α*a} = inf {y ∈ P: z ⊳α*a} for every a ∈ P.

Remark 2.6

([5]).

  1. Obviously, x ≪α*y ⟹ x ≪α y and z ⊳α*y ⟹ z ⊳α y, then an α*-doubly continuous poset is α-doubly continuous.

  2. If P is an α*-doubly continuous poset, then for each x ∈ P,

x = sup{a ∈ P : ∃a′ ∈ P, a ≪α*a′ ≪α*x} = inf{b ∈ P : ∃b′ ∈ P, b ⊳α*b′ ⊳α*x}.

Theorem 2.7

([5]). For any poset P, if O2-convergence is topological, then P is α-doubly continuous.

Theorem 2.8

([5]). If P is an α*-doubly continuous poset, then O2-convergence is topological.

That is to say, α*-doubly continuous ⟹ O2-convergence is topological⟹ α-doubly continuous.

Condition (*). Let P be a poset and x, y, z ∈ P with x ≪α y ≤ z, then x ≪α z. Let w, s, t ∈ P with s ⊳α t ≥ w, then s ⊳α w.

Corollary 2.9

([5]). For any poset P which satisfies condition (*), O2-convergence is topological if and only if P is α-doubly continuous.

3. O2-doubly continuous posets

In this section, we give a characterization of the relations ≪α and ⊳α, then introduce the concept of O2-doubly continuous poset which is strictly stronger than α-doubly continuous and strictly weaker than α*-doubly continuous.

Proposition 3.1

Let P be a poset. y ≪α x if and only if for every subset M and N such that sup M = inf N = x, there exist M0 ⊆finM and N0 ⊆fin N such that∩a∈M0∩b∈N0[a,b]⊆↑y.

Dually, z ⊳α x if and only if for every subset M and N such that sup M = inf N = x, there exist M0 ⊆finM and N0 ⊆fin N such that∩a∈M0∩b∈N0[a,b]⊆↓z.

Proof

Sufficiency, if there exists a net (xi)i ∈ I in P which O2-converges to x, then there exist subsets M and N of P such that sup M = inf N = x, and for each pair a ∈ M and b ∈ N, xi ∈ [a, b] holds eventually. By the condition there exist M0 ⊆fin M and N0 ⊆fin N such that ∩a∈M0∩b∈N0[a,b]⊆↑y. Since M0 and N0 are finite, xi∈∩a∈M0∩b∈N0[a,b]⊆↑y holds eventually, thus y ≤ xi holds eventually, we conclude that y ≪α x.

Necessary, suppose y ≪α x and sup M = inf N = x, denote Bx={∩a∈A∩b∈B[a,b]:A⊆finM and 𝒟 = {(r, U) ∈ (⋃𝓑x) × 𝓑x: r ∈ U}, we define a preorder " ≤ " on 𝒟 by: (r1, U1) ≤(r2, U2) ⇔ U2 ⊆ U1. Arbitrarily take (r1, U1) and(r2, U2) ∈ 𝒟, suppose U1=∩a∈A1∩b∈B1 [a,b] and U2=∩a∈A2∩b∈  B2[a,b] where A1, A2 ⊆fin M and B1, B2 ⊆fin N. Let A3 = A1 ⋃ A2, B3 = B1 ⋃ B2, U3=∩a∈A3∩b∈B3 [a,b], then (x, U3) ∈ 𝒟 and (r1, U1) (r2, U2) ≤ (x, U3), hence (𝒟, ≤) is directed. We construct a net (xd)d∈𝒟 = x(r, U) = r. We know sup M = inf N = x, and for each m ∈ M and n ∈ N, [m, n] ∈ 𝓑x, when (r, U) ≥ (x, [m; n], we have U ⊆ [m, n] and x(r, U) = r ∈ U ⊆[m, n], so m ≤ (xd)d∈𝒟 ≤ n holds eventually. This implies that (xd)d ∈ 𝒟O2-converges to x. Then y ≪ α x implies that y ≤ x(r, U) holds eventually, i.e., there exists (ro, U0) ∈ 𝒟 such that y ≤ x(r, U) = r for every (r, U) ≥ (ro, U0). Since (ro, U0) ∈ 𝒟, there exist M0 ⊆fin M and N0 ⊆fin N such that U0=∩a∈M0∩b∈​  N0[a,b], for each r ∈ U0, (r, U0) ≥ (ro, U0), so y ≤ x(r, U0) = r, we conclude that U0=∩a∈M0∩b∈N0[a,b]⊆↑y.

Definition 3.2

A poset P is called an O2-doubly continuous poset if it satisfies the following condition:

  1. P is α-doubly continuous, i.e. x = sup {y ∈ P : y ≪α*x} = inf {z ∈ P : z ⊳α*x} for each x ∈ P.

  2. If y ≪α x and z ⊳α x, then there exist finite A ⊆fin {a ∈ P : a ≪α x} and B ⊆fin {b ∈ P : b ⊳α x} such that y ≪α c and z ⊳αc∈∩m∈A∩m∈B[m,n].

Remark 3.3

Every finite lattice is O2-doubly continuous, every chain or antichain is O2-doubly continuous.

Proposition 3.4

If P is an α*-doubly continuous poset, then P is an O2-doubly continuous poset.

Proof

By Remark 2.6 we know an α*-doubly continuous poset is α-doubly continuous. In an α*-doubly continuous poset P, we denote Mx = {a ∈ P : ∃a′ ∈ P; a ≪α*a′ ≪α*x} and Ny = {b ∈ P : ∃ b′ ∈ P : ⊳α*b′ ⊳α*x} for each x ∈ P, then sup Mx = inf Nx = x. If y ≪α x and z ⊳α x, from Proposition 3.1 there exist finite M0 ⊆finMx and N0 ⊆finNx such that ∩a∈M0∩b∈N0[a,b]⊆[y,z]. Denote A0 = {a′ ∈ P : a ∈ M0} and B0 = {b′ ∈ P : b ∈ N0}, we have ∩a′∈A0∩a′∈B0[a′,b′]⊆∩a∈M0∩b∈M0[a,b]⊆[y,z]. Obviously A0 ⊆fin {a ∈ P : a ≪α x} and B0 ⊆fin {b ∈ P : b ⊳α x}, so we only need to prove y ≪α c and z ⊳α c hold for each c∈∩a′∈A0∩b′∈B0[a′,b′].

If there exists a net (xi)i ∈ IO2-convergence to c, for each a ∈ M0 and b ∈ N0, a ≪α c and b ⊳α c, then xi ∈ [a, b] holds eventually. Because M0 and N0 are both finite, thus xi∈∩a∈M0∩b∈N0[a,b]⊆[y,z] eventually, we conclude that y ≪α c and z ⊳α c.

An α*-doubly continuous poset implies it is O2-doubly continuous. An O2-doubly continuous poset implies it is α-doubly continuous. But conversely both are untrue – see the following examples:

Example 3.5

Let P1 = {1} ⋃ {aji ∈ ℕ} ⋃ {bi : i ∈ ℕ} ⋃ {ci: i ∈ ℕ}, ⋃ {bij: i, j ∈ ℕ} ⋃ {cij: i, j ∈ ℕ} where ℕ denotes the set of all positive integers. The order ≤ on P1is constructed as follows (see Fig. 1 for the Hasse diagram of P1:

  1. ai ≤ ai+1; bi ≤ bi+1;ci ≤ ci+1; for all i ∈ ℕ.

  2. ai ≤ bi; ai ≤ ci for all i ∈ ℕ

  3. bij ≤ bi,j+1 ≤ bi and cij ≤ ci,j+1, ≤ ci for all i, j ∈ ℕ.

  4. 1 is the greatest element in P.

It is straightforward to verify that P1is α-doubly continuous, {x ∈ P : x ≪α 1 = {ai : i ∈ ℕ} and {x ∈ P : x ⊳α 1} = {1}. Notice a1 ≪α 1, for any finite A ⊆fin {x ∈ P : x ≪α 1} = {ai : i ∈ ℕ}, there exists i0 ∈ ℕ such thatai0∈∩ai∈A[ai,1], thenbi0∈∩ai∈A[ai,1], but a1 ≪αbi0does not hold, so P1is not an O2-doubly continuous poset.

Moreover, a1 ≪αa2 ≤ b2but a1 ≪αb2does not hold, that means P1does not satisfy the condition(*).

Fig. 1
Fig. 1
Example 3.6

Let P2 = {1}⋃{a}⋃{b1, b2, b3, ...}. The order “≤” on P2is defined as follows: (see Fig. 1 for the Hasse diagram): a ≤ 1; bi ≤ 1 for all i ∈ ℕ; bi ≤ bj , whenever i ≤ j, for all i, j ∈ ℕ.

It is obvious that P is an α-doubly continuous poset and satisfies the condition (2) in Definition 3.2, then P is an O2-doubly continuous poset. But it is not α*-doubly continuous because {x ∈ P : x ≪α*a} = ϕ.

4. A sufficient and necessary condition for O2-convergence to be topological

In this section, we obtain the main result that O2-convergence on a poset P is topological if and only if P is an O2-doubly continuous poset.

We recall that saying the O2-convergence on a poset P is topological means that there exists a topology 𝒯 on P such that a net (xi)i∈I in P O2-converges to x ∈ P if and only if it converges to x with respect to the topology 𝒯. Zhao and Li proved that for any poset P, if the O2-convergence is topological, then P is α-doubly continuous in [5]. We will prove it again using another method in the following theorem because some results from the process will be used next. The following theorem is a further result of Zhao and Li’s.

Theorem 4.1

For any poset P, if the O2-convergence is topological, then P is an O2-doubly continuous poset.

Proof

For a poset P, if the O2-convergence is topological, then there exists a topology 𝒯 on P such that a net (xi)i∈IO2-converges to x ∈ P if and only if it converges to x with respect to the topology 𝒯. For every x ∈ P, let Ix = {(W,k,r) ∈ 𝒩(x)× ℕ× P : r ∈ W}, where 𝒩(x) consists of all open sets containing x in the topology 𝒯, i.e., 𝒩(x) = {W ∈ 𝒯: x ∈ W}. Now we define the lexicographic order on the first two coordinates on Ix, this means for (W1, k1, r1), (W2, k2, r2) ∈ Ix, (W1, k1, r1) ≤. (W2, k2, r2) if and only if W2 is a proper subset of W1 or W1 = W2 and k1 ≤ k2. This is a preorder. Arbitrarily take (W1, k1, r1), (W2, k2, r2) ∈ Ix, then (W1 ⋂ W2, max{k1, k2}, x) ≤ (W1, k1, r1) and (W1 ⋂ W2, max{k1, k2},x) ≤ (W2, k2, r2). Hence Ix is directed. Let xi = r for every i = (W,k,r) ∈ Ix. Then it is straightforward to verify that the net (xi)i∈Ix converges to x with respect to the topology 𝒯. Thus (xi)i∈IxO2-converges to x. By the definition of O2-convergence, there exist subsets M0 and N0 such that sup M0 = inf N0 = x, and for each m ∈ M0 and n ∈ N0 there exists i0 = (W0, k0, r0) ∈ Ix such that xi = r ∈ [m, n] holds for every i = (W,k,r) ≥ i0. In particular, we have (W0, k0 + 1; w) ≥ (W0, k0 + 1, w) ≥ (w0, k0, r0) for all w ∈ W0, hence x(W0, k0+1, w) = w ∈ [m, n] for all w ∈ W0, this means W0 ⊆ [m, n]. Since W0 is related to m and n, we denoted it by Wmn.

For arbitrary m ∈ M0 and n ∈ N0, if there exists a net (xj)j∈JO2-convergence to x, then (xj)j∈J converges to x with respect to the topology 𝒯, this means xj is eventually in Wmn ⊆ [m, n] (Wmn is an open neighborhood of x in the topology 𝒯), thus m ≪αx and n ⊳αx. We conclude that M0 ⊆ {y ∈ P : y ≪αx} and N0 ⊆ {z ∈ P : z ⊳αx}, since sup M0 = inf N0 = x, we have sup {y ∈ P : y ≪αx} = inf{z ∈ P : z ⊳αx} = x, i.e., P is an α-doubly continuous poset.

Suppose y ≪αx and z ⊳αx, by Proposition 3.1 we know there exist M1 ⊆finM0 and N1 ⊆finN0 such that ∩m∈M1∩n∈N1[m,n]⊆[y,z]. For arbitrary m ∈ M1 and n ∈ N1, there exists Wmn ∈ 𝒩(x) such that Wmn ⊆ [m, n], denote W1=∩m∈M1∩n∈N1Wmn, then W1 ∈ 𝒩(x) and W1∩m∈M1∩n∈N1[m,n]⊆[y,z] because M1 and N1 are both finite. We denote Ax = {a ∈ P : a ≪αx} and Bx = {b ∈ P : b ≪αx}. Let Dx={∩a∈A0∩b∈B0⁡[a,b]:A0⊆finAx and B0 ⊆finBx}, 𝒟 = {r, U) ∈ (∪ Dx) × Dx : r ∈ U}. Define “≤” on 𝒟 by: (r1, U1) ≤ (r2, U2) ⇔ U2 ⊆ U1, then “≤” is a preorder on 𝒟 and 𝒟 is directed. Let x(r,U) = r for (r, U) ∈ 𝒟, we will prove (x(r,U) ∈ 𝒟O2-convergence to x in the next paragraph.

Since P is an α-doubly continuous poset, we know sup Ax = inf Bx = x, for each a ∈ Ax and b ∈ Bx, [a, b] ∈ 𝒟, when (r, U) ≥ (x, [a, b]) i.e. U ⊆[a, b], x(r,U) = r ∈ U ⊆[a, b], so a ≤ (x(r,U))(r,U)∈𝒟 ≤ b holds eventually, by the definition of O2-convergence we know that (x(r,U))(r,U)∈𝒟 O2-convergence to x.

For poset P, the O2-convergence is topological, then (x(r,U))(r,U)∈𝒟 converges to x with respect to the topology 𝒯, hence x(r,U) ∈ W1 holds eventually. There exist finite subsets A1 ⊆finAx and B1 ⊆finBx such that when (r,U)≥(r0,∩a∈A1∩b∈B1[a,b]), we have x(r, U) ∈ W1. In particular for each t∈∩a∈A1∩b∈B1⁡[a,b],(t,∩a∈A1∩b∈B1⁡[a,b]≥(r0,∩a∈A1∩b∈B1⁡[a,b]), then x(t,∩a∈A1∩b∈B1[a,b])=t∈W1, we conclude that ∩a∈A1∩b∈B1[a,b]⊆W1⊆[y,z]. To prove P satisfies the condition (2) in Definition 3.2, it is sufficient to show y ≪α c for each c∈∩a∈A1∩b∈B1[a,b]. If there exists a net (xj)j∈JO2-convergence to c, then (xj)j∈J converges to c with respect to the topology 𝒯; this means xj is eventually in W1 since W1 ∈ 𝒯 and c∈∩a∈A1∩b∈B1[a,b]⊆W1, therefore xj∈W1∩m∈M1∩n∈N1[m,n]⊆[y,z] hold eventually, we conclude that y ≪αc and z ⊳αc.

Remark 4.2

The relations amongW1,∩m∈M1∩n∈N1[m,n]and∩a∈A1∩b∈B1[a,b]are revealed in Fig 2.

We review the general relation between convergence and topology. If on any set P a class 𝓛 of pairs ((xi)i∈I, x) is given, then 𝓞(𝓛) = {U ⊆ X : whenever (xi)i∈I ∈ 𝓛 and x ∈ U, xi ∈ U eventually} is a topology on X. In this paper, we denote the class 𝓛 = {((xi)i∈I, x) : (xi)i∈IO2-converges to x}. Kelly gave a standard characterization for a class 𝓛 of convergent nets to be topological:

Fig. 2
Fig. 2
Fact 4.3

([11]). Given a poset P, the class 𝓛 is topological if and only if the following axioms are satisfied.

(Constant) If (xi)i∈I is a constant net with value xi = x for every i ∈ I, then ((xi)i∈I, x) ∈ 𝓛.

(Subnets) If ((xi)i∈I, x) ∈ 𝓛 and (yj)j∈J is a subnet of (xi)i∈I, then ((yj)j∈J ((yj)j∈J, x) ∈ 𝓛.

(Divergence) If ((xi)i∈I,x) is not in 𝓛, then there exists a subnet (yj)j∈J of (xi)i∈I which has no subnet (zk)k∈K so that ((zk)k∈K, x) belongs to 𝓛.

(Iterated Limits) If ((xi)i∈I, x), ∈ 𝓛, and if ((xi, j)j∈J(i), xi) ∈ 𝓛 for all i ∈ I, then ((xi, f(i))i, f∈I×M,x) ∈ 𝓛, where M=∏i∈IJ(i) is a product of directed sets.

Lemma 4.4

For an α-doubly continuous poset P, a net (xi)i∈IO2-converges to x ∈ P if and only if for any y, z ∈ P with y ≪α x and z ⊳α x, y ≤ xi ≤ z holds eventually.

Proof

Necessary, if net (xi)i∈IO2-converges to x ∈ P and y ≪αx, z ⊳αx, by the Definition of ≪α and ⊳α we know y ≤ xi ≤ z holds eventually. Conversely, just let M = {y ∈ P : y ≪αx} and N = {z ∈ P : z ⊳αx}, then supM = infN = x. For each y ∈ M and z ∈ N, y ≤ xi ≤ z holds eventually, we have (xi)i∈IO2-converges to x.

Lemma 4.5

For an O2-doubly continuous poset P, a net (xi)i∈IO2-converges to x ∈ P if and only if for any y, z ∈ P with y ≪α x and z ⊳α x, y ≪α xi and z ⊳α xi hold eventually.

Proof

Necessary, if net (xi)i∈IO2-converges to x ∈ P and y ≪αx, z ⊳αx, by Definition 3.2 there exist finite A ⊆fin {a ∈ P : a ≪αx} and B ⊆fin {b ∈ P : b ⊳αx} such that y ≪α c and z ⊳α c for each c ∈ c ∈ c∈∩m∈A∩n∈B [m,n] Since A and B are both finite, by Lemma 4.4, xi ∈ xi∈∩m∈A∩n∈B [m,n] holds eventually, thus y ≪α xi and z ⊳α xi hold eventually. Conversely, since y ≪α xi and z ⊳α xi imply y ≤ xi ≤ z, by Lemma 4.4 we know (xi)i∈IO2-converges to x ∈ P.

Proposition 4.6

For any poset P, the class 𝓛 satisfies axiom (Constant).

Proof

Suppose xi = x for every i ∈ I, just take M = N = {x}, sup M = inf N = x and x ≤ xi = x ≤ x holds for all i ∈ I. Therefore ((xi)i∈I, x) ∈ 𝓛.

Proposition 4.7

For any poset P, the class 𝓛 satisfies axiom (Subnets).

Proof

Let ((xi)i∈I, x) ∈ 𝓛. There exist subsets M and N such that sup M = inf N = x, and for every m ∈ M and every n ∈ N there exists k ∈ I such that m ≤ xi ≤ n for all i ≥ k. For any subnet (yj)j∈J of net (xi)i∈I, there exists a mapping f : J ⟶ I such that yj = xf(j) for every j ∈ J, and for every i ∈ I there exists ji ∈ J such that f(j) ≥ i for every j ≥ ji. In particular, for k ∈ I, there exists jk ∈ J such that f(j) ≥ k for all j ≥ J such that f(j) ≤ i for every j ≤ ji . In particular, for k ∈ I, there exists jk ∈ J such that f(j) ≥ k for all j ≥ jk. Hence m ≤ yj = xf(j) ≤ n for all j ≥ jk. Therefore ((yj)j∈J, x) ∈ 𝓛.

Proposition 4.8

For an α-doubly continuous poset P, the class 𝓛 satisfies axiom (Divergence).

Proof

Assume that ((xi)i∈I, x) is not in 𝓛. By lemma 4. 4, there exist y0, z0 ∈ P with y0 ≪αx and z0 ⊳αx such that y0 ≤ xi ≤ z0 does not hold eventually. This means that for every i ∈ I we can choose a ji ∈ I such that y0 ≰ xji or z0 ≱ xji. Let J = {ji : i ∈ I} ⊆ I, then J is a cofinal subset of I. We now consider the subnet (xj)j∈J of (xi)i∈I, since y0 ≰ xj or z0 ≱ xj for every j ∈ J, by Lemma 4.4 again, (xj)j∈J has no subnet (zk)k∈K such that ((zk)k∈K, x) belongs to 𝓛. Thus the class 𝓛 satisfies axiom (Divergence).

Proposition 4.9

For an O2-doubly continuous poset P, the class 𝓛 satisfies axiom (Iterated Limits).

Proof

Let ((xi)i∈I, x) ∈ 𝓛 and ((xi, j)j∈Ji, ∈ 𝓛 for every i ∈ I. For any y, z ∈ P with y ≪αx and z ⊳αx, by Lemma 4.5 we know y ≪α xi and z ⊳α xi hold eventually, i.e., there exists i0 ∈ I such that y ≪α xi and z ⊳α xi for all i ≥ i0. Since ((xi, j)j∈Ji, xi) ∈ 𝓛, by Lemma 4.5 again, for every i ∈ I with i ≥ i0 there exists ji ∈ Ji such that y ≪αxi, j and z ⊳αxi, j for every j ≥ ji. Define f0∈M=∏i∈IJ(i) as: f0(i) = ji for every i ≥ i0 and f0(i) is an arbitrarily fixed element in Ji for every i ≱ i0. Now we claim that ((xi, f(i))(i, f)∈I×M, x) ∈ 𝓛, because if (i, f) ∈ I × M and (i, f) ≥ (i0, f0) then y ≪αxi, f(i) and z ⊳αxi, f(i), it means ((xi, f(i))(i, f)∈I×M, x) O2-converges to x by Lemma 4.4. This completes the proof.

From Propositions 4.6 –4.9 and Fact 4. 3, we have

Theorem 4.10

For any poset P, if P is O2-doubly continuous, then O2-convergence is topological.

Combining Theorem 4.1 with Theorem 4.1 0, we obtain the following theorem.

Theorem 4.11

Let P be a poset. Then the O2-convergence is topological if and only if P is O2-doubly continuous.

By Proposition 3.4 and Theorem 4.1 0, we have the following corollary.

Corollary 4.12

([5]). If P is an α*-doubly continuous poset, then O2-convergence is topological.

Example 4.13

  1. For the poset P1showed in Example 3.5, the O2-convergence is not topological.

  2. In Example 3.6, the O2-convergence on the poset P2is topological.

Acknowledgement

This work is supported by the Natural Science Foundation of China (Grant No: 11371130) and (Grant No: 2014GXNSFBA118015).

References

[1] Frink O., Topology in lattice Trans. Amer. Math. Soc., 1942, 51, 569-582.10.1090/S0002-9947-1942-0006496-XSuche in Google Scholar

[2] Mathews J.C., Anderson R.F., A comparison of two modes of order convergence, Proc. Amer. Math. Soc., 1967, 18(1), 100-104.10.1090/S0002-9939-1967-0203675-6Suche in Google Scholar

[3] Wolk E.S., On order-convergence, Proc. Amer. Math. Soc., 1961, 12(3), 379-384.10.1090/S0002-9939-1961-0136562-7Suche in Google Scholar

[4] Olejcek V., Order Convergence and Order Topology on a Poset, Int. J. Theor. Phys., 1999, 38, 557-561.10.1023/A:1026690820346Suche in Google Scholar

[5] Zhao B., Li J., O2-convergence in posets, Topology Appl., 2006, 153, 2971-2975.10.1016/j.topol.2006.01.004Suche in Google Scholar

[6] Wang K.Y., Zhao B., Some further result on order-convergence in posets, Topology Appl., 2013, 160, 82-86.10.1016/j.topol.2012.09.018Suche in Google Scholar

[7] Zhao B., Zhao D., Lim-inf-convergence in partially ordered sets, J. Math. Anal. Appl., 2005, 309, 701-708.10.1016/j.jmaa.2004.11.028Suche in Google Scholar

[8] Zhou Y., Zhao B., Order-convergence and Lim-inf-convergence in poset, J. Math. Anal. Appl., 2007, 325, 655-664.10.1016/j.jmaa.2006.02.016Suche in Google Scholar

[9] Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S., Continuous Lattices and Domain, Camberidge University Press, Cambridge, 2003.10.1017/CBO9780511542725Suche in Google Scholar

[10] Birkhoff G., Lattice Theory, Amer. Math. Soc. Colloq. Publ., New York, 1940.10.1090/coll/025Suche in Google Scholar

[11] Kelly J.L., General topology, Van Nostrand, New York, 1955.Suche in Google Scholar

Received: 2016-1-6
Accepted: 2016-3-9
Published Online: 2016-4-23
Published in Print: 2016-1-1

© 2016 Li and Zou, published by De Gruyter Open

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

Artikel in diesem Heft

  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-2016-0018
Schlagwörter für diesen Artikel
O2-convergence; O2-doubly continuous poset; Topological, α-doubly continuous poset; α-doubly continuous poset
Creative Commons
BY-NC-ND 3.0

Artikel in diesem Heft

  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
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 9.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/math-2016-0018/html
Button zum nach oben scrollen