Generalized Virtually Stable Maps and Their Associated Sequences

Abstract

We extend the concept of virtual stability of continuous self-maps to arbitrary selfmaps and investigate the structure of sequences associated with uniformly virtually stable selfmaps. We also obtain a necessary and sufficient condition for a uniformly virtually stable selfmap to have the largest possible associated sequence. Examples of a uniformly virtually stable selfmap having the prescribed largest sequence and a uniformly virtually stable selfmap having no largest sequence are given.

1. Introduction

Virtual stability of a self-map was first introduced in [1], where it was proved to unify various (continuous) types of nonexpansiveness in fixed-point theory. The main feature of a virtually stable self-map is that its fixed-point set is always a retract of its convergence set, and this fact immediately allows us to connect topological structures (e.g., connectedness and contractibility) of the fixed-point set to those of the convergence set. Although the original definition of virtual stability requires the continuity of a self-map, it seems that the connection between the fixed-point set and the convergence set also remains valid to some degree in discontinuous settings as hinted in [2, Corollary 2.6] and the work on mean-type mappings in [3–5] (see Example 4 below). A careful investigation of this fact will definitely result in a more powerful notion of virtual stability that enables us to study fixed-point sets of interesting discontinuous self-maps (e.g., Suzuki generalized nonexpansive maps [6]). Moreover, since all well-known self-maps in metric fixed-point theory are always uniformly virtually stable with respect to the same sequence (𝑛) of all natural numbers, it is quite natural to ask whether this also holds in general for a uniformly virtually stable self-map. It turns out that the answer to this simple question is highly nontrivial and interestingly involves algebraic properties of a certain object. Therefore, the purpose of this work is to revise the concept of virtual stability as well as to take care of the previos question. We will first redefine the notion of virtual stability to include discontinuous self-maps and investigate some properties of these newly defined virtually stable self-maps and prove the expected connection between the convergence set and the fixed-point set. Then we will show that the sequence associated with a uniformly virtually stable self-map can be made, in some sense, largest possible provided that the set of all associated sequences has a certain structure. We will also see that when the largest sequence exists, it may not be (𝑛) unless the map is continuous. For the sake of completeness, since the existence of the largest possible sequence associated with a given self-map cannot be guaranteed in general, we will provide two constructive examples of uniformly virtually stable self-maps: one having the prescribed largest sequence and the other having no largest sequence.

2. Virtual Stability without Continuity

Let 𝑋 be a (nonempty) Hausdorff space and 𝑓∶𝑋→𝑋 a self-map (that may not be continuous). We will also let 𝒩(𝑥) and 𝒩_𝑌(𝑥) denote the sets of (open) neighborhoods of 𝑥 in 𝑋 and in 𝑌⊆𝑋, respectively. Recall that the fixed-point set and the convergence set of 𝑓 are defined, respectively, to be 𝐹(𝑓)={𝑥∶𝑓(𝑥)=𝑥} and 𝐶𝑓=𝑥∶thesequence𝑓^𝑛(𝑥)converges,(1) where 𝑓^𝑛 denotes the 𝑛-th iterate of 𝑓. We also define the map 𝑓^∞∶𝐶(𝑓)→𝑋 by 𝑓^∞(𝑥)=lim_𝑛→∞𝑓^𝑛(𝑥),(2) for all 𝑥∈𝐶(𝑓). Notice that 𝑓^∞ may not be continuous, and since we do not assume the continuity of 𝑓, we only have 𝐹(𝑓)⊆𝑓^∞(𝐶(𝑓)). However, in this work, we will always assume that ∅≠𝐹𝑓=𝑓^∞𝐶𝑓.(3) This condition is quite natural in the sense that it is automatically satisfied whenever 𝑓 is continuous with 𝐹(𝑓)≠∅, and it is also required for the iterative sequence (𝑓^𝑛) to be a scheme according to [2, Definition 2.1]. With the above assumption in mind, we are able to previous define the concept of virtual stability just as in [1, Definition 2.1].

Definition 1. A fixed point 𝑥 of 𝑓 is said to be virtually 𝑓-stable if for each 𝑈∈𝒩(𝑥), there exist 𝑉∈𝒩(𝑥) together with an increasing sequence (𝑎_𝑛) of positive integers such that 𝑓^𝑎𝑛(𝑉)⊆𝑈 for all 𝑛∈ℕ. We simply call 𝑓virtually stable if every fixed point of 𝑓 is virtually stable. Moreover, we will call a fixed point 𝑥 of 𝑓uniformly virtually 𝑓-stable with respect to an increasing sequence (𝑎_𝑛) of positive integers if for each 𝑈∈𝒩(𝑥), there exists 𝑉∈𝒩(𝑥) such that 𝑓^𝑎𝑛(𝑉)⊆𝑈 for all 𝑛∈ℕ. When every fixed point of 𝑓 is uniformly virtually 𝑓-stable with respect to the same sequence, we will simply call 𝑓uniformly virtually stable.

Clearly, a uniformly virtually stable self-map is always virtually stable, and a continuously (uniformly) virtually stable self-map as defined in [1] is also (uniformly) virtually stable according to our new definition. Moreover, if 𝑓 is uniformly virtually stable with respect to (𝑎_𝑛), it is immediately uniformly virtually stable with respect to any subsequence of (𝑎_𝑛).

Let us recall that, when (𝑋,𝑑) is a metric space, a self-map 𝑓∶𝑋→𝑋 is called (1)virtually nonexpansive [7] if 𝑓 is continuous and the family {𝑓^𝑛} of all iterates of 𝑓 is equicontinuous on 𝐹(𝑓) (equivalently, on 𝐶(𝑓)), (2)uniformly quasi-Lipschitzian if there is 𝐿>0 such that 𝑑𝑓^𝑛(𝑥),𝑝≤𝐿𝑑𝑥,𝑝,(4) for all (𝑥,𝑝,𝑛)∈𝑋×𝐹(𝑓)×ℕ.

Notice that a virtually nonexpansive self-map is always continuous and uniformly virtually stable with respect to the sequence (𝑛) of all natural numbers, while a uniformly quasi-lipschitzian self-map may not be continuous in general.

Proposition 2. Suppose 𝑓 is a self-map on a metric space (𝑋,𝑑). (1) If 𝑓 is continuous and uniformly virtually stable with respect to (𝑛), then it is virtually nonexpansive. (2) If 𝑓 is uniformly quasi-lipschitzian, then it is uniformly virtually stable with respect to (𝑛).

Proof. Suppose 𝑓 is a self-map on a metric space (𝑋,𝑑). (1) Let 𝑝∈𝐹(𝑓) and 𝜖>0. Then there is 𝐻∈𝒩(𝑝) such that 𝑓^𝑛(𝐻)⊆𝐵(𝑝;𝜖) for all 𝑛∈ℕ. Thus, for 𝛿>0 whose 𝐵(𝑝;𝛿)⊆𝐻, we have 𝑓^𝑛(𝐵(𝑝;𝛿))⊆𝑓^𝑛(𝐻)⊆𝐵(𝑝;𝜖) for all 𝑛∈ℕ. This implies that {𝑓^𝑛} is equicontinuous on 𝐹(𝑓) and hence 𝑓 is virtually nonexpansive. (2) Suppose there is 𝐿>0 such that 𝑑(𝑓^𝑛(𝑥),𝑝)≤𝐿𝑑(𝑥,𝑝) for all (𝑥,𝑝,𝑛)∈𝑋×𝐹(𝑓)×ℕ. It follows that 𝑓^𝑛(𝐵(𝑝;𝑟/𝐿))⊆𝐵(𝑝;𝑟) for all 𝑝∈𝐹(𝑓), 𝑛∈ℕ and 𝑟>0. Thus, 𝑓 is uniformly virtually stable with respect to (𝑛).

It is not difficult to see that the class of uniformly quasi-lipschitzian self-maps includes various important (possibly discontinuous) self-maps in metric fixed-point theory such as nonexpansive maps, Kannan maps [8], Suzuki generalized nonexpansive maps [6], quasi-nonexpansive maps, and even asymptotically quasi-nonexpansive maps. Hence, the previous proposition immediately implies that those maps are all uniformly virtually stable with respect to (𝑛). The following theorem directly extends [1, Theorem 2.6] to include discontinuous maps.

Theorem 3. Suppose that 𝑋 is a regular space. If 𝑓 is virtually stable and 𝑓^𝑘 is continuous for some 𝑘∈ℕ, then 𝑓^∞ is continuous and hence 𝐹(𝑓) is a retract of 𝐶(𝑓).

Proof. Let 𝑥∈𝐶(𝑓) and 𝑈∈𝒩_𝐹(𝑓)(𝑓^∞(𝑥)). Since 𝑋 is regular, so is 𝐹(𝑓) and then there is 𝑊∈𝒩(𝑓^∞(𝑥)) such that 𝑊∩𝐹(𝑓)⊆𝑊∩𝐹(𝑓)⊆𝑈. By virtual stability of 𝑓, 𝑓^𝑎𝑛(𝑉)⊆𝑊 for all 𝑛∈ℕ, for some sequence (𝑎_𝑛) and 𝑉∈𝒩(𝑓^∞(𝑥)). Also there is, by the fact that 𝑉∈𝒩(𝑓^∞(𝑥)), 𝑛₀∈ℕ such that 𝑓^𝑛(𝑥)∈𝑉 for all 𝑛≥𝑛₀. Then 𝐴=𝑓^−𝑘𝑛0(𝑉)∩𝐶(𝑓)∈𝒩_𝐶(𝑓)(𝑥) and for each 𝑎∈𝐴, 𝑓^∞(𝑎)=lim_𝑛→∞𝑓^𝑛(𝑎)=lim_𝑛→∞𝑓^𝑎𝑛𝑓^𝑘𝑛0(𝑎)∈𝑊∩𝐹𝑓⊆𝑈.(5) Thus, 𝑓^∞ is continuous and hence 𝐹(𝑓) is a retract of 𝐶(𝑓).

Example 4. Let 𝑝∈ℕ and 𝑝≥2, and let 𝐼⊆ℝ be an interval. Consider the subspace 𝐼^𝑝 of ℝ^𝑝 equipped with the maximum norm ‖(𝑥₁,…,𝑥_𝑝)‖=max{|𝑥₁|,…,|𝑥_𝑝|}. Since the maximum norm on ℝ^𝑝 also induces the usual topology, 𝐼^𝑝 is a regular space. Recall that a self-map 𝑓∶𝐼^𝑝→𝐼^𝑝 is called mean-type if 𝑓=(𝑀₁,…,𝑀_𝑝), where each 𝑀_𝑖∶𝐼^𝑝→ℝ is a mean on 𝐼; that is, for each (𝑥₁,…,𝑥_𝑝)∈𝐼^𝑝, min𝑥₁,…,𝑥_𝑝≤𝑀_𝑖𝑥₁,…,𝑥_𝑝≤max𝑥₁,…,𝑥_𝑝.(6) Notice that 𝑓 may not be continuous in general. However, if 𝑓 satisfies 𝑀_𝑖𝑥₁,…,𝑥_𝑝−𝑧_𝑖≤max𝑥₁−𝑧₁,…,𝑥_𝑝−𝑧_𝑝(∗) for all (𝑥₁,…,𝑥_𝑝)∈𝐼^𝑝, (𝑧₁,…,𝑧_𝑝)∈𝐹(𝑓) and 𝑖=1,…,𝑝, it is always quasi-nonexpansive (with respect to the maximum norm), and hence uniformly virtually stable (with respect to the sequence (𝑛)). For if (𝑥₁,…,𝑥_𝑝)∈𝐼^𝑝 and (𝑧₁,…,𝑧_𝑝)∈𝐹(𝑓), one has 𝑓𝑥₁,…,𝑥_𝑝−𝑓𝑧₁,…,𝑧_𝑝=max𝑀_𝑖𝑥₁,…,𝑥_𝑝−𝑧_𝑖𝑥∶𝑖=1,…,𝑝≤max₁−𝑧₁,…,𝑥_𝑝−𝑧_𝑝(∗=𝑥by)₁,…,𝑥_𝑝−𝑧₁,…,𝑧_𝑝.(7)
In particular, the mean-type mapping 𝑓∶(0,∞)²→(0,∞)² defined by 𝑓(𝑥,𝑦)=((𝑥+𝑦)/2, 2𝑥𝑦/(𝑥+𝑦)) satisfies (∗), and hence it is uniformly virtually stable. Since 𝑓 is also continuous, 𝐹(𝑓) is a retract of 𝐶(𝑓) by the previous theorem. In fact, it follows from [4] that 𝐹(𝑓)={(𝑥,𝑥)∶𝑥>0}, 𝐶(𝑓)=(0,∞)², and 𝑓^∞(𝑥,𝑦)=(√𝑥𝑦,√𝑥𝑦) for (𝑥,𝑦)∈(0,∞)². Therefore, 𝐹(𝑓) is clearly a retract of 𝐶(𝑓).

Example 5. Consider that ℝ² equipped with the maximum norm, and a discontinuous self-map 𝑓∶ℝ²→ℝ² given by 𝑓=()=();(|𝑥𝑥,𝑦0,1if𝑥,𝑦0,3𝑥,|)otherwise.(8) It is straightforward to verify that 𝑓 is quasi-nonexpansive (hence, uniformly virtually stable) and 𝑓² is continuous. Therefore, by the previous theorem, 𝑓^∞ is continuous and hence 𝐹(𝑓) is a retract of 𝐶(𝑓). Notice also that 𝐹(𝑓)={(𝑥,|𝑥|)∶𝑥∈ℝ} is not convex.

The following examples show that if we drop the continuity assumption of some iterate 𝑓^𝑘 in Theorem 3, the map 𝑓^∞ may or may not be continuous.

Example 6. Let 𝑓∶[0,1]→[0,1] be defined by 𝑓(𝑥)=1𝑥if𝑥<2;1otherwise.(9) Observe that 𝑓 is uniformly virtually stable and 𝑓^𝑛=𝑓 is not continuous for all 𝑛∈ℕ. Moreover, 𝐹(𝑓)=[0,1/2)∪{1} and 𝐶(𝑓)=[0,1]. Hence 𝑓^∞ is not continuous.

Example 7. Consider that ℝ² equipped with the supremum norm. Define 𝑓∶ℝ²→ℝ² by 𝑓𝑥,𝑦=0,3^𝑛−1if𝑥,𝑦=0,3^𝑛(|𝑥forsome𝑛∈ℕ;𝑥,|)otherwise.(10) It is straightforward to verify that 𝑓 is quasi-nonexpansive (hence, uniformly virtually stable) and 𝑓^𝑛 is not continuous for all 𝑛∈ℕ. However, 𝑓^∞ is continuous because 𝑓^∞(𝑥,𝑦)=(𝑥,|𝑥|) for all (𝑥,𝑦)∈ℝ².

The next theorem gives a condition allowing us to enlarge the sequence associated with a given uniformly virtually stable self-map. As a result, we immediately obtain a refinement and a generalization of [1, Theorem 2.17].

Definition 8. An increasing sequence (𝑎_𝑛) of natural numbers is said to satisfy the sup-finite condition if sup_𝑛(𝑎_𝑛+1−𝑎_𝑛)∶=sup{𝑎_𝑛+1−𝑎_𝑛∶𝑛∈ℕ}<∞.

Theorem 9. If, for some 𝑘∈ℕ, 𝑓^𝑘 is continuous on 𝐹(𝑓) and 𝑓 is uniformly virtually stable with respect to (𝑎_𝑛𝑘) where (𝑎_𝑛) satisfies the sup-finite condition, then 𝑓 is uniformly virtually stable with respect to (𝑛𝑘).

Proof. Let 𝑠=sup_𝑛(𝑎_𝑛+1−𝑎_𝑛), 𝑝∈𝐹(𝑓), and 𝑈∈𝒩(𝑝). Because 𝑓 is uniformly virtually stable with respect to (𝑎_𝑛𝑘), there is 𝑊∈𝒩(𝑝) such that 𝑓^𝑎𝑛𝑘(𝑊)⊆𝑈 for all 𝑛∈ℕ. By continuity at 𝑝 of 𝑓^𝑖𝑘, 𝑖∈ℕ∪{0}, there exists 𝑉∈𝒩(𝑝) for which 𝑓^𝑖𝑘(𝑉)⊆𝑊∩𝑈 for all 𝑖=0,…,max{𝑠,𝑎₁}. Now let 𝑛∈ℕ. If 𝑛≤𝑎₁, 𝑓^𝑛𝑘(𝑉)⊆𝑊∩𝑈⊆𝑈, whereas if 𝑛>𝑎₁, there is 𝑗∈ℕ such that 𝑛=𝑎_𝑗+𝑖 for some 𝑖=0,…,𝑠 and so 𝑓^𝑛𝑘(𝑉)=𝑓^𝑎𝑗𝑘𝑓^𝑖𝑘(𝑉)⊆𝑓^𝑎𝑗𝑘(𝑊∩𝑈)⊆𝑓^𝑎𝑗𝑘(𝑊)⊆𝑈.(11)

Corollary 10. Suppose that 𝑋 is a metric space. If 𝑓 is continuous and uniformly virtually stable with respect to (𝑎_𝑛), where (𝑎_𝑛) satisfies the sup-finite condition, then 𝑓 is virtually nonexpansive. In particular, 𝐶(𝑓) is a 𝐺_𝛿-set when 𝑋 is complete.

Proof. By setting 𝑘=1 in the previous theorem, 𝑓 is uniformly virtually stable with respect to (𝑛) and hence virtually nonexpansive by Proposition 2 (1). For, if 𝑋 is complete, by [9, Theorem 1.2], 𝐶(𝑓) is then a 𝐺_𝛿-set.

Corollary 11. Suppose that 𝑋 is a complete metric space. If, for some 𝑘∈ℕ, 𝑓 is uniformly virtually stable with respect to (𝑛𝑘) and 𝑓^𝑘 is continuous, then 𝐶(𝑓^𝑘) is a 𝐺_𝛿-set.

Proof. Let 𝑔=𝑓^𝑘. Then 𝑔 is uniformly virtually stable with respect to (𝑛). Thus, 𝐶(𝑓^𝑘)=𝐶(𝑔) is a 𝐺_𝛿-set by the previous corollary.

3. Sequences Associated with Uniformly Virtually Stable Self-Maps

In view of Proposition 2 (2) and Theorem 9, the sequence (𝑎_𝑛) associated with a given uniformly virtually stable self-map 𝑓 seems to have an implicit structure. In fact, some questions may naturally arise. (i)Can we replace (𝑎_𝑛) with the larger (in some sense) sequence (𝑎^′_𝑛) so that 𝑓 is still uniformly virtually stable with respect to (𝑎^′_𝑛)? (ii)If (𝑎_𝑛) can be enlarged to (𝑎^′_𝑛), can we make (𝑎^′_𝑛) largest possible? and how does it look like? (iii)Does the largest possible sequence (𝑎^′_𝑛) satisfy the sup-finite condition? (iv)Is the largest possible sequence (𝑎^′_𝑛) always (𝑛)?

As we will see later on in this section, it turns out that the answers to these questions are interestingly nontrivial and require some careful considerations in both continuous and discontinuous settings.

Throughout this section, 𝑋 is a Hausdorff space and 𝑓 is a self-map of 𝑋. Every sequence is assumed to be an increasing sequence of natural numbers, so it is uniquely determined by its image. We then always identify a sequence (𝑎_𝑛) with the infinite set {𝑎_𝑛∶𝑛∈ℕ}, and with this identification in mind, the following notations become natural: (i)(𝑎_𝑛)⊆(𝑏_𝑛) if (𝑎_𝑛) is a subsequence of (𝑏_𝑛); (ii)(𝑎_𝑛)∪{𝑐} represents the sequence whose image is {𝑎_𝑛∶𝑛∈ℕ}∪{𝑐}; (iii)(𝑎_𝑛+𝑐) represents the sequence whose image is {𝑎_𝑛+𝑐∶𝑛∈ℕ}; (iv)(𝑎_𝑛)∪(𝑏_𝑛) represents the sequence whose image is {𝑎_𝑛∶𝑛∈ℕ}∪{𝑏_𝑛∶𝑛∈ℕ}, and so (𝑎_𝑛)∪⋯∪(𝑏_𝑛) represents the sequence whose image is {𝑎_𝑛∶𝑛∈ℕ}∪⋯∪{𝑏_𝑛∶𝑛∈ℕ}; (v)(𝑎_𝑛)⊕(𝑏_𝑛) represents the sequence whose image is {𝑎_𝑖+𝑏_𝑗∶𝑖,𝑗∈ℕ}, and ⊕^𝑘_𝑖=1(𝑎_𝑛)=(𝑎_𝑛)⊕⋯⊕(𝑎_𝑛)_{𝑘copiesof(𝑎𝑛)}.

For a nonempty subset 𝐴 of ℕ, the natural number 𝑑 satisfying (1)𝑑∣𝑎 for all 𝑎∈𝐴 and (2)if 𝑐∣𝑎 for all 𝑎∈𝐴, then 𝑑∣𝑐

will be called the greatest common divisor of 𝐴 and denoted by gcd𝐴. Also, we let ⟨𝐴⟩ denote the set of all finite sums of elements in 𝐴; that is, ⟨𝐴⟩=𝑥∶𝑥=𝑥₁+⋯+𝑥_𝑘forsome𝑥₁,…,𝑥_𝑘∈𝐴.(12) Notice that 𝐴⊆⟨𝐴⟩ and if 1∈𝐴, we have ⟨𝐴⟩=ℕ. As usual, gcd(𝑎_𝑛) will represent gcd{𝑎_𝑛∶𝑛∈ℕ}, and ⟨𝑎_𝑛⟩ will represent at the same time both the set ⟨{𝑎_𝑛∶𝑛∈ℕ}⟩ and the sequence whose image is ⟨{𝑎_𝑛∶𝑛∈ℕ}⟩. We may simply write ⟨𝑎₁,…,𝑎_𝑛⟩ for ⟨{𝑎₁,…,𝑎_𝑛}⟩.

Moreover, for a uniformly virtually stable self-map 𝑓, we let 𝒮_𝑓=𝑎_𝑛𝑎∶𝑓isuniformlyvirtuallystablewithrespectto_𝑛.(13) Clearly, 𝒮_𝑓 is partially ordered by ⊆, and a maximal element in (𝒮_𝑓,⊆) will be called a maximal sequence associated with 𝑓. The following are some basic properties of 𝒮_𝑓.

Proposition 12. Suppose 𝑓 is uniformly virtually stable with respect to (𝑎_𝑛). one has the following: (1)If (𝑏_𝑛)⊆(𝑎_𝑛), then (𝑏_𝑛)∈𝒮_𝑓. (2)If (𝑏_𝑛)∈𝒮_𝑓, then (𝑎_𝑛)∪(𝑏_𝑛)∈𝒮_𝑓. (3)⊕^𝑘_𝑖=1(𝑎_𝑛)∈𝒮_𝑓 for any 𝑘∈ℕ. (4)(𝑎_𝑛+𝑐),(𝑎_𝑛)∪{𝑐}∈𝒮_𝑓 for any 𝑐∈⟨𝑎_𝑛⟩.

Proof. Let 𝑥∈𝐹(𝑓) and 𝑈∈𝒩(𝑥).(1)It is straightforward from the definition of virtual stability.(2)There exist 𝑉,𝑊∈𝒩(𝑥) such that 𝑓^𝑎𝑛(𝑉)⊆𝑈 and 𝑓^𝑏𝑛(𝑊)⊆𝑈 for all 𝑛∈ℕ. Hence 𝑓^𝑐𝑛(𝑉∩𝑊)⊆𝑈 for all 𝑛∈ℕ where (𝑐_𝑛)=(𝑎_𝑛)∪(𝑏_𝑛).(3)By induction, it suffices to show that (𝑎_𝑛)⊕(𝑎_𝑛)∈𝒮_𝑓. There are, by virtual stability of 𝑓, 𝑉,𝑊∈𝒩(𝑥) such that 𝑓^𝑎𝑛(𝑉)⊆𝑈 and 𝑓^𝑎𝑛(𝑊)⊆𝑉 for all 𝑛. This implies that 𝑓^{𝑎𝑛+𝑎𝑚}(𝑊)⊆𝑓^𝑎𝑛(𝑉)⊆𝑈 for all 𝑛,𝑚∈ℕ.(4)By letting 𝑐∈⟨𝑎_𝑛⟩, we have 𝑐=𝑎_𝑖1+⋯+𝑎_𝑖𝑘∈⊕^𝑘_𝑖=1(𝑎_𝑛) for some 𝑘∈ℕ, and hence, (𝑎_𝑛+𝑐)⊆⊕^𝑘+1_𝑖=1(𝑎_𝑛)∈𝒮_𝑓 and (𝑎_𝑛)∪{𝑐}⊆⋃^𝑘_𝑗=1⊕^𝑗_𝑖=1(𝑎_𝑛)∈𝒮_𝑓 by (1) and (2).

Remark 13. From (2) in the previous proposition, if (𝑎_𝑛) and (𝑏_𝑛) are maximal elements in (𝒮_𝑓,⊆), then (𝑎_𝑛)=(𝑎_𝑛)∪(𝑏_𝑛)=(𝑏_𝑛). This implies that a maximal sequence associated with 𝑓, if exists, is always unique.

Before considering the existence, we will first investigate the structure of the maximal sequence associated with a given self-map.

Lemma 14. Suppose 𝑓 is uniformly virtually stable. For a sequence (𝑎_𝑛), one has (1)gcd(𝑎_𝑛)=gcd⟨𝑎_𝑛⟩=gcd{𝑎_𝑖∶𝑖≤𝑚} for some 𝑚∈ℕ, (2)there are 𝑏∈⟨𝑎_𝑛⟩ and 𝑚∈ℕ such that 𝑘gcd(𝑎_𝑛)+𝑏∈⟨𝑎₁,…,𝑎_𝑚⟩ for all 𝑘≥0, (3)if (𝑎_𝑛)∈𝒮_𝑓 and (𝑎_𝑛) satisfies the sup-finite condition, then ⟨𝑎_𝑛⟩∈𝒮_𝑓.

Proof. (1) The fact that gcd(𝑎_𝑛)=gcd⟨𝑎_𝑛⟩ follows directly from the definition. Now, let 𝑑=min{gcd{𝑎_𝑖∶𝑖≤𝑘}∶𝑘∈ℕ}. Then, 𝑑=gcd{𝑎_𝑖∶𝑖≤𝑚} for some 𝑚∈ℕ. If 𝑑∤𝑎_𝑛 for some 𝑛>𝑚, then gcd{𝑎₁,…,𝑎_𝑛}<𝑑, a contradiction. Also, for any 𝑐>0 satisfying 𝑐∣𝑎_𝑛 for all 𝑛∈ℕ, we have 𝑐∣𝑎_𝑖 for all 1≤𝑖≤𝑚, and hence, 𝑐≤𝑑. This implies gcd(𝑎_𝑛)=gcd{𝑎_𝑖∶𝑖≤𝑚}.
(2) If gcd(𝑎_𝑛)=𝑎₁, the proof is clear. So, we assume that gcd(𝑎_𝑛)≠𝑎₁. By (1), gcd(𝑎_𝑛)=gcd{𝑎_𝑖∶𝑖≤𝑚} for some 𝑚∈ℕ. Then, for some partition {𝑃,𝑁} of {1,…,𝑚}, gcd(𝑎_𝑛)=∑_𝑖∈𝑃𝑛_𝑖𝑎_𝑖−∑_𝑖∈𝑁𝑛_𝑖𝑎_𝑖 where 𝑛_𝑖≥0. Set 𝐴=∑_𝑖∈𝑃𝑛_𝑖𝑎_𝑖 and 𝐵=∑_𝑖∈𝑁𝑛_𝑖𝑎_𝑖. Notice that 𝐵≠0 because gcd(𝑎_𝑛)≠𝑎₁, and 𝑙𝐵∈⟨𝑎₁,…,𝑎_𝑚⟩ for all 𝑙∈ℕ. Then for each 𝑙∈ℕ and 0≤𝑟≤𝐵, 𝑟gcd𝑎_𝑛+𝐵²+𝑙𝐵=𝑟𝐴+(𝐵−𝑟+𝑙)𝐵∈𝑎₁,…,𝑎_𝑚.(14) Since for 𝑘≥0, we have 𝑘=𝑞𝐵+𝑟 for some 𝑞≥0 and 0≤𝑟<𝐵, 𝑘gcd𝑎_𝑛+𝐵²𝑎+𝐵=𝑟gcd_𝑛+𝐵²+1+𝑞gcd𝑎_𝑛𝐵∈𝑎₁,…,𝑎_𝑚.(15) Therefore, the result follows by letting 𝑏=𝐵²+𝐵.
(3) Assume that (𝑎_𝑛)∈𝒮_𝑓 and Δ∶=sup_𝑛(𝑎_𝑛+1−𝑎_𝑛)<∞. By (2), there is 𝑏∈⟨𝑎_𝑛⟩ for which 𝑘gcd(𝑎_𝑛)+𝑏∈⟨𝑎_𝑛⟩ for all 𝑘≥0. By Proposition 12, we have 𝑏_𝑛∶=𝑎_𝑛∪𝑎_𝑛+gcd𝑎_𝑛∪𝑎+𝑏_𝑛+2gcd𝑎_𝑛𝑎+𝑏∪⋯∪_𝑛+Δgcd𝑎_𝑛+𝑏∈𝒮_𝑓,(16) and hence (𝑐_𝑛)∶=(𝑏_𝑛)∪{𝑐∈⟨𝑎_𝑛⟩∶𝑐≤𝑎₁+𝑏}∈𝒮_𝑓. The proof will immediately follow by Proposition 12 (1) once we can show that ⟨𝑎_𝑛⟩⊆(𝑐_𝑛). To see this, let 𝑐∈⟨𝑎_𝑛⟩. The case of 𝑐≤𝑎₁+𝑏 is clear. Otherwise, 𝑎_𝑚+𝑏≤𝑐<𝑎_𝑚+1+𝑏 for some 𝑚≥1. Since gcd(𝑎_𝑛)∣𝑐−𝑎_𝑚−𝑏, we have 𝑐=𝑎_𝑚+𝑏+𝑟gcd(𝑎_𝑛) for some 0≤𝑟<𝑎_𝑚+1−𝑎_𝑚≤Δ. This implies that 𝑐∈(𝑎_𝑛+𝑟gcd(𝑎_𝑛)+𝑏)⊆(𝑏_𝑛)⊆(𝑐_𝑛).

Theorem 15. Suppose 𝑓 is uniformly virtually stable and (𝑎_𝑛) is the maximal sequence associated with 𝑓. One has(1)(𝑎_𝑛)=⟨𝑎_𝑛⟩, (2)(𝑎_𝑛)=⟨𝐴⟩ for some finite subset 𝐴 of (𝑎_𝑛), (3)if 𝑓 is continuous, then (𝑎_𝑛)=(𝑛).

Proof. (1) It suffices to show that ⟨𝑎_𝑛⟩⊆(𝑎_𝑛). Let 𝑐∈⟨𝑎_𝑛⟩. By Proposition 12, (𝑎_𝑛)∪{𝑐}∈𝒮_𝑓, and hence, 𝑐∈(𝑎_𝑛) by maximality of (𝑎_𝑛).
(2) From (1), we have (𝑎_𝑛)=⟨𝑎_𝑛⟩. Then by Lemma 14 (2), there exist 𝑙,𝑚∈ℕ such that 𝑘gcd(𝑎_𝑛)+𝑎_𝑙∈⟨𝑎₁,…,𝑎_𝑚⟩ for all 𝑘≥0. By setting 𝐴={𝑎_𝑖∶𝑖≤max{𝑙,𝑚}}, we clearly have ⟨𝐴⟩⊆⟨𝑎_𝑛⟩=(𝑎_𝑛). On the other hand, for each 𝑖∈ℕ, notice that (i)if 𝑖≤max{𝑙,𝑚}, then 𝑎_𝑖∈𝐴, and (ii)if 𝑖>max{𝑙,𝑚}, then 𝑎_𝑖=𝑎_𝑙+𝑘gcd(𝑎_𝑛)∈⟨𝑎₁,…,𝑎_𝑚⟩⊆⟨𝐴⟩ for some 𝑘∈ℕ.
Therefore, (𝑎_𝑛)⊆⟨𝐴⟩.
(3) Suppose that 𝑓 is continuous. It follows that (𝑎_𝑛)∪{1}∈𝒮_𝑓, and by maximality of (𝑎_𝑛), 1∈(𝑎_𝑛). Therefore, by (1), we have (𝑎_𝑛)=⟨𝑎_𝑛⟩=(𝑛).

The next theorem gives a necessary and sufficient condition for the existence of the maximal sequence.

Theorem 16. Suppose 𝑓 is uniformly virtually stable. Then the maximal sequence associated with 𝑓 exists if and only if there is (𝑎_𝑛)∈𝒮_𝑓 satisfying the sup-finite condition.

Proof. (⇒) If (𝑎_𝑛) is maximal, then (𝑛𝑎₁)⊆(𝑎_𝑛) and hence sup_𝑛𝑎_𝑛+1−𝑎_𝑛≤sup_𝑛(𝑛+1)𝑎₁−𝑛𝑎₁=𝑎₁<∞.(17)(⇐) Assume that Δ∶=sup_𝑛(𝑎_𝑛+1−𝑎_𝑛)<∞ for some (𝑎_𝑛)∈𝒮_𝑓. By Lemma 14 (3), ⟨𝑎_𝑛⟩∈𝒮_𝑓. If 𝑎₁=1, we are done. Suppose that 𝑎₁>1. For 0≤𝑖<𝑎₁=min⟨𝑎_𝑛⟩, let 𝐴_𝑖=𝑚∶𝑚≡𝑖mod𝑎₁and𝑎_𝑛∪{𝑚}∈𝒮_𝑓,(18) and 𝑚_𝑖=min𝐴_𝑖 if 𝐴_𝑖≠∅, otherwise 𝑚_𝑖=𝑎₁. Then {𝑐∶⟨𝑎_𝑛⟩∪{𝑐}∈𝒮_𝑓}=𝐴₀∪⋯∪𝐴_𝑎1−1. By Proposition 12 (2), ⟨𝑎_𝑛⟩∪{𝑚_𝑖∶0≤𝑖<𝑎₁}∈𝒮_𝑓 and again by Lemma 14 (3), since sup_𝑛(𝑏_𝑛+1−𝑏_𝑛)≤sup_𝑛(𝑎_𝑛+1−𝑎_𝑛)<∞ where (𝑏_𝑛)∶=⟨⟨𝑎_𝑛⟩∪{𝑚_𝑖∶0≤𝑖<𝑎₁}⟩, (𝑏_𝑛)∈𝒮_𝑓. Notice that (𝑏_𝑛) is maximal because whenever (𝑏_𝑛)∪{𝑐}∈𝒮_𝑓 for some 𝑐, there is 0≤𝑖<𝑎₁ such that 𝑐∈𝐴_𝑖⊆𝐴₀∪⋯∪𝐴_𝑎1−1⊆(𝑏_𝑛).

Towards the end of this section, we will construct some interesting examples including a uniformly virtually stable self-map whose maximal sequence is prescribed as well as a uniformly virtually stable self-map having no maximal sequence.

Example 17. For 𝑘∈ℕ∪{0}, let 𝜃_𝑘=𝜋∑^𝑘_𝑖=0(1/2^𝑖). Then 𝜋≤𝜃_𝑘<2𝜋 for all 𝑘. Consider the sequence (𝑎_𝑛)=⟨𝑎_𝑛⟩. Set 𝛿₀=𝑎₁ and 𝛿_𝑛=𝑎_𝑛+1−𝑎_𝑛 for any 𝑛∈ℕ. Define 𝑓∶ℂ→ℂ by 𝑓(𝑧)=𝑓𝑟𝑒^𝑖𝜙=(𝑟+1)𝑒^𝑖𝜙if𝜙=𝜃_𝑘,0<𝑟<𝛿_𝑘(⌊𝑟𝑒−1forsome𝑘;𝑟−⌋)^{𝑖(𝜋+𝜙/2)}if𝜙=𝜃_𝑘,𝑟≥𝛿_𝑘−1forsome𝑘;0otherwise.(19) Then, 𝑓 can be illustrated in Figure 1 and it is uniformly virtually stable with respect to the maximal sequence (𝑎_𝑛).
To see this, note that 𝐹(𝑓)={0}, 𝜋+𝜃_𝑘/2=𝜋(1+∑^𝑘+1_𝑖=1(1/2^𝑖))=𝜋∑^𝑘+1_𝑖=0(1/2^𝑖)=𝜃_𝑘+1 for all 𝑘≥0, and for each 𝑛∈ℕ and 0<𝑟<1, 𝑓^𝑎𝑛𝑟𝑒^𝑖𝜃0=𝑓^{𝛿𝑛−1+⋯+𝛿0}𝑟𝑒^𝑖𝜃0=𝑓^{𝛿𝑛−1+⋯+𝛿1}𝑟𝑒^𝑖𝜃1=⋯=𝑟𝑒^𝑖𝜃𝑛.(20) To show the virtual stability of 𝑓 with respect to (𝑎_𝑛), it suffices to prove that 𝑓^𝑎𝑛(𝐵(0;𝑟))⊆𝐵(0;𝑟) for all 0<𝑟<1. Let 0<𝑟<1. For each 𝑧=𝑠𝑒^𝑖𝜙∈𝐵(0;𝑟), 𝑓^𝑎𝑛(𝑧)≤𝑓^𝑎𝑛𝑠𝑒^𝑖𝜃𝑘=𝑓^𝑎𝑛𝑓^𝑎𝑘𝑠𝑒^𝑖𝜃0=𝑓^𝑎𝑚𝑠𝑒^𝑖𝜃0=𝑠𝑒^𝑖𝜃𝑚=𝑠<𝑟(21) for some 𝑘, and 𝑎_𝑚=𝑎_𝑛+𝑎_𝑘. Thus, we get the claim. Moreover (𝑎_𝑛) is maximal because whenever 𝑏∉(𝑎_𝑛), either 𝑏<𝑎₁ or 𝑎_𝑚<𝑏<𝑎_𝑚+1 for some 𝑚∈ℕ (and so 𝑏=𝑎_𝑚+𝑘 for some 1≤𝑘<𝛿_𝑚) holds, which the first case implies that 𝑓^𝑏(𝑟𝑒^𝑖𝜃0)=(𝑟+𝑏)𝑒^𝑖𝜃0∉𝐵(0;1) for all 0<𝑟<1, while the latter implies that 𝑓^𝑏𝑟𝑒^𝑖𝜃0=𝑓^𝑘𝑓^𝑎𝑚𝑟𝑒^𝑖𝜃0=𝑓^𝑘𝑟𝑒^𝑖𝜃𝑚=(𝑟+𝑘)𝑒^𝑖𝜃𝑚=𝑟+𝑘>1(22) for all 0<𝑟<1; that is, 𝑓^𝑏(𝑟𝑒^𝑖𝜃0)∉𝐵(0;1). Thus (𝑎_𝑛)∪{𝑏}∉𝒮_𝑓 for all 𝑏∉(𝑎_𝑛), which induces the maximality of (𝑎_𝑛) as desired.

Figure 1 

The map 𝑓 in Example 17.

Lemma 18. Let 𝐴₁={2^𝑛∶𝑛∈ℕ} and for 𝑚>1, 𝐴_𝑚=2^𝑛1+⋯+2^𝑛𝑚∶𝑛_𝑖∈ℕ−_𝑖<𝑚𝐴_𝑖.(23) Then {𝐴_𝑛∶𝑛∈ℕ} forms a partition of 2ℕ∶={2𝑛∶𝑛∈ℕ}, and 𝐴_𝑛 is infinite for each 𝑛∈ℕ.

Proof. Following from the fact that 2^𝑛+2^𝑛=2^𝑛+1 for all 𝑛∈ℕ, if ∑^𝑚_𝑖=12^𝑛𝑖∈𝐴_𝑚, then 𝑛_𝑖≠𝑛_𝑗 for all 𝑖≠𝑗. Also for each 𝑛∈ℕ, since 2𝑛=∑^𝑛_𝑖=12∈⋃^𝑛_𝑖=1𝐴_𝑖, we have 2𝑛∈𝐴_𝑖 for some 𝑖≤𝑛. Hence, {𝐴_𝑛∶𝑛∈ℕ} forms a partition of 2ℕ. For the next implication, notice that 𝐴₁ is clearly infinite. Now, assume that 𝑛>1. We first claim that 2^𝑛−2∈𝐴_𝑛−1. Since 2^𝑛−2∈2ℕ, we have 2^𝑛−2∈𝐴_𝑘 for some 𝑘∈ℕ. It follows that 𝑘<𝑛 and for if 2^𝑛−2=∑^𝑘_𝑖=12^𝑛𝑖, then 𝑛_𝑖<𝑛 and ∑^𝑘_𝑖=12^𝑛𝑖=2^𝑛−2=∑^𝑛−1_𝑖=12^𝑖, and hence 𝑘=𝑛−1. Next, we claim that 2^𝑛+𝑚+2^𝑛−2∈𝐴_𝑛 for all 𝑚∈ℕ. Let 𝑚∈ℕ. By the previous claim, we have 2^𝑛+𝑚+(2^𝑛−2)∈⋃^𝑛_𝑖=1𝐴_𝑖. Hence, 2^𝑛+𝑚+2^𝑛−2∈𝐴_𝑘 for some 1≤𝑘≤𝑛; that is, 2^𝑛+𝑚+2^𝑛−2=∑^𝑘_𝑖=12^𝑛𝑖. Since ∑^{𝑛+𝑚−1}_𝑖=12^𝑖=2^𝑛+𝑚−2<2^𝑛+𝑚+2^𝑛−2<2^𝑛+𝑚+1, we must have 2^𝑛𝑖=2^𝑛+𝑚 for some 1≤𝑖≤𝑘. Thus, 2^𝑛+𝑚+2^𝑛−2∈𝐴_𝑛 as claimed. Finally, since {2^𝑛+𝑚+2^𝑛−2∶𝑚∈ℕ}⊆𝐴_𝑛, then 𝐴_𝑛 is infinite as desired.

Example 19. For each 𝑚∈ℕ, let 𝐴_𝑚 be defined as in the previous lemma, 𝑝₁=3 and for each 𝑛>1, let 𝑝_𝑛 be the smallest prime number such that 𝑝_𝑛>max{𝑛²+𝑛,𝑝_𝑛−1}. For examples, 𝑝₂=7 and 𝑝₃=13. Clearly, 𝑝_𝑛≠𝑝_𝑚 whenever 𝑛≠𝑚. Now, let 𝑋=[0,2]×[0,2] be a subspace of ℝ² equipped with the maximum norm, 𝟎∶=(0,0)∈𝑋 and let 𝐵(𝑎;𝑟) denote the open ball {𝑥∈𝑋∶‖𝑥−𝑎‖<𝑟}.
The goal of this example is to find a uniformly virtually stable self-map having no maximal sequence. This can be done by constructing a uniformly virtually stable self-map 𝑓∶𝑋→𝑋 with respect to (2^𝑛) but not (2𝑛). For if (𝑎_𝑛) is the maximal sequence associated with such an 𝑓, we must have (2^𝑛)⊆(𝑎_𝑛) and hence (2𝑛)=⟨2^𝑛⟩⊆⟨𝑎_𝑛⟩=(𝑎_𝑛), but this contradicts the property that 𝑓 is not uniformly virtually stable with respect to (2𝑛).
To obtain the desired self-map, it suffices to require the map 𝑓∶𝑋→𝑋 to satisfy the following three conditions: (C1) 𝐹(𝑓)={𝟎},(C2) 𝑓^2𝑛(𝐵(𝟎;1/(𝑖+1)))⊆𝐵(𝟎;1/𝑖) for each 𝑖,𝑛∈ℕ, (C3) For each 𝑘∈ℕ, there exists 𝑎∈2ℕ such that 𝑓^𝑎(1/(𝑘+2),0)∉𝐵(𝟎;1).
Notice that (C1) and (C2) imply the virtual stability of 𝑓 with respect to (2^𝑛), while (C1) and (C3) imply that, for each 𝑘∈ℕ, there exists 𝑎∈2ℕ such that 𝑓^𝑎(𝐵(𝟎;1/(𝑘+1)))⊆𝐵̸(𝟎;1), and hence 𝑓 is not uniformly virtually stable with respect to (2𝑛).
We are now ready to give the explicit description of such an 𝑓. Consider the following subsets of 𝑋:  𝑇₁={(1/(𝑘+2),0)∶𝑘∈ℕ},  𝑇₂={(1,1/𝑝^𝑙_𝑘)∶𝑘,𝑙∈ℕ  and  𝑙+1∈𝐴_𝑚 for some 𝑚≤𝑘},  𝑇₃={(1/(𝑘+2−𝑚),1/𝑝^𝑙_𝑘)∶𝑘,𝑙∈ℕ and 𝑙∈𝐴_𝑚 for some 𝑚≤𝑘},  𝑇₄={(1,1/𝑝^𝑙_𝑘)∶𝑘,𝑙∈ℕand either 𝑙∈𝐴_𝑚 or 𝑙+1∈𝐴_𝑚 for some 𝑚>𝑘}.
Observe that (O1)𝑇_𝑖∩𝑇_𝑗=∅ whenever 𝑖≠𝑗. All cases are trivial except for the case of 𝑇₂∩𝑇₄, where it follows from the fact that {𝐴_𝑚∶𝑚∈ℕ} forms a partition of 2ℕ;(O2)‖(𝑥,𝑦)‖=𝑥 for all (𝑥,𝑦)∈⋃⁴_𝑖=1𝑇_𝑖, because for each 𝑘,𝑚,𝑙∈ℕ with 𝑚≤𝑘, we have 𝑝^𝑙_𝑘≥𝑝_𝑘>𝑘²+𝑘≥𝑘+1 and hence 1/𝑝^𝑙_𝑘<1/(𝑘+1)≤1/(𝑘+2−𝑚).
Define 𝑓∶𝑋→𝑋 by

𝑓=1𝑥,𝑦1,𝑝_𝑘=1if𝑥,𝑦𝑘+2,0∈𝑇₁;1,1𝑘+2−𝑚𝑝_𝑘^𝑙+1=1if𝑥,𝑦1,𝑝^𝑙_𝑘∈𝑇₂where𝑙+1∈𝐴_𝑚;11,𝑝_𝑘^𝑙+1=1if𝑥,𝑦,1𝑘+2−𝑚𝑝^𝑙_𝑘∈𝑇₃=1or𝑥,𝑦1,𝑝^𝑙_𝑘∈𝑇₄;𝟎otherwise.(24)

Clearly, 𝑓 is well-defined by (O1). The following properties are also satisfied.(P1)𝑓(𝑇₁)⊆𝑇₂, 𝑓(𝑇₂)⊆𝑇₃ and 𝑓(𝑇₃∪𝑇₄)⊆𝑇₂∪𝑇₄. So, ⋃⁴_𝑖=1𝑇_𝑖 is 𝑓-invariant. Moreover, since 𝑝_𝑖’s are all distinct primes, we have 𝑝^𝑘_𝑖=𝑝^𝑙_𝑗 if and only if (𝑖,𝑘)=(𝑗,𝑙), and hence (𝜋₂∘𝑓)|_⋃⁴_{𝑖=1𝑇𝑖} is injective, where 𝜋₂ denotes the second-coordinate projection 𝜋₂(𝑥,𝑦)=𝑦. (P2)For each 𝑘,𝑙∈ℕ, one can verify that 𝑓^𝑙1=1𝑘+2,0,1𝑘+2−𝑚𝑝^𝑙_𝑘if𝑙∈𝐴_𝑚1forsome𝑚≤𝑘;1,𝑝^𝑙_𝑘otherwise.(25) In particular, we have the following. (i)𝑓^2𝑙(1/(𝑘+2),0)=(1/(𝑘+2−1),1/𝑝^2𝑙_𝑘)=(1/(𝑘+1),1/𝑝^2𝑙_𝑘) since 2^𝑙∈(2^𝑛)=𝐴₁ and 1≤𝑘. (ii)𝑓^𝑐(1/(𝑘+2),0)=(1,1/𝑝^𝑐_𝑘), where 𝑐=min𝐴_𝑘+1, since 𝑐∈𝐴_𝑘+1 and 𝑘+1>𝑘. (P3)For each 𝑘,𝑙,𝑚,𝑛∈ℕ with 𝑚<𝑘 and 𝑙∈𝐴_𝑚, we have 𝑚+1≤𝑘, 𝑓^𝑙(1/(𝑘+2), 0)=(1/(𝑘+2−𝑚),1/𝑝^𝑙_𝑘) by (P2), and 2^𝑛+𝑙∈𝐴_𝑗⊆⋃^𝑚+1_𝑖=1𝐴_𝑖 for some 𝑗≤𝑚+1. Again by (P2), it follows that 1,1𝑘+2−𝑗𝑝^2𝑛_𝑘^+𝑙=𝑓^2𝑛+𝑙1𝑘+2,0=𝑓^2𝑛1,1𝑘+2−𝑚𝑝^𝑙_𝑘.(26)

Clearly, (C1) is implied by (P1). To prove (C2), let 𝑖,𝑛∈ℕ, and it suffices to consider the following two cases.(1)If (𝑥,𝑦)=(1/(𝑘+2),0)∈𝑇₁∩𝐵(𝟎;1/(𝑖+1)), then 1/(𝑘+2)<1/(𝑖+1), and hence 𝑓^2𝑛𝑥,𝑦=𝑓^2𝑛1𝑘+2,0_(P2)=1,1𝑘+1𝑝^2𝑛_𝑘^(O2)=1<1𝑘+1𝑖.(27)(2)If (𝑥,𝑦)=(1/(𝑘+2−𝑚),1/𝑝^𝑙_𝑘)∈𝑇₃∩𝐵(𝟎;1/(𝑖+1)), then 𝑙∈𝐴_𝑚 for some 𝑚≤𝑘, and 1/(𝑘+2−𝑚)=^(O2)‖(1/(𝑘+2−𝑚),1/𝑝^𝑙_𝑘)‖<1/(𝑖+1). If 𝑚=𝑘, we have 1/2=1/(𝑘+2−𝑚)<1/(𝑖+1), a contradiction. Thus 𝑚<𝑘, and by (P3), there is some 𝑗≤𝑚+1 such that 𝑓^2𝑛1,1𝑘+2−𝑚𝑝^𝑙_𝑘^(P3)=1,1𝑘+2−𝑗𝑝^2𝑛_𝑘^+𝑙_(O2)=1≤1𝑘+2−𝑗()<1𝑘+2−𝑚+1𝑖.(28)From above cases, it follows that 𝑓^2𝑛(𝐵(𝟎;1/(𝑖+1)))⊆𝐵(𝟎;1/𝑖).

Finally, for (C3), let 𝑘∈ℕ and 𝑐=min𝐴_𝑘+1∈2ℕ. Then by (P2), we have 𝑓^𝑐(1/(𝑘+2),0)=(1,1/𝑝_𝑘^𝑘+1)∉𝐵(𝟎;1).

Remark 20. From the previous example, we can easily check that 𝑓 is uniformly virtually stable with respect to (𝑎_𝑛)=𝐴₁∪⋯∪𝐴_𝑚 for any 𝑚∈ℕ, just by showing that 𝑓^𝑎𝑛(𝐵(𝟎;1/(𝑘+𝑚)))⊆𝐵(𝟎;1/𝑘) for any 𝑘,𝑛∈ℕ. Hence, Proposition 12 (2) does not hold in general for an arbitrary union of sequences in 𝒮_𝑓.

Acknowledgments

The first author is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author wishes to thank the H. M. King Bhumibol Adulyadej’s 72nd Birthday Anniversary Scholarship. The authors are grateful to Professor Pimpen Vejjajiva and the anonymous referee(s) for their valuable comments and suggestions for improving this paper.