A Study on 𝑁_𝜽-Quasi-Cauchy Sequences

Abstract

Recently, the concept of 𝑁_𝜃-ward continuity was introduced and studied. In this paper, we prove that the uniform limit of 𝑁_𝜃-ward continuous functions is 𝑁_𝜃-ward continuous, and the set of all 𝑁_𝜃-ward continuous functions is a closed subset of the set of all continuous functions. We also obtain that a real function 𝑓 defined on an interval 𝐸 is uniformly continuous if and only if (𝑓(𝛼_𝑘)) is 𝑁_𝜃-quasi-Cauchy whenever (𝛼_𝑘) is a quasi-Cauchy sequence of points in 𝐸.

1. Introduction

The concept of continuity and any concept involving continuity play a very important role not only in pure mathematics but also in other branches of sciences involving mathematics especially in computer science, information theory, and biological science.

A real function 𝑓 is continuous if and only if it preserves convergent sequences. A subset 𝐸 of 𝐑, the set of real numbers, is compact if any sequence of points in 𝐸 has a convergent subsequence whose limit is in 𝐸. Using the idea of continuity of a real function and the idea of compactness in terms of sequences, many kinds of continuities and compactness were introduced and investigated, not all but some of them we recall in the following: slowly oscillating continuity, slowly oscillating compactness [1], quasi-slowly oscillating continuity, quasi-slowly oscillating compactness [2], Δ-quasi-slowly oscillating continuity, Δ-quasi-slowly oscillating compactness [3–5], ward continuity, ward compactness [6, 7], 𝛿-ward continuity, 𝛿-ward compactness [8], statistical ward continuity, and lacunary statistical ward continuity [9, 10].

In [11], the notion of 𝑁_𝜃 convergence was introduced, and studied by Freedman et al. Using the main idea for continuity and compactness given above the concepts of 𝑁_𝜃-ward compactness of a subset 𝐸 of 𝐑 and 𝑁_𝜃-ward continuity of a real function are introduced and investigated recently in [12].

The purpose of this paper is to continue the investigation given in [12] and obtain further interesting results on 𝑁_𝜃-ward continuity.

2. Preliminaries

Boldface letters 𝜶, 𝐱, 𝐲, 𝐳,… will be used for sequences 𝜶=(𝛼_𝑘), 𝜶=(𝑥_𝑛), 𝐲=(𝑦_𝑛), 𝐳=(𝑧_𝑛),… of points in the set of real numbers 𝐑 for the sake of abbreviation. Sums of the form ∑^𝑘𝑟_{𝑘𝑟−1+1}|𝛼_𝑘| frequently occur and will often be written for convenience as ∑_{𝑘∈𝐼𝑟}|𝛼_𝑘|.

The concept of a Cauchy sequence involves far more than that the distance between successive terms is tending to zero. Nevertheless, sequences which satisfy this weaker property are interesting in their own right. A sequence (𝛼_𝑛) of points in 𝐑 is quasi-Cauchy if (Δ𝛼_𝑛) is a null sequence where Δ𝛼_𝑛=𝛼_𝑛+1−𝛼_𝑛. These sequences were named as quasi-Cauchy by Burton and Coleman [13, page 328], while they were called as forward convergent to 0 sequences in [7, page 226]. A sequence (𝛼_𝑘) of points in 𝐑 is called 𝑁_𝜃-convergent to an element 𝐿 of 𝐑 if lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝛼_𝑘−𝐿=0, (1) where 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟] and 𝜃=(𝑘_𝑟) is a lacunary sequence, that is, an increasing sequence of positive integers such that 𝑘₀=0 and ℎ_𝑟∶𝑘_𝑟−𝑘_𝑟−1→∞. The intervals determined by 𝜃 are denoted by 𝐼_𝑟=(𝑘_𝑟−1,𝑘_𝑟], and the ratio 𝑘_𝑟/𝑘_𝑟−1 is abbreviated by 𝑞_𝑟. A sequence (𝛼_𝑛) of points in 𝐑 is called 𝑁_𝜃-quasi-Cauchy if (Δ𝛼_𝑛) is 𝑁_𝜃-convergent to 0. A function defined on a subset 𝐴 of 𝐑 is called 𝑁_𝜃-ward continuous if it preserves 𝑁_𝜃-quasi-Cauchy sequences, that is, (𝑓(𝛼_𝑘)) is an 𝑁_𝜃-quasi-Cauchy sequence whenever (𝛼_𝑘) is. |Δ𝑁⁰_𝜃| will denote the set of 𝑁_𝜃-quasi-Cauchy sequences of points in 𝐑. Any subsequence of a Cauchy sequence is Cauchy. The analogous property fails for 𝑁_𝜃-quasi-Cauchy sequences. A counterexample for the case is the sequence (𝑎_𝑛)=(√𝑛) with the subsequence (𝑎_𝑛2)=(𝑛).

A sequence (𝛼_𝑛) of points in 𝐑 is slowly oscillating [14, Definition 2, page 947] if lim_𝜆→1+lim_𝑛max_{[]𝑛+1≤𝑘≤𝜆𝑛}𝛼_𝑘−𝛼_𝑛=0, (2) where [𝜆𝑛] denotes the integer part of 𝜆𝑛 (see also [14]).

An ideal 𝐼 is a family of subsets of positive integers 𝐍 which is closed under taking finite unions and subsets of its elements. A sequence (𝛼_𝑛) of real numbers is said to be ideal convergent to a real number 𝐿, if for each 𝜀>0 the set {𝑛∶|𝛼_𝑛−𝐿|≥𝜀} belongs to 𝐼. Ideal ward compactness of a subset of 𝐑 and ideal ward continuity of a real function were recently introduced by Çakalli and Hazarika in [15].

3. Results

Any quasi-Cauchy sequence is 𝑁_𝜃-quasi-Cauchy, so any slowly oscillating sequence is 𝑁_𝜃-quasi-Cauchy, and so any Cauchy sequence is. A sequence 𝜶=(𝛼_𝑘) is called Cesaro summable to a real number 𝐿 if lim_𝑛→∞(1/𝑛)∑^𝑛_𝑖=1𝛼_𝑖=𝐿. This is denoted by 𝐶₁−lim𝛼_𝑘=𝐿, and the set of all Cesaro sequences is denoted by 𝜎₁. We call a sequence 𝜶=(𝛼_𝑘) Cesaro quasi-Cauchy if 𝐶₁−limΔ𝛼_𝑘=0. The set of all Cesaro quasi-Cauchy sequences is denoted by Δ𝜎⁰₁. A sequence 𝜶=(𝛼_𝑘) is called strongly Cesaro summable to a real number 𝐿 if lim_𝑛→∞(1/𝑛)∑^𝑛_𝑖=1|𝛼_𝑖−𝐿|=0. This is denoted by |𝐶₁|−lim𝛼_𝑘=𝐿. The set of all strongly Cesaro summable sequences is denoted by |𝜎₁|. We call a sequence 𝜶=(𝛼_𝑘) strongly Cesaro quasi-Cauchy if |𝐶₁|−limΔ𝛼_𝑘=0. The set of all strongly Cesaro quasi-Cauchy sequences is denoted by |Δ𝜎⁰₁|. The following inclusions are satisfied: |𝜎₁|⊂𝜎₁ and |Δ𝜎⁰₁|⊂Δ𝜎⁰₁.

Using a similar idea to that of [11], one can easily find out the following inclusion properties between the set of strongly Cesaro quasi-Cauchy sequences and the set of 𝑁_𝜃-quasi-Cauchy sequences (see also [16]).(i)|Δ𝑁⁰_𝜃|⊂|Δ𝜎⁰₁| if and only if limsup𝑞_𝑟<∞ for any lacunary sequence 𝜃.(ii)|Δ𝜎⁰₁|⊂|Δ𝑁⁰_𝜃| if and only if liminf𝑞_𝑟>1 for any lacunary sequence 𝜃.

 Combining these facts, for any lacunary sequence 𝜃, we have the following:(iii)|Δ𝑁⁰_𝜃|=|Δ𝜎⁰₁| if and only if 1<liminf𝑞_𝑟≤limsup𝑞_𝑟<∞;(iv)|Δ𝑁⁰_𝜃|=|Δ𝜎⁰₁| if and only if |𝜎₁|=𝑁⁰_𝜃.

In the sequel, we will always assume that liminf_𝑟𝑞_𝑟>1.

We observe that 𝑁_𝜃-summability is a kind of strong 𝐴-summability where 𝐴=(𝑎_𝑟𝑘) is a regular matrix generated by the lacunary sequence 𝜃=(𝑘_𝑟) as follows: 𝑎_𝑟𝑘=1ℎ_𝑟if𝑘∈𝐼_𝑟,𝑎_𝑟𝑘=0otherwise. (3) On the other hand, we see that 𝑁_𝜃-ward continuity cannot be given as a strong 𝐴-continuity by any kind of regular summability matrix (related to continuity for strong matrix methods see [17]).

As far as ideal continuity is considered, we note that any 𝑁_𝜃-ward continuous function is ideal continuous; furthermore any 𝑁_𝜃 continuous function is ideal continuous for an admissible ideal.

Theorem 1. If a function 𝑓 is uniformly continuous on a subset 𝐸 of 𝐑, then, (𝑓(𝛼_𝑘)) is 𝑁_𝜃-quasi-Cauchy whenever (𝛼_𝑘) is a quasi-Cauchy sequence of points in 𝐸.

Proof. Let 𝐸 be a subset of 𝐑, and let 𝑓 be a uniformly continuous function on 𝐸. Take any quasi-Cauchy sequence (𝛼_𝑘) of points in 𝐸, and let 𝜀 be any positive real number. By uniform continuity of 𝑓, there exists a 𝛿>0 such that |𝑓(𝛼)−𝑓(𝛽)|<𝜀 whenever |𝛼−𝛽|<𝛿 and 𝛼,𝛽∈𝐸. Since (𝛼_𝑘) is a quasi-Cauchy sequence, there exists a positive integer 𝑘₀ such that |𝛼_𝑘+1−𝛼_𝑘|<𝛿 for 𝑘≥𝑘₀. Hence, 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘+1−𝑓𝛼_𝑘<1ℎ_𝑟𝑘_𝑟−𝑘_𝑟−1𝜀=𝜀, (4) for 𝑟≥𝑘₀. Thus, (𝑓(𝛼_𝑘)) is an 𝑁_𝜃-quasi-Cauchy sequence. This completes the proof of the theorem.

We have much more below for a real function 𝑓 defined on an interval that 𝑓 is uniformly continuous if and only if (𝑓(𝛼_𝑘)) is 𝑁_𝜃-quasi-Cauchy whenever (𝛼_𝑘) is a quasi-Cauchy sequence of points in 𝐸. First, we give the following lemma.

Lemma 2. If (𝜉_𝑛,𝜂_𝑛) is a sequence of ordered pairs of points in an interval such that lim_𝑛→∞|𝜉_𝑛−𝜂_𝑛|=0, then, there exists an 𝑁_𝜃-quasi-Cauchy sequence (𝛼_𝑛) with the property that for any positive integer 𝑖 there exists a positive integer 𝑗 such that (𝜉_𝑖,𝜂_𝑖)=(𝛼_𝑗−1,𝛼_𝑗).

Proof. Although the following proof is similar to that of [13], we give it for completeness. For each positive integer 𝑘, we can fix 𝑧^𝑘₀,𝑧^𝑘₁,…,𝑧^𝑘_𝑛𝑘 in 𝐸 with 𝑧^𝑘₀=𝜂_𝑘, 𝑧^𝑘_𝑛𝑘=𝜉_𝑘+1, and |𝑧^𝑘_𝑖−𝑧^𝑘_𝑖−1|<1/𝑘 for 1≤𝑖≤𝑛_𝑘. Now write 𝜉₁,𝜂₁,𝑧¹₁,…,𝑧¹_𝑛1−1,𝜉₂,𝜂₂,𝑧²₁,…,𝑧²_𝑛2−1,𝜉₃,𝜂₃𝜉,…,_𝑘,𝜂_𝑘,𝑧^𝑘₁,…,𝑧^𝑘_𝑛𝑘−1,𝜉_𝑘+1,𝜂_𝑘+1,…. (5) Then denoting this sequence by (𝛼_𝑛), we obtain that for any positive integer 𝑖 there exists a positive integer 𝑗 such that (𝜉_𝑖,𝜂_𝑖)=(𝛼_𝑗−1,𝛼_𝑗). The sequence constructed is a quasi-Cauchy sequence, and it is an 𝑁_𝜃-quasi-Cauchy sequence, since any quasi-Cauchy sequence is an 𝑁_𝜃-quasi-Cauchy sequence. This completes the proof of the lemma.

Theorem 3. If a function 𝑓 defined on an interval 𝐸 is 𝑁_𝜃-ward continuous, then, it is uniformly continuous.

Proof. Suppose that 𝑓 is not uniformly continuous on 𝐸. Then, there is an 𝜀₀>0 such that for any 𝛿>0 there exist 𝑥,𝑦∈𝐸 with |𝑥−𝑦|<𝛿 but |𝑓(𝑥)−𝑓(𝑦)|≥𝜀₀. For every integer 𝑛≥1 fix 𝜉_𝑛, 𝜂_𝑛∈𝐸 with |𝜉_𝑛−𝜂_𝑛|<1/𝑛 and |𝑓(𝜉_𝑛)−𝑓(𝜂_𝑛)|≥𝜀₀. By the lemma, there exists an 𝑁_𝜃-quasi-Cauchy sequence (𝛼_𝑖) such that for any integer 𝑖≥1 there exists a 𝑗 with 𝜉_𝑖=𝛼_𝑗 and 𝜂_𝑖=𝛼_𝑗+1. This implies that |𝑓(𝛼_𝑗+1)−𝑓(𝛼_𝑗)|≥𝜀₀; hence, (𝑓(𝛼_𝑖)) is not 𝑁_𝜃-quasi-Cauchy. Thus, 𝑓 does not preserve 𝑁_𝜃-quasi-Cauchy sequences. This completes the proof of the theorem.

Observing that the sequence, constructed in the proof of the preceding theorem, is also a quasi-Cauchy sequence, we obtain that a real function 𝑓 defined on an interval 𝐸 is uniformly continuous if (𝑓(𝛼_𝑘)) is 𝑁_𝜃-quasi-Cauchy whenever (𝛼_𝑘) is a quasi-Cauchy sequence of points in 𝐸. Combining this with Theorem 1, we have that a real function 𝑓 defined on an interval 𝐸 is uniformly continuous if and only if (𝑓(𝛼_𝑘)) is 𝑁_𝜃-quasi-Cauchy whenever (𝛼_𝑘) is a quasi-Cauchy sequence of points in 𝐸.

Corollary 4. If a function defined on an interval is 𝑁_𝜃-ward continuous, then, it is ward continuous.

Proof. The proof follows from Theorem 3 and [7, Theorem 6] so it is omitted.

Corollary 5. If a function defined on an interval is 𝑁_𝜃-ward continuous, then, it is slowly oscillating continuous.

Proof. The proof follows from Theorem 3 and [7, Theorem 5] so it is omitted.

It is a well-known result that uniform limit of a sequence of continuous functions is continuous. This is also true in case of 𝑁_𝜃-ward continuity; that is, uniform limit of a sequence of 𝑁_𝜃-ward continuous functions is 𝑁_𝜃-ward continuous.

Theorem 6. If (𝑓_𝑛) is a sequence of 𝑁_𝜃-ward continuous functions on a subset 𝐸 of 𝐑 and (𝑓_𝑛) is uniformly convergent to a function 𝑓, then, 𝑓 is 𝑁_𝜃-ward continuous on 𝐸.

Proof. Let (𝛼_𝑘) be any 𝑁_𝜃-quasi-Cauchy sequence of points in 𝐸, and let 𝜀 be any positive real number. By uniform convergence of (𝑓_𝑛), there exists an 𝑛₁∈𝐍 such that |𝑓(𝛼)−𝑓_𝑘(𝛼)|<𝜀/3 for 𝑛≥𝑛₁ and every 𝛼∈𝐸. Hence, 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓(𝛼)−𝑓_𝑘(𝛼<1)ℎ_𝑟𝑘_𝑟−𝑘_𝑟−1𝜀3=𝜀3, (6) for 𝑟≥𝑛₁ and every 𝛼∈𝐸. As 𝑓_𝑛1 is 𝑁_𝜃-ward continuous on 𝐸, there exists an 𝑛₂∈𝐍 such that for 𝑟≥𝑛₂1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑛1𝛼_𝑘+1−𝑓_𝑛1𝛼_𝑘<𝜀3. (7) Now write 𝑛₀=max{𝑛₁,𝑛₂}. Thus for 𝑟≥𝑛₀, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘+1−𝑓𝛼_𝑘≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘+1−𝑓_𝑛1𝛼_𝑘+1+1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑛1𝛼_𝑘+1−𝑓_𝑛1𝛼_𝑘+1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑛1𝛼_𝑘−𝑓𝛼_𝑘<𝜀3+𝜀3+𝜀3=𝜀. (8) Hence, lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘+1−𝑓𝛼_𝑘=0. (9) Thus, 𝑓 preserves 𝑁_𝜃-quasi-Cauchy sequences. This completes the proof of the theorem.

Theorem 7. The set of all 𝑁_𝜃-ward continuous functions on a subset 𝐸 of 𝐑 is a closed subset of the set of all continuous functions on 𝐸, that is, Δ𝑁_𝜃𝑊𝐶(𝐸)=Δ𝑁_𝜃𝑊𝐶(𝐸), where Δ𝑁_𝜃𝑊𝐶(𝐸) is the set of all 𝑁_𝜃-ward continuous functions on 𝐸 and Δ𝑁_𝜃𝑊𝐶(𝐸) denotes the set of all cluster points of Δ𝑁_𝜃𝑊𝐶(𝐸).

Proof. Let 𝑓 be any element in the closure of Δ𝑁_𝜃𝑊𝐶(𝐸). Then, there exists a sequence (𝑓_𝑛) of points in Δ𝑁_𝜃𝑊𝐶(𝐸) such that lim_𝑘→∞𝑓_𝑘=𝑓. To show that 𝑓 is 𝑁_𝜃-ward continuous, take any 𝑁_𝜃-quasi-Cauchy sequence (𝛼_𝑘) of points in 𝐸. Let 𝜀>0. Since (𝑓_𝑛) converges to 𝑓, there exists an 𝑛₁∈𝐍 such that |𝑓(𝛼_𝑘)−𝑓_𝑛1(𝛼_𝑘)|<𝜀/3 for all 𝑘∈𝐍. Hence, 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘−𝑓_𝑛1𝛼_𝑘<1ℎ_𝑟𝑘_𝑟−𝑘_𝑟−1𝜀3=𝜀3, (10) for 𝑟≥𝑛₁. As 𝑓_𝑛1 is 𝑁_𝜃-ward continuous on 𝐸, there exists a positive integer 𝑛₂∈𝐍 such that 𝑟≥𝑛₂ implies that 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑛1𝛼_𝑘+1−𝑓_𝑛1𝛼_𝑘<𝜀3. (11) Now write 𝑛₀=max{𝑛₁,𝑛₂}. Thus for 𝑟≥𝑛₀, we have 1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘+1−𝑓𝛼_𝑘≤1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘+1−𝑓_𝑛1𝛼_𝑘+1+1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑛1𝛼_𝑘+1−𝑓_𝑛1𝛼_𝑘+1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓_𝑛1𝛼_𝑘−𝑓𝛼_𝑘<𝜀3+𝜀3+𝜀3=𝜀; (12) Hence, lim_𝑟→∞1ℎ_{𝑟𝑘∈𝐼𝑟}𝑓𝛼_𝑘+1−𝑓𝛼_𝑘=0. (13)
Thus, 𝑓 preserves 𝑁_𝜃-quasi-Cauchy sequences. This completes the proof of the theorem.

Corollary 8. The set of all 𝑁_𝜃-ward continuous functions on a subset 𝐸 of 𝐑 is a complete subspace of the space of all continuous functions on 𝐸.

Proof. The proof follows from the preceding theorem.

4. Conclusion

In this paper, new results concerning 𝑁_𝜃-ward continuity are obtained namely; a real function 𝑓 defined on an interval 𝐸 is uniformly continuous if and only if (𝑓(𝛼_𝑘)) is 𝑁_𝜃-quasi-Cauchy whenever (𝛼_𝑘) is a quasi-Cauchy sequence of points in 𝐸, the uniform limit of 𝑁_𝜃-ward continuous functions is 𝑁_𝜃-ward continuous, and the set of all 𝑁_𝜃-ward continuous functions is a closed subset of the set of all continuous functions. We also prove that if a function 𝑓 is uniformly continuous on a subset 𝐸 of 𝐑, then, (𝑓(𝛼_𝑘)) is 𝑁_𝜃-quasi-Cauchy whenever (𝛼_𝑘) is a quasi-Cauchy sequence of points in 𝐸.

As a further study one can find out if Theorem 3 is valid when the set 𝐸 is replaced by a 𝐺-sequentially connected subset of 𝐑 for a regular sequential method 𝐺 [18]. For another further study, we suggest to investigate the present work for the fuzzy case. However, due to the change in settings, the definitions and methods of proofs will not always be analogous to those of the present work (see [19] for the definitions in the fuzzy setting). One can introduce and give an investigation of 𝑁_𝜃-quasi-Cauchy sequences in cone normed spaces (see [20] for basic concepts in cone normed spaces).