Second Order Ideal-Ward Continuity

Abstract

The main aim of the paper is to introduce a concept of second order ideal-ward continuity in the sense that a function 𝑓 is second order ideal-ward continuous if 𝐼−lim_𝑛→∞Δ²𝑓(𝑥_𝑛)=0 whenever 𝐼−lim_𝑛→∞Δ²𝑥_𝑛=0 and a concept of second order ideal-ward compactness in the sense that a subset 𝐸 of 𝐑 is second order ideal-ward compact if any sequence 𝑥=(𝑥_𝑛) of points in 𝐸 has a subsequence 𝑧=(𝑧_𝑘)=(𝑥_𝑛𝑘) of the sequence x such that 𝐼−lim_𝑘→∞Δ²𝑧_𝑘=0 where Δ²𝑧_𝑘=𝑧_𝑘+2−2𝑧_𝑘+1+𝑧_𝑘. We investigate the impact of changing the definition of convergence of sequences on the structure of ideal-ward continuity in the sense of second order ideal-ward continuity and compactness of sets in the sense of second order ideal-ward compactness and prove related theorems.

1. Introduction

Let us start with basic definitions from the literature. Let 𝐾⊆ℕ, the set of all natural numbers, and 𝐾_𝑛={𝑘≤𝑛∶𝑘∈𝐾}. Then the natural density of 𝐾 is defined by 𝛿(𝐾)=lim_𝑛𝑛⁻¹|𝐾_𝑛| if the limit exists, where the vertical bars indicate the number of elements in the enclosed set.

Fast [1] presented the following definition of statistical convergence for sequences of real numbers. The sequence 𝑥=(𝑥_𝑘) is said to be statistically convergent to 𝐿 if for every 𝜖>0, the set 𝐾_𝜖∶={𝑘∈𝐍∶|𝑥_𝑘−𝐿|≥𝜖} has natural density zero; that is, for each 𝜖>0, lim_𝑛1𝑛𝑗≤𝑛∶𝑥_𝑗−𝐿≥𝜖=0. (1) In this case, we write 𝑆−lim𝑥=𝐿 or 𝑥_𝑘→𝐿(𝑆) and 𝑆 denotes the set of all statistically convergent sequences. Note that every convergent sequence is statistically convergent but not conversely.

Some basic properties related to the concept of statistical convergence were studied in [2, 3]. In 1985, Fridy [4] presented the notion of statistically Cauchy sequence and determined that it is equivalent to statistical convergence. Caserta et al. [5] studied statistical convergence in function spaces, while Caserta and Koc̆inac [6] investigated statistical exhaustiveness.

Kostyrko et al. [7] introduced the notion of ideal convergence. It is a generalization of statistical convergence. For details on ideal convergence we refer to [8–13].

Let 𝑋 be a nonempty set; then a family of sets 𝐼⊂𝑃(𝑋) (power sets of 𝑋) is called an ideal on 𝑋 if and only if(a)𝜙∈𝐼, (b)for each 𝐴,𝐵∈𝐼, we have 𝐴∪𝐵∈𝐼,(c)for each 𝐴∈𝐼 and each 𝐵⊂𝐴, we have 𝐵∈𝐼.

A nonempty family of sets 𝐹⊂𝑃(𝑋) is a filter on𝑋if and only if(a)𝜙∉𝐹, (b)for each 𝐴,𝐵∈𝐹, we have 𝐴∩𝐵∈𝐹,(c)each 𝐴∈𝐹 and each 𝐵⊃𝐴, we have 𝐵∈𝐹.

An ideal 𝐼 is called nontrivial ideal if 𝐼≠𝜙 and 𝑋∉𝐼. Clearly 𝐼⊂𝑃(𝑋) is a nontrivial ideal if and only if 𝐹=𝐹(𝐼)={𝑋−𝐴∶𝐴∈𝐼} is a filter on 𝑋. A nontrivial ideal 𝐼⊂𝑃(𝑋) is called admissible if and only if {{𝑥}∶𝑥∈𝑋}⊂𝐼. A nontrivial ideal 𝐼 is maximal if there cannot exist any nontrivial ideal 𝐽≠𝐼 containing 𝐼 as a subset.

Definition 1 (see [7]). A sequence 𝑥=(𝑥_𝑛) of points in 𝐑 is said to be 𝐼-convergent to the number ℓ if, for every 𝜀>0, the set {𝑛∈𝐍∶|𝑥_𝑛−ℓ|≥𝜀}∈𝐼. One writes 𝐼−lim𝑥_𝑛=ℓ. One sees that a sequence 𝑥=(𝑥_𝑛) being 𝐼-convergent implies that 𝐼−lim_𝑛→∞Δ𝑥_𝑛=0.

Burton and Coleman [14] introduced the concept of quasi-Cauchy sequences as a sequence (𝑥_𝑛) of points of 𝐑 is said to be a quasi-Cauchy sequence if (Δ𝑥_𝑛) is a null sequence where Δ𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛. Çakallı and Hazarika [15] introduced the concept of ideal quasi-Cauchy sequences. Recall from [15] that a sequence (𝑥_𝑛) of points of 𝐑 is called ideal quasi-Cauchy if 𝐼−lim_𝑛→∞Δ𝑥_𝑛=0.

We say that a sequence 𝑥=(𝑥_𝑛) is ward convergent to a number ℓ if lim_𝑛→∞Δ𝑥_𝑛=ℓ where Δ𝑥_𝑛=𝑥_𝑛+1−𝑥_𝑛. Using the idea of continuity of a real function and the idea of compactness in terms of sequences, Çakallı [16] introduced the concept of ward continuity in the sense that a function 𝑓 is ward continuous if it transforms ward convergent to 0 sequences to ward convergent to 0 sequences; that is, (𝑓(𝑥_𝑛)) is ward convergent to 0 whenever (𝑥_𝑛) is ward convergent to 0, and Çakallı [17] introduced the concept of ward compactness in the sense that a subset 𝐸 of 𝐑 is ward compact if any sequence 𝑥=(𝑥_𝑛) of points in 𝐸 has a subsequence 𝑧=(𝑧_𝑘)=(𝑥_𝑛𝑘) of the sequence 𝑥 such that lim_𝑘→∞Δ𝑧_𝑘=0 where Δ𝑧_𝑘=𝑧_𝑘+1−𝑧_𝑘. Throughout the paper 𝑐, 𝑆, 𝐼, and Δ𝐼 will denote the set of all convergent sequences, statistically convergent sequences, 𝐼-convergent sequences, and the set of all 𝐼-ward convergent to 0 sequences of points in 𝐑 where a sequence 𝑥=(𝑥_𝑛) is called 𝐼-ward convergent to 0 if 𝐼−lim_𝑛→∞Δ𝑥_𝑛=0.

Throughout the paper we assume 𝐼 is a nontrivial admissible ideal of 𝐍.

2. Second Order Ideal-Ward Continuity

We introduce the notion of second order ward convergent sequences as follows.

Definition 2. A sequence 𝑥=(𝑥_𝑛) is said to be second order ward convergent to a number ℓ if lim_𝑛→∞Δ²𝑥_𝑛=ℓ where Δ²𝑥_𝑛=𝑥_𝑛+2−2𝑥_𝑛+1+𝑥_𝑛. For the special case ℓ=0, 𝑥 is called second order ward convergent to 0.

We note that any ward convergent to 0 sequence is also second order ward convergent to 0, but the opposite is not always true as it can be considering the sequence (𝑛).

Definition 3. A sequence 𝑥=(𝑥_𝑛) is said to be second order ideal-ward convergent to a number ℓ if 𝐼−lim_𝑛→∞Δ²𝑥_𝑛=ℓ where Δ²𝑥_𝑛=𝑥_𝑛+2−2𝑥_𝑛+1+𝑥_𝑛. For the special case ℓ=0, 𝐱 is called second order ideal-ward convergent to 0. One denotes by Δ²𝐼 the set of all second order ideal-ward convergent sequences.

Now we give the definition of second order ideal-ward continuous function on a subset of 𝐑.

Definition 4. A function 𝑓 is called ward continuous on 𝐸 if the sequence (𝑓(𝑥_𝑛)) is ward convergent to 0 whenever 𝑥=(𝑥_𝑛) is a ward convergent to 0 sequence of terms in 𝐸.

Definition 5. A function 𝑓 is called second order ideal-ward continuous on 𝐸 if 𝐼−lim_𝑛→∞Δ²𝑓(𝑥_𝑛)=0 whenever 𝐼−lim_𝑛→∞Δ²𝑥_𝑛=0, for a sequence 𝑥=(𝑥_𝑛) of terms in 𝐸.

Theorem 6. If 𝑓 is second order ideal-ward continuous on a subset 𝐸 of 𝐑, then it is an ideal-ward continuous on 𝐸.

Proof. Suppose that 𝑓 is a second order ideal-ward continuous function on a subset 𝐸 of 𝐑. Let (𝑥_𝑛) be a sequence with 𝐼−lim_𝑛→∞Δ𝑥_𝑛=0. Then we have the sequence 𝑥₁,𝑥₁,𝑥₂,𝑥₂,𝑥₃,𝑥₃,…,𝑥_𝑛−1,𝑥_𝑛−1,𝑥_𝑛,𝑥_𝑛,… (2) such that 𝐼−lim_𝑛→∞Δ²𝑥_𝑛=0. Since 𝑓 is second order ideal-ward continuous, then we get the sequence 𝑦_𝑛=𝑓𝑥₁,𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₂𝑓𝑥,…,_𝑛,𝑓𝑥_𝑛,… (3) such that 𝐼−lim_𝑛→∞Δ²𝑦_𝑛=0. It follows that 𝑓𝑥₁,𝑓𝑥₁,𝑓𝑥₂,𝑓𝑥₂,…,𝑓𝑥_𝑛,𝑓𝑥_𝑛,… (4) is second order ideal-ward convergent to 0. This completes the proof of the theorem.

The converse is not always true for this we consider the function𝑓(𝑥)=sin𝑥which is ideal-ward continuous but not second order ideal-ward continuous.

Corollary 7. If 𝑓 is second order ward continuous, then it is an ideal continuous.

Theorem 8. If 𝑓 is second order ideal-ward continuous, then it is second order ward continuous.

Proof. The proof is easy, so omitted.

Definition 9. A subset 𝐸 of 𝐑 is called second order ward compact if 𝑥=(𝑥_𝑛) is a sequence of points in 𝐸and there is a subsequence 𝑧=(𝑧_𝑘)=(𝑥_𝑛𝑘) of 𝐱 such that lim_𝑘→∞Δ²𝑧_𝑘=0.

Now we give the definition of second order ideal-ward compactness of a subset of 𝐑.

Definition 10. A subset 𝐸 of 𝐑 is called second order ideal-ward compact if 𝑥=(𝑥_𝑛) is a sequence of points in 𝐸and there is a subsequence 𝐳=(𝑧_𝑘)=(𝑥_𝑛𝑘) of 𝐱 such that 𝐼−lim_𝑘→∞Δ²𝑧_𝑘=0.

Theorem 11. A second order ideal-ward continuous image of any second order ideal-ward compact subset of 𝐑 is second order ideal-ward compact.

Proof. Suppose that 𝑓 is a second order ideal-ward continuous function on a subset 𝐸 of 𝐑 and 𝐸 is a second order ideal-ward compact subset of 𝐑. Let (𝑦_𝑛) be a sequence of points in 𝑓(𝐸). Write 𝑦_𝑛=𝑓(𝑥_𝑛) where 𝑥_𝑛∈𝐸 for each 𝑛∈𝐍. A second order ideal-ward compactness of 𝐸 implies that there is a subsequence 𝐳=(𝑧_𝑘)=(𝑥_𝑛𝑘) of (𝑥_𝑛) with 𝐼−lim_𝑘→∞Δ²𝑧_𝑘=0. Write (𝑡_𝑘)=(𝑓(𝑧_𝑘)). Since 𝑓 is second order ideal-ward continuous, so we have 𝐼−lim_𝑘→∞Δ²𝑓(𝑧_𝑘)=0. Thus we have obtained a subsequence (𝑡_𝑘) of the sequence (𝑓(𝑥_𝑛)) with 𝐼−lim_𝑘→∞Δ²𝑡_𝑘=0. Thus 𝑓(𝐸) is second order ideal-ward compact. This completes the proof of the theorem.

Corollary 12. A second order ideal-ward continuous image of any compact subset of 𝐑 is compact.

Proof. The proof of this theorem follows from the preceding theorem.

Theorem 13. If (𝑓_𝑛) is a sequence of second order ideal-ward continuous functions defined on a subset 𝐸 of 𝐑 and (𝑓_𝑛) is uniformly convergent to a function 𝑓, then 𝑓 is second order ideal-ward continuous on 𝐸.

Proof. Let 𝜀>0 and (𝑥_𝑛) be a sequence of points in 𝐸 such that 𝐼−lim_𝑛→∞Δ²𝑥_𝑛=0. By the uniform convergence of (𝑓_𝑛) there exists a positive integer 𝑁 such that |𝑓_𝑛(𝑥)−𝑓(𝑥)|<𝜀/8 for all 𝑥∈𝐸 whenever 𝑛≥𝑁. Since 𝑓_𝑁 is second order ideal-ward continuous on 𝐸, then we have 𝑛∈𝐍∶𝑓_𝑁𝑥_𝑛+2−2𝑓_𝑁𝑥_𝑛+1+𝑓_𝑁𝑥_𝑛≥𝜀8∈𝐼. (5) But 𝑛∈𝐍∶𝑓𝑥_𝑛+2−2𝑓𝑥_𝑛+1+𝑓𝑥_𝑛⊆𝑓𝑥≥𝜀𝑛∈𝐍∶_𝑛+2−𝑓_𝑁𝑥_𝑛+2≥𝜀8∪𝑛∈𝐍∶𝑓_𝑁𝑥_𝑛+1−𝑓𝑥_𝑛+1≥𝜀4∪𝑛∈𝐍∶𝑓_𝑁𝑥_𝑛−𝑓𝑥_𝑛≥𝜀8∪𝑛∈𝐍∶𝑓_𝑁𝑥_𝑛+2−2𝑓_𝑁𝑥_𝑛+1+𝑓_𝑁𝑥_𝑛≥𝜀8. (6) Since 𝐼 is an admissible ideal, the right-hand side of relation (6) belongs to 𝐼, and we have 𝑛∈𝐍∶𝑓𝑥_𝑛+2−2𝑓𝑥_𝑛+1+𝑓𝑥_𝑛≥𝜀∈𝐼. (7) This completes the proof of the theorem.

Theorem 14. The set of all second order ideal-ward continuous functions on a subset 𝐸 of 𝐑 is a closed subset of the set of all continuous functions on 𝐸; that is, Δ²𝑖𝑤𝑐(𝐸)=Δ²𝑖𝑤𝑐(𝐸) where Δ²𝑖𝑤𝑐(𝐸) is the set of all second order ideal-ward continuous functions on 𝐸,Δ²𝑖𝑤𝑐(𝐸) denotes the set of all cluster points of Δ²𝑖𝑤𝑐(𝐸).

Proof. Let 𝑓 be an element in Δ²𝑖𝑤𝑐(𝐸). Then there exists sequence (𝑓_𝑛) of points in Δ²𝑖𝑤𝑐(𝐸) such that lim_𝑛→∞𝑓_𝑛=𝑓. To show that 𝑓 is second order ideal-ward continuous, consider a sequence (𝑥_𝑛) of points in 𝐸 such that 𝐼−lim_𝑛→∞Δ²𝑥_𝑛=0. Since (𝑓_𝑛) converges to 𝑓, there exists a positive integer 𝑁 such that, for all 𝑥∈𝐸 and for all 𝑛≥𝑁, |𝑓_𝑛(𝑥)−𝑓(𝑥)|<𝜀/8. Since 𝑓_𝑁 is second order ideal-ward continuous on 𝐸 we have 𝑛∈𝐍∶𝑓_𝑁𝑥_𝑛+2−2𝑓_𝑁𝑥_𝑛+1+𝑓_𝑁𝑥_𝑛≥𝜀4∈𝐼. (8) But 𝑛∈𝐍∶𝑓𝑥_𝑛+2−2𝑓𝑥_𝑛+1+𝑓𝑥_𝑛⊆𝑓𝑥≥𝜀𝑛∈𝐍∶_𝑛+2−𝑓_𝑁𝑥_𝑛+2≥𝜀8∪𝑛∈𝐍∶𝑓_𝑁𝑥_𝑛+1−𝑓𝑥_𝑛+1≥𝜀4∪𝑛∈𝐍∶𝑓𝑥_𝑛−𝑓_𝑁𝑥_𝑛≥𝜀8∪𝑛∈𝐍∶𝑓_𝑁𝑥_𝑛+2−2𝑓_𝑁𝑥_𝑛+1+𝑓𝑥_𝑛≥𝜀8. (9) Since 𝐼 is an admissible ideal, the right-hand side of the relation (9) belongs to 𝐼, and we have 𝑛∈𝐍∶𝑓𝑥_𝑛+2−2𝑓𝑥_𝑛+1+𝑓𝑥_𝑛≥𝜀∈𝐼. (10) This completes the proof of the theorem.

Corollary 15. The set of all second order ideal-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.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.