Second Order Ideal-Ward Continuity

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→∞ Δ 2 𝑓(𝑥 𝑛 ) = 0 whenever 𝐼 − lim 𝑛→∞ Δ 2 𝑥 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 𝑛 𝑘 ) of the sequence x such that 𝐼 − lim 𝑘→∞ Δ 2 𝑧 𝑘 = 0 where Δ 2 𝑧 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.


Introduction
Let us start with basic definitions from the literature. Let ⊆ N, the set of all natural numbers, and = { ≤ : ∈ }. Then the natural density of is defined by ( ) = lim −1 | | 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 = ( ) is said to be statistically convergent to if for every > 0, the set := { ∈ N : | − | ≥ } has natural density zero; that is, for each > 0, In this case, we write − lim = or → ( ) and 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 Koc inac [6] investigated statistical exhaustiveness.
Let be a nonempty set; then a family of sets ⊂ ( ) (power sets of ) is called an ideal on if and only if (a) ∈ , (b) for each , ∈ , we have ∪ ∈ , (c) for each ∈ and each ⊂ , we have ∈ . Definition 1 (see [7]). A sequence = ( ) of points in R is said to be -convergent to the number ℓ if, for every > 0, the set { ∈ N : | − ℓ| ≥ } ∈ . One writes − lim = ℓ. One sees that a sequence = ( ) being -convergent implies that − lim → ∞ Δ = 0.
Burton and Coleman [14] introduced the concept of quasi-Cauchy sequences as a sequence ( ) of points of R is said to be a quasi-Cauchy sequence if (Δ ) is a null sequence where Δ = +1 − . Ç akallı and Hazarika [15] introduced the concept of ideal quasi-Cauchy sequences. Recall from [15] that a sequence ( ) of points of R is called ideal quasi-Cauchy if − lim → ∞ Δ = 0.
We say that a sequence = ( ) is ward convergent to a number ℓ if lim → ∞ Δ = ℓ where Δ = +1 − . Using the idea of continuity of a real function and the idea of compactness in terms of sequences, Ç akallı [16] introduced the concept of ward continuity in the sense that a function is ward continuous if it transforms ward convergent to 0 sequences to ward convergent to 0 sequences; that is, ( ( )) is ward convergent to 0 whenever ( ) is ward convergent to 0, and Ç akallı [17] introduced the concept of ward compactness in the sense that a subset of R is ward compact if any sequence = ( ) of points in has a subsequence = ( ) = ( ) of the sequence such that lim → ∞ Δ = 0 where Δ = +1 − . Throughout the paper , , , and Δ will denote the set of all convergent sequences, statistically convergent sequences, -convergent sequences, and the set of all -ward convergent to 0 sequences of points in R where a sequence Throughout the paper we assume is a nontrivial admissible ideal of N.

Second Order Ideal-Ward Continuity
We introduce the notion of second order ward convergent sequences as follows.
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 ( ).

Definition 3.
A sequence = ( ) is said to be second order ideal-ward convergent to a number ℓ if − lim → ∞ Δ 2 = ℓ where Δ 2 = +2 −2 +1 + . For the special case ℓ = 0, x is called second order ideal-ward convergent to 0. One denotes by Δ 2 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 is called ward continuous on if the sequence ( ( )) is ward convergent to 0 whenever = ( ) is a ward convergent to 0 sequence of terms in .

Theorem 6. If is second order ideal-ward continuous on a subset of R, then it is an ideal-ward continuous on .
Proof. Suppose that is a second order ideal-ward continuous function on a subset of R. Let ( ) be a sequence with − lim → ∞ Δ = 0. Then we have the sequence such that − lim → ∞ Δ 2 = 0. Since is second order idealward continuous, then we get the sequence 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 function ( ) = sin which is ideal-ward continuous but not second order ideal-ward continuous.

Theorem 8. If is second order ideal-ward continuous, then it is second order ward continuous.
Proof. The proof is easy, so omitted. Proof. Suppose that is a second order ideal-ward continuous function on a subset of R and is a second order ideal-ward compact subset of R. Let ( ) be a sequence of points in ( ). Write = ( ) where ∈ for each International Journal of Analysis 3 ∈ N. A second order ideal-ward compactness of implies that there is a subsequence z = ( ) = ( ) of ( ) with − lim → ∞ Δ 2 = 0. Write ( ) = ( ( )). Since is second order ideal-ward continuous, so we have − lim → ∞ Δ 2 ( ) = 0. Thus we have obtained a subsequence ( ) of the sequence ( ( )) with − lim → ∞ Δ 2 = 0. Thus ( ) 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.
Since is an admissible ideal, the right-hand side of relation (6) belongs to , and we have This completes the proof of the theorem. Proof. Let be an element in Δ 2 ( ). Then there exists sequence ( ) of points in Δ 2 ( ) such that lim → ∞ = .
To show that is second order ideal-ward continuous, consider a sequence ( ) of points in such that − lim → ∞ Δ 2 = 0. Since ( ) converges to , there exists a positive integer such that, for all ∈ and for all ≥ , | ( ) − ( )| < /8. Since is second order ideal-ward continuous on we have Since is an admissible ideal, the right-hand side of the relation (9) belongs to , and we have This completes the proof of the theorem. 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.