Variations on Statistical Quasi Cauchy Sequences

In this paper, we introduce a concept of statistically p-quasi-Cauchyness of a real sequence in the sense that a sequence (αk) is statistically p-quasi-Cauchy if limn→∞ 1 n |{k ≤ n : |αk+p − αk| ≥ ε}| = 0 for each ε > 0. A function f is called statistically p-ward continuous on a subset A of the set of real umbers R if it preserves statistically p-quasi-Cauchy sequences, i.e. the sequence f(x) = (f(αn)) is statistically p-quasi-Cauchy whenever α = (αn) is a statistically p-quasi-Cauchy sequence of points in A. It turns out that a real valued function f is uniformly continuous on a bounded subset A of R if there exists a positive integer p such that f preserves statistically p-quasi-Cauchy sequences of points in A.

The purpose of this paper is to introduce statistically p-quasi-Cauchy sequences, and prove interesting theorems.

Variations on statistical ward compactness
The concept of a Cauchy sequence involves far more than that the distance between successive terms is tending to 0 and statistically tending to zero, and more generally speaking, than that the distance between p-successive terms is statistically tending to zero, by p-successive terms we mean α k+p and α k .Nevertheless, sequences which satisfy this weaker property are interesting in their own right.
Before giving our main definition we recall basic concepts.A sequence (α n ) is called quasi Cauchy if lim n→∞ ∆α n = 0, where ∆α n = α n+1 − α n for each n ∈ N ( [4], [17]).The set of all bounded quasi-Cauchy sequences is a closed subspace of the space of all bounded sequences with respect to the norm defined for bounded sequences ( [50]).A sequence (α k ) of points in R is slowly oscillating if [16], [33], and [12]).Recently in [26] it was proved that a real valued function is uniformly continuous whenever it is p-ward continuous on a bounded subset of R. Now we introduce the concept of a statistically p-quasi-Cauchy sequence.We will denote the set of all statistically p-quasi-Cauchy sequences by ∆ s p for each p ∈ N. The sum of two statistically p-quasi-Cauchy sequences is statistically p-quasi-Cauchy, the product of a statistically p-quasi-Cauchy sequence and a constant real number is statistically p-quasi-Cauchy, so that the set of all statistically p-quasi-Cauchy sequences ∆ s p is a vector space.We note that a sequence is statistically quasi-Cauchy when p = 1, i.e. statistically 1-quasi-Cauchy sequences are statistical quasi-Cauchy sequences.It follows from the inclusion {k ≤ n : p } that any statistically quasi-Cauchy sequence is also statistically p-quasi-Cauchy, but the converse is not always true as it can be seen by considering the the sequence (α k ) defined by (α k ) = (0, 1, 0, 1, ..., 0, 1, ...) is 2-statistically quasi Cauchy which is not statistically quasi Cauchy.More examples can be seen in [50,Section 1.4].It is clear that any Cauchy sequence is in p=1 ∞ ∆ s p , so that each ∆ s p is a sequence space containing the space C of Cauchy sequences.It is also to be noted that C is a proper subset of ∆ s p for each p ∈ N.
Definition 2.2.A subset A of R is called statistically p-ward compact if any sequence of points in A has a statistically p-quasi-Cauchy subsequence.
We note that this definition of statistically p-ward compactness cannot be obtained by any summability matrix in the sense of [15] (see also [13], and [22]).
Since any statistically quasi-Cauchy sequence is statistically p-quasi-Cauchy we see that any ward compact subset of R is statistically p-ward compact for any p ∈ N. A finite subset of R is statistically p-ward compact, the union of finite number of statistically p-ward compact subsets of R is statistically p-ward compact, and the intersection of any family of statistically p-ward compact subsets of R is statistically p-ward compact.Furthermore any subset of a statistically p-ward compact set of R is statistically p-ward compact and any bounded subset of R is statistically p-ward compact.These observations above suggest to us the following.
Theorem 2.1.A subset A of R is bounded if and only if there exists a p ∈ N such that A is statistically p-ward compact.
Proof: The bounded subsets of R are statistically p-ward compact, since any bounded sequence of points in a bounded subset of R is bounded and any bounded sequence has a convergent subsequence which is statistically p-quasi-Cauchy for any p ∈ N. To prove the converse, suppose that A is not bounded.If it is unbounded above, pick an element α 1 of A greater than p.Then we can find an element α 2 of A such that α 2 > 2p + α 1 .Similarly, choose an element α 3 of A such that α 3 > 3p + α 2 .So we can construct a sequence (α j ) of numbers in A such that α j+1 > (j + 1)p + α j for each j ∈ N. Let (α j k ) be any subsequence of (α j ).Since {k ≤ n : This means that (α j k ) is not statistically p-quasi-Cauchy.Then the sequence (α j ) does not have any statistically p-quasi-Cauchy subsequence.If A is bounded above and unbounded below, then pick an element β 1 of A less than −p.Then we can find an element β 2 of A such that β 2 < −2p + β 1 .Similarly, choose an element β 3 of A such that β 3 < −3p + β 2 .Thus one can construct a sequence (β i ) of points in A such that β i+1 < −(i + 1)p + β i for each i ∈ N. Let (β i k ) be any subsequence of (β i ).Since {k ≤ n : (see [8]), and [19,Theorem 3]), ideal ward compactness ( [9, Theorem 8]), Abel ward compactness ( [7, Theorem 5]).
Corollary 2.2.A subset of R is statistically p ward compact if and only if it is statistically q ward compact for any p, q ∈ N. Proof: The proof follows from [11, Theorem 1.3 and Theorem 1.9], so is omitted.✷

Variations on statistical ward continuity
In this section, we investigate connections between uniformly continuous functions and statistically p-ward continuous functions.A function f : R −→ R is continuous if and only if it preserves statistically convergent sequences.Using this idea, we introduce statistical p-ward continuity.Definition 3.1.A function f is called statistically p-ward continuous on a subset A of R if it preserves statistically p-quasi-Cauchy sequences, i.e. the sequence We see that this definition of statistically p-ward continuity can not be obtained by any summability matrix A (see [13]).
We note that the sum of two statistically p-ward continuous functions is statistically p-ward continuous, and for any constant c ∈ R, cf is statistically p-ward continuous whenever f is a statistically p-ward continuous function, so that the set of all statistically p ward continuous functions is a vector space.The composite of two statistically p-ward continuous functions is statistically p-ward continuous, but the product of two statistically p-ward continuous functions need not be statistically p-ward continuous as it can be seen by considering product of the statistically p-ward continuous function f (x) = x with itself.If f is a statistically p-ward continuous function, then |f | is also statistically p-ward continuous since Proof: Suppose that f is not uniformly continuous on A so that there exist an ǫ 0 > 0 and sequences (α n ) and ( Since A is statistically p-ward compact, there is a subsequence (α n k ) of (α n ) that is statistically p-quasi-Cauchy.On the other hand, there is a subsequence (β n k j ) of (β n k ) that is statistically p-quasi-Cauchy as well.It is clear that the corresponding sequence (a n k j ) is also statistically p-quasi-Cauchy, since {j ≤ n : } for every n ∈ N, and for every ε > 0. Now the sequence is statistically p-quasi-Cauchy while the sequence is not statistically p-quasi-Cauchy where same term repeats p-times.Hence this establishes a contradiction, so completes the proof of the theorem.✷ Corollary 3.8.If a function defined on a bounded subset of R is statistically pward continuous, then it is uniformly continuous.
We note that when the domain of a function is restricted to a bounded subset of R, statistically p-ward continuity implies not only ward continuity, but also slowly oscillating continuity.

Conclusion
In this paper, we introduce statistically p-quasi Cauchy sequences, and investigate conditions for a statistically p ward continuous real function to be uniformly continuous, and prove some other results related to these kinds of continuities and some other kinds of continuities.It turns out that statistically p-ward continuity implies uniform continuity on a bounded subset of R. The results in this paper not only involves the related results in [21] as a special case for p = 1, but also some interesting results which are also new for the special case p = 1.The statistically p-quasi Cauchy concept for p > 1 might find more interesting applications than statistical quasi Cauchy sequences to the cases when statistically quasi Cauchy does not apply.For a further study, we suggest to investigate statistically p-quasi-Cauchy sequences of soft points and statistically p-quasi-Cauchy sequences of fuzzy

Theorem 3 . 7 .
If a function is statistically p-ward continuous on a statistically p-ward compact subset of R, then it is uniformly continuous on A.
Corollary 2.3.A subset of R is statistically p ward compact if and only if it is both statistically upward half compact and statistically downward half compact.Proof: The proof follows from [27, Corollary 3.9], so is omitted.
✷Corollary 2.4.A subset of R is statistically p ward compact for a p ∈ N if and only if it is both lacunary statistically upward half compact and lacunary statistically downward half compact.