δ-QUASI-SLOWLY OSCILLATING SEQUENCES IN LOCALLY NORMAL RIESZ SPACES

In this paper, we introduce the notion of δ-quasi-slowly oscillating sequences, study on δ-quasi-slowly oscillating compactness and δ-quasi-slowly oscillating continuous functions in locally normal Riesz space.


Introduction
Using the idea of continuity of a real function and the idea of compactness in terms of sequences, many kinds of continuities were introduced and investigated, not all but some of them we recall in the following: forward continuity [5], slowly oscillating continuity [8,12,13,14,34], statistical ward continuity [6], δ-ward continuity [10], ideal ward continuty [4,24], ideally slowly oscillating continuity [25]. 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 (x n ) of points in R is called quasi-Cauchy if (∆x n ) is a null sequence where ∆x n = x n+1 − x n . In [3] Burton and Coleman named these sequences as "quasi-Cauchy" and in [7] Ç akallı used the term "ward convergent to 0" sequences. In terms of quasi-Cauchy we restate the definitions of ward compactness and ward continuity as follows: a function f is ward continuous if it preserves quasi-Cauchy sequences, i.e. (f (x n )) is quasi-Cauchy whenever (x n ) is, and a subset E of R is ward compact if any sequence x = (x n ) of points in E has a quasi-Cauchy subsequence z = (z k ) = (x n k ) of the sequence x.
The idea of statistical convergence first appeared, under the name of almost convergence, in the first edition Zygmund [36] of celebrated monograph [37] of Zygmund. Later, this idea was introduced by Fast [18] and Steinhaus [33] and many authors. Actually, this concept is based on the natural density of subsets of N of positive integers. A subset E of N is said to have natural or asymptotic where E(n) = {k ≤ n : k ∈ E} and |E| denotes the cardinality of the set E. The notion of lacunary statistical convergence was introduced by Fridy and Orhan [21] and has been investgated for the real case in [22].
Kostyrko et al. [27] introduced the notion of ideal convergence which is a generalization of statistical convergence (see [18,20]) based on the structure of the admissible ideal I of subsets of natural numbers N. A family of sets I ⊂ P (N) (the power sets of N) is said to be an ideal on N if and only if ϕ ∈ I for each A, B ∈ I, we have A ∪ B ∈ I for each A ∈ I and each B ⊂ A, we have B ∈ I. A non-empty family of sets F ⊂ P (N) is said to be a filter on N if and only if ϕ / A Riesz space is an ordered vector space which is a lattice at the same time. It was first introduced by F. Riesz [31] in 1928. Riesz spaces have many applications in measure theory, operator theory and optimization. They have also some applications in economics (see [2]), and we refer to [1,23,26,28,29,32,35] for more details.

Preliminaries and Notations
It is known that a sequence x = (x n ) of points in R, the set of real numbers, is slowly oscillating, denoted by x ∈ so, if where [λn] denotes the integer part of λn. This is equivalent to the following if (x m − x n ) → 0 whenever 1 ≤ m n → 1 as m, n → ∞. Using ε > 0 and δ this is also equivalent to the case when for any given ε > 0, there exists δ = δ(ε) > 0 and N = N (ε) such that |x m −x n | < ε if n ≥ N (ε) and n ≤ m ≤ (1+δ)n (see [8,15,17]).
A function defined on a subset E of R is called slowly oscillating continuous if it preserves slowly oscillating sequences, i.e. (f (x n )) is slowly oscillating whenever (x n ) is.
Connor and Grosse-Erdman [16] gave sequential definitions of continuity for real functions calling G-continuity instead of A-continuity and their results covers the earlier works related to A-continuity where a method of sequential convergence, or briefly a method, is a linear function G defined on a linear subspace of s, space of all sequences, denoted by c G , into R. A sequence x = (x n ) is said to be Gconvergent to ℓ if x ∈ c G and G(x) = ℓ. In particular, lim denotes the limit function lim x = lim n x n on the linear space c.
A method G is called regular if every convergent sequence x = (x n ) is Gconvergent with G(x) = lim x. A method is called subsequential if whenever x is G-convergent with G(x) = ℓ, then there is a subsequence (x n k ) of x with lim k x n k = ℓ (for details see [9]).
Let X be a real vector space and ≤ be a partial order on this space. Then X is said to be an ordered vector space if it satisfies the following properties: (i) if x, y ∈ X and y ≤ x, then y + z ≤ x + z for each z ∈ X.
(ii) if x, y ∈ X and y ≤ x, then ay ≤ ax for each a ≥ 0.
If, in addition, X is a lattice with respect to the partial ordered, then X is said to be a Riesz space (or a vector lattice)(see [35]).
For an element x of a Riesz space X, the positive part of x is defined by x + = x ∨ 0 = sup{x, 0}, the negative part of x by x − = −x ∨ 0 and the absolute value of x by |x| = x ∨ (−x), where 0 is the zero element of X.
A subset S of a Riesz space X is said to be normal if y ∈ S and |x| ≤ |y| implies x ∈ S.
A topological vector space (X, τ ) is a vector space X which has a topology (linear) τ, such that the algebraic operations of addition and scalar multiplication in X are continuous. Continuity of addition means that the function f : X × X → X defined by f (x, y) = x + y is continuous on X × X, and continuity of scalar multiplication means that the function f : Every linear topology τ on a vector space X has a base N for the neighborhoods of θ satisfying the following properties: (1) Each Y ∈ N is a balanced set, that is, ax ∈ Y holds for all x ∈ Y and for every a ∈ R with |a| ≤ 1. (2) Each Y ∈ N is an absorbing set , that is , for every x ∈ X, there exists a > 0 such that ax ∈ Y.
A linear topology τ on a Riesz space X is said to be locally normal if τ has a base at zero consisting of normal sets. A locally normal Riesz space (X, τ ) is a Riesz space equipped with a locally normal topology τ.
Recall that a first countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first countable if each point has a countable neighborhood basis (local base). That is, for each point x in X there exists a sequence V 1 , V 2 , · · · of open neighborhoods of x such that for any open neighborhood V of x there exists an integer j with V j contained in V.
A sequence x = (x n ) of points in a locally normal Riesz space X is said to be statistically convergent (see [1] A sequence x = (x n ) in a locally normal Riesz space X is called lacunary statistically convergent to an element L in X (see [30]) if for every τ -neighborhood V of zero, is an increasing sequence of positive integers.
A sequence x = (x n ) of points in a locally normal Riesz space X is said to be ideally convergent to x 0 ∈ X if for every τ -neighborhood V of zero, the set {n ∈ N : In this case we write x n Iτ → ℓ i.e. I τ -lim x n = ℓ (for details see [23]).
Throughout the article, the symbol N nor we will denote any base at zero consisting of normal sets and satisfying the conditions (1), (2) and (3) in a locally normal topology. Also (X, τ ) a locally normal Riesz space (in short LNRS) and N and R will denote the set of all positive integers, and the set of all real numbers, respectively. We will use boldface letters x, y, z, ... for sequences x = (x n ), y = (y n ), z = (z n ), ... of points in X.

δ-quasi-slowly oscillating sequences in LNRS
In this section we introduce the concepts of δ-quasi-slowly oscillating continuity and δ-quasi-slowly oscillating compactness in LNRS and establish some intresting results related to these notions.
It is clear that Cauchy sequences are slowly oscillating not only the real case but also in the LNRS setting. It is easy to see that any slowly oscillating sequence of points in X is quasi-Cauchy and therefore Cauchy sequence is quasi-Cauchy. The converses are not always true. There are quasi-Cauchy sequences which are not Cauchy. There are quasi-Cauchy sequences which are not slowly oscillating. Any subsequence of Cauchy sequence is Cauchy. The analogous property fails for quasi-Cauchy sequences and slowly oscillating sequences as well. A sequence x = (x n ) of points in X is said to be slowly oscillating, denoted by x ∈ so(X) if (x n ) is a slowly oscillating sequences, i.e. for each τ -neighborhood V of zero, there exist δ = δ(V ) > 0 and m = m(V ) such that Now we introduce the notion of δ-quasi-slowly oscillating sequences in LNRS.

Theorem 3.2. If a sequence is quasi-slowly oscillating then it is a δ-quasi-slowly oscillating.
Proof. Let (x n ) be a quasi-slowly oscillating sequence.
Since (x n ) is a quasi-slowly oscillating sequence, there exist δ = δ(W ) > 0 and a positive integer n 1 = n 1 (W ) such that ∆x k − ∆x n ∈ W for all n ≥ n 1 and n ≤ k ≤ (1 + δ)n.
Hence for all n ≥ n 1 (W ) and n ≤ k ≤ (1 + δ)n we have It implies that (x n ) is a δ-quasi-slowly oscillating sequence.
We introduce the notion of δ-quasi-slowly oscillating continuity in LNRS.

Definition 3.3.
A function f defined on a subset E of X is called δ-quasi-slowly oscillating continuous if it transforms δ-quasi-slowly oscillating sequences to δ-quasislowly oscillating sequences of points in E, that is, (f (x n )) is δ-quasi-slowly oscillating whenever (x n ) is δ-quasi-slowly oscillating sequences of points in E.
We note that sum of two δ-quasi-slowly oscillating continuous functions is δquasi-slowly oscillating continuous and the composite of two δ-quasi-slowly oscillating continuous functions is δ-quasi-slowly oscillating continuous in LNRS.
In connection with quasi-slowly oscillating sequences, δ-quasi-slowly oscillating sequences and convergent sequences the problem arises to investigate the following types of continuity of functions on X.

Theorem 3.4.
If f is δ-quasi-slowly oscillating continuous on a subset E of X then it is continuous on E in the ordinary sense.
Proof. Suppose that f is δ-quasi-slowly oscillating continuous on E and let (x n ) be any convergent sequence of points in E with lim n→∞ x n = x 0 . Then the sequence (y n ) = ( is also convergent to x 0 and hence (y n ) is δ-quasi-slowly oscillating. Since f is δ-quasi-slowly oscillating continuous, the sequence is also δ-quasi-slowly oscillating. Therefore (∆z n ) is a quasi-slowly oscillating. Since any slowly oscillating sequence is quasi-Cauchy, then the sequence (∆ 2 z n ) is slowly oscillating and so is quasi-Cauchy. Now it follows that if for each τ -neighborhood V of zero, there exists m = m(V ) such that This completes the proof of theorem. Proof. Let f be uniformly continuous function and x = (x n ) be any δ-quasi-slowly oscillating sequence in E. Let W be a τ -neighborhood of zero. Since f is uniformly continuous on E, then there exists a τ -neighborhood V of zero such that f ( It follows that (f (x n )) is δ-quasi-slowly oscilatting. This completes the proof of theorem.
Definition 3.10. [19] A sequence (x n ) of points in X is called Cesàro δ-quasislowly oscillating if (t n ) is δ-quasi-slowly oscillating, where t n = 1 n ∑ n k=1 x k , is the Cesàro means of the sequence (x n ). Also a function f defined on a subset E of X is called Cesàro δ-quasi-slowly oscillating continuous if it preserves Cesàro δ-quasi-slowly oscillating sequences of points in E.
By using the similar argument used in proof of Theorem 3.9, we immediately have the following result. Theorem 3.11. If f is a uniformly continuous on a subset E of X and (x n ) is a δ-quasi-slowly oscillating sequence in E, then (f (x n )) is Cesàro δ-quasi-slowly oscillating. Proof. Let (x n ) be any δ-quasi-slowly oscillating sequence of points in E. By uniform convergence of (f n ), if for each τ -neighborhood V of zero, there exists a Y ∈ N nor such that Y ⊆ V. Choose W ∈ N nor such that W + W + W + W + W + W + W ⊆ Y. Then there exists n 1 = n 1 (W ) such that

Definition 3.12. A sequence of functions (f n ) defined on a subset E of X is said to be uniformly convergent to a function f if for each τ -neighborhood V of zero, there exists an integer
for each x ∈ E and for all n ≥ n 1 (W ). Also since f n1 is δ-quasi-slowly oscillating continuous, there exist n 2 = n 2 > n 1 and δ = δ(W ) > 0 such that whenever n ≥ n 2 (W ) and n ≤ k ≤ (1 + δ)n. Therefore if n ≥ n 1 (W ) and n ≤ k ≤ (1 + δ)n we have It follows that (f (x n )) is a δ-quasi-slowly oscillating sequences of points in E which completes the proof of theorem.
Using the same techniques as in the Theorem 3.9, the following result can be obtained easily.  Proof. Let f be any element of δqso(E). Then there exists a sequence of points (f n ) in δqso(E) such that lim n→∞ f n = f. To show that f is δ-quasi-slowly oscillating sequence on E. Now let (x n ) be any δ-quasi-slowly oscillating sequence in E. Let V be an arbitrary τ -neighborhood of zero. There exists a Y ∈ N nor such that Y ⊆ V. Choose W ∈ N nor such that W + W + W + W + W + W + W + W + W ⊆ Y. Since (f n ) converges to f, there exists a positive integer n 1 such that for all x ∈ E and for all n ≥ n 1 , f n (x) − f (x) ∈ W. Also since f n1 is δ-quasi-slowly oscillating continuous, there exist an integer n 2 = n 2 > n 1 and δ > 0 such that whenever n ≥ n 2 and n ≤ k ≤ (1 + δ)n. Hence, for all n ≥ n 1 and n ≤ k ≤ (1 + δ)n we have Thus it implies that ∆ 2 f (x k ) − ∆ 2 f (x n ) ∈ V for all n ≥ n 1 and n ≤ k ≤ (1 + δ)n. Thus f is δ-quasi-slowly oscillating continuous function on E and this completes the proof of theorem.

Corollary 3.16. The set of all δ-quasi-slowly oscillating continuous functions defined on a subset E of X is a complete subspace of the space of all continuous functions on E.
An element x 0 in X is called an ideal limit point of a subset E of X if there is an E-valued sequence of points with ideal limit x 0 . It follows that the set of all ideal limit points of E is equal to the set of all limit points of E in the ordinary sense.
An element x 0 in X is called an ideal accumulation point of a subset E if it is an ideal limit point of the set E − {x 0 }. The set of all ideal accumulation points of E is equal to the set of all accumulation points of E in the ordinary sense.
A function f on X is said to have an ideally sequential limit at a point x 0 of X if the image sequence (f (x n )) is ideally convergent to x 0 for any ideally convergent sequence x = (x n ) with ideal limit x 0 and a function f is to be ideally sequentially continuous at a point x 0 of X if the sequence (f (x n )) is ideally convergent to f (x 0 ) for any ideally convergent sequence x = (x n ) with ideal limit x 0 (for details see [4]).

Lemma 3.17.
A function f on X has an ideally sequential limit at a point x 0 of X if and only if it has an ideal limit at a point x 0 of X in ordinary sense.
Proof. The proof follows from the fact that any ideally convergent sequence has a convergent subsequence (also see [4]).
Next we define the concept of δ-quasi-slowly oscillating compactness in LNRS.

Definition 3.18. A subset E of X is called δ-quasi-slowly oscillating compact if any sequence of points in E has a quasi-slowly oscillating subsequence.
We see that any compact subset of X is δ-quasi-slowly oscilatting compact, union of two δ-quasi-slowly oscillating compact subsets of X is δ-quasi-slowly oscillating compact. Any subset of δ-quasi-slowly oscillating compact set is also δ-quasi-slowly oscillating compact and so intersection of any δ-quasi-slowly oscillating compact subsets of X is δ-quasi-slowly oscillating compact.

Theorem 3.19.
A δ-quasi-slowly oscillating continuous image of a δ-quasi-slowly oscillating compact subset of X is δ-quasi-slowly oscillating compact.
Proof. Let f be a δ-quasi-slowly oscillating continuous function on X and E be a δ-quasi-slowly oscillating compact subset of X. Let y = (y n ) be a sequence of points in f (E). Then we can write y n = f (x n ) where (x n ) is sequence of points in E for each n ∈ N. Since E is δ-quasi-slowly oscillating compact, there is a δ-quasi-slowly oscillating subsequence z = (z k ) = (x n k ) of (x n ). Then, δ-quasi-slowly oscillating continuity of f implies that f (z k ) is a δ-quasi-slowly oscillating subsequence of f (x n ). Hence f (E) is δ-quasi-slowly oscillating compact.
Theorem 3.23. Let (X, τ 1 ) and (Y, τ 2 ) be LNR spaces. If a function f : X → Y is uniformly continuous, then f is ideally continuous.
Proof. Let f : X → Y be uniformly continuous and x n Iτ 1 → x 0 in X. Let θ 1 and θ 2 be denote the zeros in X and Y, respectively. Let W be an arbitrary τ 2 -neighborhood of θ 2 . Since f is uniformly continuous, there exists some τ 1 -neighborhood V of θ 1 such that x i.e. we have f (x n ) Iτ 2 → f (x 0 ), which shows that f is ideally continuous. Proof. The proof follows from the fact that any ideally convergent sequence has a convergent subsequence and from Lemma 3.17.
Theorem 3.25. Let f : X → X be any function and (x n ) be a sequence of points in X such that I τ − lim n→∞ x n = x 0 implies lim n→∞ f (x n ) = f (x 0 ), then it is a constant function.
Proof. For the proof of the theorem follows form Theorem 3 in [12].

Theorem 3.26. If a function is δ-quasi-slowly oscillating continuous on a subset E of X, then it is ideally sequentially continuous on E.
Proof. Let f be any δ-quasi-slowly oscillating continuous on E. By Theorem 3.4, we have f is continuous on E. Also from Theorem 3.24, we see that f is ideally sequentially continuous on E. This completes the proof. Proof. Let f be any δ-ward continuous function on E. It follows from Corollary 2 in [10] that f is continuous. By Theorem 3. 24 we obtain that f is ideally sequentially continuous on E. This completes the proof of the theorem.