# Strongly Lacunary Ward Continuity in 2-Normed Spaces

Authors: Hüseyin Çakalli; Sibel Ersan
Journal: The Scientific World Journal
 (2014)

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## Abstract

A functionf defined on a subset E of a 2-normed space X is strongly lacunary ward continuous if it preserves strongly lacunary quasi-Cauchy sequences of points in E; that is, (f(xk)) is a strongly lacunary quasi-Cauchy sequence whenever (xk) is strongly lacunary quasi-Cauchy. In this paper, not only strongly lacunary ward continuity, but also some other kinds
of continuities are investigated in 2-normed spaces.


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## Body

## 1. Introduction

Menger [1] introduced a notion called a generalized metric in 1928, and ten years later Vulich [2] defined a notion of a higher dimensional norm in linear spaces. Unfortunately, these studies had been neglected by many analysts for a long time. The concept of a 2-normed space was developed by Gähler in the middle of 1960s [3–5]. Since then, Mashadi [6], Gurdal [7], Mazaheri and Kazemi [8], and Sahiner [9] have studied this concept and obtained various results.Using the main idea in the definition of sequential continuity, many kinds of continuities were introduced and investigated, and not all but some of them are in [10–13]. The concept of Nθ-convergence was introduced in [14] and further studied in [15]. Strongly lacunary ward continuity of a real function was introduced by Cakalli in [11] as named Nθ-ward continuity and further studied in [16].The aim of this paper is to investigate strongly lacunary ward continuity in 2-normed spaces and prove interesting theorems.


## 2. Preliminaries

Throughout this paper,N and R will denote the set of all positive integers and the set of all real numbers, respectively. Now we recall the definition of a two-normed space. Let X be a real linear space with dim⁡X>1 and ∥·,·∥:X×X→R a function. Then (X,∥·,·∥) is called a linear 2-normed space if (i) ∥x,y∥=0⇔x and y are linearly dependent, (ii) ∥x,y∥=∥y,x∥, (iii) ∥αx,y∥=|α|∥x,y∥, and (iv) ∥x,y+z∥≤∥x,y∥+∥x,z∥ for α∈R and x,y,z∈X. The function ∥·,·∥ is called a 2-norm on X. Observe that in any 2-normed space (X,∥·,·∥) we have that ∥·,·∥ is nonnegative, ∥x-z,x-y∥=∥x-z,y-z∥, and ∥x,y+αx∥=∥x,y∥ for all x,y∈X, α∈R. Throughout this paper by X we will mean a 2-normed space with a two-norm ∥·,·∥. As an example of a 2-normed space we may take X=R2 being equipped with the 2-norm
(1)∥x1,x2∥=abs(|x11x12x21x22|),
where x1=(x11,x12), x2=(x21,x22).A sequence(xn) of points in X is said to converge to a point L of X in the 2-normed space X if lim⁡n→∞∥xn-L,z∥=0 for every z∈X. This is denoted by lim⁡n→∞∥xn,z∥=∥L,z∥. A sequence (xn) of points in X is said to be a Cauchy sequence with respect to the 2-norm if lim⁡n,m→∞∥xn-xm,z∥=0 for every z∈X. A sequence of functions (fn) is said to be uniformly convergent to a function f on a subset E of X if for each ε>0, an integer N can be found such that ∥fn(x)-f(x),z∥<ε for n≥N and for all x,z∈X [17]. A lacunary sequence θ=(kr) is an increasing sequence of positive integers such that k0=0 and hr=kr-kr-1→∞ as r→∞. The intervals determined by θ will be denoted by Ir=(kr-1,kr], and the ratio kr/kr-1 will be abbreviated by qr. A sequence (xk) of points in X is called strongly lacunary convergent, or Nθ-convergent to an element L of X if
(2)lim⁡r→∞1hr∑k∈Ir‍∥xk-L,z∥=0,
for every z∈X and it is denoted by Nθ-lim⁡k→∞∥xk,z∥=∥L,z∥ for every z∈X [9]. Throughout the paper we will use the word “Nθ” instead of “strongly lacunary” and assume that liminf⁡rqr>1.


## 3. Results

The set of seminorms{pz} defined by pz(x)=∥x,z∥, ∀x, for each z∈X, forms a locally convex topological vector space, and the topology formed by this family of seminorms gives the required topology on X. Since dim⁡X≥2, for each x∈X there exists a y∈X such that x and y are linearly independent, and hence by (i), py(x)=∥x,y∥≠0. Thus the locally convex topological vector space induced by the set {pz:z∈X} of seminorms is Hausdorff so X is a Hausdorff space [18]. In this section, we investigate the concepts of a strongly lacunary quasi-Cauchy sequence and strongly lacunary ward continuity of a function on a 2-normed space.Definition 1.
A subsetE of X is called Nθ-sequentially compact if any sequence of points in E has an Nθ-convergent sequence with an Nθ-limit in E.We note that union of twoNθ-sequentially compact subsets of X is Nθ-sequentially compact, intersection of any Nθ-sequentially compact subsets is Nθ-sequentially compact, any compact subset of X is Nθ-sequentially compact, and any finite subset of X is Nθ-sequentially compact. Sum of Nθ-sequentially compact subsets of X is Nθ-sequentially compact where sum of two subsets A and B is defined as A+B={a+b:a∈A,b∈B}.Definition 2.
A functionf defined on a subset E of X is said to be strongly lacunary sequentially continuous or Nθ-sequentially continuous at a point x0 of E if (f(xk)) is an Nθ-convergent sequence to f(x0) whenever (xk) is an Nθ-convergent to x0 sequence of points in E. If f is strongly lacunary sequentially continuous at every point of E, then it is said to be strongly lacunary sequentially continuous on E.If a functionf defined on a subset E of X is lacunary statistically sequentially continuous at a point x0, then (f(xk)) is an Nθ-convergent sequence with Nθ-lim⁡∥f(xk),z∥=∥f(x0),z∥ for every z∈X whenever (xk) is an Nθ-convergent sequence with Nθ-lim⁡∥xk,z∥=∥x0,z∥ for every z∈X. We see that a function f defined on a subset E of X is strongly lacunary sequentially continuous if and only if it preserves strongly lacunary convergent sequences without stating limit of the sequence. We note that sum of two Nθ-sequentially continuous functions at a point x0 of X is Nθ-sequentially continuous at x0, and composite of two Nθ-sequentially continuous functions at a point x0 of X is Nθ-sequentially continuous at x0. In the classical case, that is in the single normed case, it is known that uniform limit of sequentially continuous function is sequentially continuous; now we see that it is also true that not only uniform limit of sequentially continuous function is sequentially continuous, but also uniform limit of Nθ-sequentially continuous function is Nθ-sequentially continuous in 2-normed spaces. Now we give the latter in the following.Theorem 3.
Uniform limit ofNθ-sequentially continuous functions is Nθ-sequentially continuous.Proof.
Let(fk) be a uniformly convergent sequence of each term defined on a subset E of X with uniform limit f and let (xk) be any Nθ-convergent sequence of points in E with Nθ-lim⁡∥xk,z∥=∥x,z∥ for every z∈X. Take any ε>0. By uniform convergence of (fk), there exists an n1∈N such that ∥f(x)-fk(x),z∥<ε/3 for k≥n1 and every x∈E and z∈X. Hence,
(3)1hr∑k∈Ir‍∥f(x)-fk(x),z∥<1hr(kr-kr-1)ε3=ε3,
for r≥n1 and every x∈E and z∈X. As fn1 is Nθ-sequentially continuous on E, there exists an n2∈N such that, for r≥n2,
(4)1hr∑k∈Ir‍∥fn1(x)-fn1(xk),z∥<ε3,
for every z∈X. Now write n0=max⁡{n1,n2}. Thus for r≥n0 we have
(5)1hr∑k∈Ir‍∥f(x)-f(xk),z∥≤1hr∑k∈Ir‍∥vk(x),z∥+1hr∑k∈Ir‍∥fn1(x)-fn1(xk),z∥+1hr∑k∈Ir‍∥fn1(xk)-f(xk),z∥<ε3+ε3+ε3=ε,
where vk(x)=f(x)-fn1(x) for every k∈N. Hence
(6)lim⁡r→∞1hr∑k∈Ir‍∥f(x)-f(xk),z∥=0.
This completes the proof of the theorem.Theorem 4.
N
θ-sequentially continuous image of any Nθ-sequentially compact subset of X is Nθ-sequentially compact.Proof.
Assume thatf is an Nθ-sequentially continuous function on a subset E of X and A is an Nθ-sequentially compact subset of E. Let (f(xn)) be any sequence of points in f(A) where xn∈A for each positive integer n. Nθ-sequentially compactness of A implies that there is a subsequence (γk)=(xnk) of (xn) with Nθ-lim⁡k→∞∥γk,z∥=∥l,z∥ for every z∈E. Write (tk)=(f(γk)). As f is Nθ-sequentially continuous, (f(γk)) is Nθ-sequentially convergent which is a subsequence of the sequence (f(xn)) with Nθ-lim⁡k→∞∥tk,z∥=∥l,z∥ for ∀z∈E. This completes the proof of the theorem.The concept of a quasi-Cauchy sequence in a 2-normed space was studied in [19]. Now we give the following definition of an Nθ-quasi-Cauchy sequence.Definition 5.
A sequence(xn) of points in a subset E of X is called Nθ-quasi-Cauchy if (Δxk) is Nθ-convergent to 0, that is,
(7)Nθ-lim⁡k→∞∥Δxk,z∥=0,
where Δxk=xk+1-xk.We note that any quasi-Cauchy sequence isNθ-quasi-Cauchy, so any convergent sequence is Nθ-quasi-Cauchy in X. Any Cauchy sequence is Nθ-quasi-Cauchy, but the converse is not always true. However the converse is not always true, that is, there are Nθ-quasi-Cauchy sequences which are not convergent. Sum of two Nθ-quasi-Cauchy sequences is Nθ-quasi-Cauchy. Subsequence of an Nθ-quasi-Cauchy sequence needs not be Nθ-quasi-Cauchy. Now we give the definition of Nθ-ward compactness of a subset of X.Definition 6.
A subsetE of X is called Nθ-ward compact if any sequence of points in E has an Nθ-quasi-Cauchy subsequence.Union of twoNθ-ward compact subset of X is Nθ-ward compact, intersection of any Nθ-ward compact subsets is Nθ-ward compact, sum of two Nθ-ward compact subset of X is Nθ-ward compact, and any finite subset of X is Nθ-ward compact.Definition 7.
A function defined on a subsetE of X is called Nθ-ward continuous if it preserves Nθ-quasi-Cauchy sequences, that is, (f(xk)) is an Nθ-quasi-Cauchy sequence whenever (xk) is.We note that a composite of twoNθ-ward continuous functions is Nθ-ward continuous, and sum of two Nθ-ward continuous functions is Nθ-ward continuous.Theorem 8.
N
θ-ward continuous image of any Nθ-ward compact subset of X is Nθ-ward compact.Proof.
Assume thatf is an Nθ-ward continuous function on a subset E of X and E is a Nθ-ward compact subset of A. Let (f(xn)) be any sequence of points in f(E) where xn∈E for each positive integer n. Nθ-ward compactness of E implies that there is a subsequence (γk)=(xnk) of (xn) with Nθ-lim⁡k→∞∥Δγk,z∥=0 for every z∈E. Write (tk)=(f(γk)). As f is Nθ-ward continuous, (f(γk)) is Nθ-quasi-Cauchy which is a subsequence of the sequence (f(xn)) with Nθ-lim⁡k→∞∥Δtk,z∥=0 for ∀z∈E. This completes the proof of the theorem.Corollary 9.
N
θ-ward continuous image of any compact subset of X is Nθ-ward compact.Proof.
The proof follows from the preceding theorem.A functionf defined on a subset E of X is sequentially continuous at x0, if for any sequence (xn) of points in E converging to x0, we have (f(xn)) converges to f(x0). f is sequentially continuous on E if it is sequentially continuous at every point of E (see [19] for the infinite dimensional case and [20] for the finite dimensional case).ConcerningNθ-quasi-Cauchy sequences, Nθ convergent sequences, and convergent sequences the problem arises to investigate the following types of continuity of functions on X. In the following Nθ, ΔNθ, and c will denote the set of Nθ-convergent sequence, the set of all Nθ-quasi-Cauchy sequences, and the set of convergent sequences of points in X, respectively:(1)
(xk)∈ΔNθ⇒(f(xk))∈ΔNθ,(2)
(xk)∈ΔNθ⇒(f(xk))∈c,(3)
(xk)∈c⇒(f(xk))∈c,(4)
(xk)∈c⇒(f(xk))∈ΔNθ,(5)
(xk)∈Nθ⇒(f(xk))∈Nθ,(6)
(xk)∈ΔNθ⇒(f(xn))is quasi-Cauchy,(7)
(xk)∈ΔNθ⇒(f(xn))∈Nθ.We see that (1) isNθ-ward continuity of f, (5) is Nθ-sequentially continuity of f, and (3) is the ordinary sequential continuity of f. It is easy to see that (2) implies (1), and (1) does not imply (2); and (1) implies (4), and (4) does not imply (1); (2) implies (3), and (3) does not imply (2). (6) implies (1), but (1) does not imply (6) and (7) implies (1), but (1) does not imply (7). Now we give that the implication (1) implies (5), that is, any Nθ-ward continuous function is Nθ-sequentially continuous.Theorem 10.
Iff is Nθ-ward continuous function on a subset E of X, then it is Nθ-sequentially continuous on E.Proof.
Assume thatf is an Nθ-ward continuous function on a subset E of X. Let (xn) be any Nθ-convergent sequence in A with Nθ-lim⁡k→∞∥xk,z∥=∥x0,z∥ for all z∈X. Then the sequence x=(xn) defined by
(8)xn={xk,ifn=2k-1forapositiveintegerkx0,ifniseven
is also Nθ-convergent to x0. Hence it is Nθ-quasi-Cauchy sequence. As f is Nθ-ward continuous on E, the transformed sequence (yn) obtained by
(9)yn={f(xk),ifn=2k-1forapositiveintegerkf(x0),ifniseven
is also Nθ-quasi-Cauchy. Thus,
(10)lim⁡r→∞1hr∑k∈Ir‍∥f(xk)-f(x0),z∥=0
for all z∈X. It follows from this that the sequence (f(xk)) is Nθ convergent to (f(x0)).The converse of this theorem is not valid in general, a counterexample can be easily constructed via the functionf(x,y)=(x2,y2) on the 2-normed space R2 with the usual 2-norm.Theorem 11.
If(fn) is a sequence of Nθ-ward continuous functions on a subset E of X and (fn) is uniformly convergent to a function f, then f is Nθ-ward continuous on E.Proof.
Let(xk) be any Nθ-quasi-Cauchy sequence of points in E, and let ε be any positive real number. By uniform convergence of (fk), there exists an n1∈N such that ∥f(x)-fk(x),z∥<ε/3 for k≥n1 and every x∈E and z∈X. Hence,
(11)1hr∑k∈Ir‍∥f(x)-fk(x),z∥<1hr(kr-kr-1)ε3=ε3,
for r≥n1 and every x∈E and z∈X. As fn1 is Nθ-ward continuous on E, there exists an n2∈N such that, for r≥n2,
(12)1hr∑k∈Ir‍∥fn1(xk+1)-fn1(xk),z∥<ε3,
for every z∈X. Now write n0=max⁡{n1,n2}. Thus, for r≥n0, we have
(13)1hr∑k∈Ir‍∥f(xk+1)-f(xk),z∥≤1hr∑k∈Ir‍f∥(xk+1)-fn1(xk+1),z∥+1hr∑k∈Ir‍∥fn1(xk+1)-fn1(xk),z∥+1hr∑k∈Ir‍∥fn1(xk)-f(xk),z∥<ε3+ε3+ε3=ε.
Hence,
(14)lim⁡r→∞1hr∑k∈Ir‍∥f(xk+1)-f(xk),z∥=0.
Thus f preserves Nθ-quasi-Cauchy sequences. This completes the proof of the theorem.


## 4. Conclusion

In this paper, we investigate strongly lacunary continuity and some other kinds of continuities defined via a lacunary sequence and we prove interesting theorems related to these kinds of continuities. The results in this paper are extensively deeper than existing related results in the literature. We note that the notion of a strongly lacunary quasi-Cauchy sequence coincides with the notion of a strongly lacunary convergent sequence in a complete non-Archimedean 2-normed space, and so the set of strongly lacunary ward continuous functions coincides with the set of strongly lacunary sequentially continuous functions in a complete non-Archimedean 2-normed space (see [21] for the related concepts in an ultrametric field). For a further study, we suggest to investigate strongly lacunary quasi-Cauchy sequences of points for the fuzzy functions in a 2-normed fuzzy spaces. 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 [22, 23] for the definitions and related concept in fuzzy setting). We note that the study in this paper can be generalized to n-normed spaces as another further study.



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