Strongly Lacunary Ward Continuity in 2-Normed Spaces

A function f 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(x k)) is a strongly lacunary quasi-Cauchy sequence whenever (x k) 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.


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][4][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][11][12][13]. The concept of -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 -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.
for every ∈ and it is denoted by − lim → ∞ ‖ , ‖ = ‖ , ‖ for every ∈ [9]. Throughout the paper we will use the word " " instead of "strongly lacunary" and assume that lim inf > 1.

Results
The set of seminorms { } defined by ( ) = ‖ , ‖, ∀ , for each ∈ , forms a locally convex topological vector space, and the topology formed by this family of seminorms gives the required topology on . Since dim ≥ 2, for each ∈ there exists a ∈ such that and are linearly independent, and hence by ( ), ( ) = ‖ , ‖ ̸ = 0. Thus the locally convex topological vector space induced by the set { : ∈ } of seminorms is Hausdorff so 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 subset of is called -sequentially compact if any sequence of points in has an
-convergent sequence with an -limit in .
We note that union of two -sequentially compact subsets of is -sequentially compact, intersection of any -sequentially compact subsets is -sequentially compact, any compact subset of is -sequentially compact, and any finite subset of is -sequentially compact. Sum of sequentially compact subsets of is -sequentially compact where sum of two subsets and is defined as + = { + : ∈ , ∈ }.

Definition 2.
A function defined on a subset of is said to be strongly lacunary sequentially continuous orsequentially continuous at a point 0 of if ( ( )) is an -convergent sequence to ( 0 ) whenever ( ) is an convergent to 0 sequence of points in . If is strongly lacunary sequentially continuous at every point of , then it is said to be strongly lacunary sequentially continuous on .
If a function defined on a subset of is lacunary statistically sequentially continuous at a point 0 , then ( ( )) is an -convergent sequence with − lim ‖ ( ), ‖ = ‖ ( 0 ), ‖ for every ∈ whenever ( ) is an -convergent sequence with − lim ‖ , ‖ = ‖ 0 , ‖ for every ∈ . We see that a function defined on a subset of 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 -sequentially continuous functions at a point 0 of is -sequentially continuous at 0 , and composite of two -sequentially continuous functions at a point 0 of is -sequentially continuous at 0 . 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 -sequentially continuous function issequentially continuous in 2-normed spaces. Now we give the latter in the following.

Theorem 3. Uniform limit of -sequentially continuous functions is -sequentially continuous.
Proof. Let ( ) be a uniformly convergent sequence of each term defined on a subset of with uniform limit and let ( ) be any -convergent sequence of points in with − lim ‖ , ‖ = ‖ , ‖ for every ∈ . Take any > 0. By uniform convergence of ( ), there exists an 1 ∈ N such that ‖ ( ) − ( ), ‖ < /3 for ≥ 1 and every ∈ and ∈ . Hence, for ≥ 1 and every ∈ and ∈ . As 1 is -sequentially continuous on , there exists an 2 ∈ N such that, for for every ∈ . Now write 0 = max{ 1 , 2 }. Thus for ≥ 0 we have where V ( ) = ( ) − 1 ( ) for every ∈ N. Hence This completes the proof of the theorem. Proof. Assume that is an -sequentially continuous function on a subset of and is an -sequentially compact subset of . Let ( ( )) be any sequence of points in ( ) where ∈ for each positive integer .
The Scientific World Journal 3 The concept of a quasi-Cauchy sequence in a 2-normed space was studied in [19]. Now we give the following definition of an -quasi-Cauchy sequence.

Definition 5. A sequence ( ) of points in a subset of is called -quasi-Cauchy if
where Δ = +1 − .
We note that any quasi-Cauchy sequence is -quasi-Cauchy, so any convergent sequence is -quasi-Cauchy in . Any Cauchy sequence is -quasi-Cauchy, but the converse is not always true. However the converse is not always true, that is, there are -quasi-Cauchy sequences which are not convergent. Sum of two -quasi-Cauchy sequences is -quasi-Cauchy. Subsequence of an -quasi-Cauchy sequence needs not be -quasi-Cauchy. Now we give the definition of -ward compactness of a subset of .

Definition 6.
A subset of is called -ward compact if any sequence of points in has an -quasi-Cauchy subsequence.
Union of two -ward compact subset of is -ward compact, intersection of any -ward compact subsets isward compact, sum of two -ward compact subset of is -ward compact, and any finite subset of is -ward compact.

Definition 7. A function defined on a subset
of is called -ward continuous if it preserves -quasi-Cauchy sequences, that is, ( ( )) is an -quasi-Cauchy sequence whenever ( ) is.
We note that a composite of two -ward continuous functions is -ward continuous, and sum of two -ward continuous functions is -ward continuous.

Theorem 8.
-ward continuous image of any -ward compact subset of is -ward compact.
Proof. Assume that is an -ward continuous function on a subset of and is a -ward compact subset of . Let ( ( )) be any sequence of points in ( ) where ∈ for each positive integer .

Corollary 9.
-ward continuous image of any compact subset of is -ward compact.
Proof. The proof follows from the preceding theorem.
A function defined on a subset of is sequentially continuous at 0 , if for any sequence ( ) of points in converging to 0 , we have ( ( )) converges to ( 0 ). is sequentially continuous on if it is sequentially continuous at every point of (see [19] for the infinite dimensional case and [20] for the finite dimensional case).
is also -convergent to 0 . Hence it is -quasi-Cauchy sequence. As is -ward continuous on , the transformed sequence ( ) obtained by is also -quasi-Cauchy. Thus, for all ∈ . It follows from this that the sequence ( ( )) is convergent to ( ( 0 )).
The converse of this theorem is not valid in general, a counterexample can be easily constructed via the function ( , ) = ( 2 , 2 ) on the 2-normed space R 2 with the usual 2-norm. 4 The Scientific World Journal

Theorem 11. If ( ) is a sequence of -ward continuous functions on a subset of and ( ) is uniformly convergent to a function , then is -ward continuous on .
Proof. Let ( ) be any -quasi-Cauchy sequence of points in , and let be any positive real number. By uniform convergence of ( ), there exists an 1 ∈ N such that ‖ ( ) − ( ), ‖ < /3 for ≥ 1 and every ∈ and ∈ . Hence, for ≥ 1 and every ∈ and ∈ . As 1 is -ward continuous on , there exists an 2 ∈ N such that, for ≥ 2 , for every ∈ . Now write 0 = max{ 1 , 2 }. Thus, for ≥ 0 , we have Hence, Thus preserves -quasi-Cauchy sequences. This completes the proof of the theorem.

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 tonormed spaces as another further study.