Strongly Lacunary Ward Continuity in 2-Normed Spaces

Cakalli, Huseyin; Ersan, Sibel

doi:10.1155/2014/479679

Cited by 17 publications

(12 citation statements)

References 17 publications

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“…For another further study, we suggest to investigate arithmetically convergent sequences in a abstract metric space (see [25], [33], [23], and [38]). Yet another further study, our suggestion is to investigate the theory in 2-normed spaces (see [19], [26] for the related concepts).…”

Section: Resultsmentioning

confidence: 99%

A variation on arithmetic continuity

Cakalli

2017

B Soc Paran Mat

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A sequence (x k ) of points in R, the set of real numbers, is called arithmetically convergent if for each ε > 0 there is an integer n such that for every integer m we have |xm − x| < ε, where k|n means that k divides n or n is a multiple of k, and the symbol < m, n > denotes the greatest common divisor of the integers m and n. We prove that a subset of R is bounded if and only if it is arithmetically compact, where a subset E of R is arithmetically compact if any sequence of point in E has an arithmetically convergent subsequence. It turns out that the set of arithmetically continuous functions on an arithmetically compact subset of R coincides with the set of uniformly continuous functions where a function f defined on a subset E of R is arithmetically continuous if it preserves arithmetically convergent sequences, i.e., (f (xn) is arithmetically convergent whenever (xn) is an arithmetic convergent sequence of points in E.

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Section: Resultsmentioning

confidence: 99%

A variation on arithmetic continuity

Cakalli

2017

B Soc Paran Mat

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“…A function defined on a subset E of X is called strongly lacunary ward continuous if it preserves N θ -quasi-Cauchy sequences, i.e. (f (x k )) is an N θ -quasi-Cauchy sequence whenever (x k ) is [30].…”

Section: Strongly Lacunary δ-Ward Continuitymentioning

confidence: 99%

“…Proof. Although the proof could be seen by using Theorem 10 in [30], we give a direct proof for completeness. Assume that f is N θ -δ ward continuous function on a subset E of X.…”

Section: Strongly Lacunary δ-Ward Continuitymentioning

confidence: 99%

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A variation on strongly lacunary delta ward continuity in 2-normed spaces

Ersan

2019

B Soc Paran Mat

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A sequence (x k ) of points in a subset E of a 2-normed space X is called strongly lacunary δ-quasi-Cauchy, or N θ -δ-quasi-Cauchy if (∆x k ) is N θconvergent to 0, that is limr→∞ 1 hr k∈Ir ||∆ 2 x k , z|| = 0 for every fixed z ∈ X. A function defined on a subset E of X is called strongly lacunary δ-ward continuous if it preserves N θ -δ-quasi-Cauchy sequences, i.e. (f (x k )) is an N θ -δ-quasi-Cauchy sequence whenever (x k ) is. In this study we obtain some theorems related to strongly lacunary δ-quasi-Cauchy sequences.

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“…In 1928, Menger ([22]) introduced a concept of a generalized metric, and later on, Vulich ([31]) gave a notion of a higher dimensional normed linear space which had been neglected by many analysists until it was developed by Gähler in the mid of 1960's ( [15], [16], and [17]). Recently, Mashadi [20], and many others ( [8,21,23]) have studied this concept and obtained various results.…”

Section: Introductionmentioning

confidence: 99%

Lacunary ward continuity in 2-normed spaces

Cakalli

Ersan

2015

Filomat

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In this paper, we introduce lacunary statistical ward continuity in a 2-normed space. A function f defined on a subset E of a 2-normed space X is lacunary statistically ward continuous if it preserves lacunary statistically quasi-Cauchy sequences of points in E where a sequence (x k ) of points in X is lacunary statistically quasi-Cauchy iffor every positive real number ε and z ∈ X, and (k r ) is an increasing sequence of positive integers such that k 0 = 0 and h r = k r − k r−1 → ∞ as r → ∞, I r = (k r−1 , k r ]. We investigate not only lacunary statistical ward continuity, but also some other kinds of continuities in 2-normed spaces.

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Strongly Lacunary Ward Continuity in 2-Normed Spaces

Cited by 17 publications

References 17 publications

A variation on arithmetic continuity

A variation on arithmetic continuity

A variation on strongly lacunary delta ward continuity in 2-normed spaces

Lacunary ward continuity in 2-normed spaces

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