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Huseyin Cakalli - One of the best experts on this subject based on the ideXlab platform.
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${N}_{\theta }$-Ward Continuity
Abstract and Applied Analysis, 2020Co-Authors: Huseyin CakalliAbstract:A function is continuous if and only if preserves convergent Sequences; that is, is a convergent Sequence whenever is convergent. The concept of -ward continuity is defined in the sense that a function is -ward continuous if it preserves -quasi-Cauchy Sequences; that is, is an -quasi-Cauchy Sequence whenever is -quasi-Cauchy. A Sequence of points in , the set of real numbers, is -quasi-Cauchy if , where , and is a lacunary Sequence, that is, an increasing Sequence of positive integers such that and . A new type compactness, namely, -ward compactness, is also, defined and some new results related to this kind of compactness are obtained.
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A new study on the strongly lacunary quasi Cauchy Sequences
2018Co-Authors: Huseyin Cakalli, Huseyin KaplanAbstract:In this paper, the concept of a strongly lacunary δ2 quasi-Cauchy Sequence is introduced. We proved interesting theorems related to strongly lacunary δ2-quasi-Cauchy Sequences. A real valued function f defined on a subset A of the set of real numbers, is strongly lacunary δ2 ward continuous on A if it preserves strongly lacunary δ2 quasi-Cauchy Sequences of points in A, i.e. (f(αk)) is a strongly lacunary δ2 quasi-Cauchy Sequence whenever (αk) is a strongly lacunary δ2 quasi-Cauchy Sequences of points in A, where a Sequence (αk) is called strongly lacunary δ2 quasi-Cauchy if (Δ2αk) is a strongly lacunary δ2 quasi-Cauchy Sequence where Δ2αk = αk+2 − 2αk+1 + αk for each positive integer k.
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A study on downward half Cauchy Sequences
arXiv: Functional Analysis, 2018Co-Authors: Huseyin CakalliAbstract:In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy Sequences, i.e. the Sequence $(f(\alpha_{n}))$ is downward half Cauchy whenever $(\alpha_{n})$ is a downward half Cauchy Sequence of points in $E$, where a Sequence $(\alpha_{ k})$ of points in $\R$ is called downward half Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\N}$ such that $\alpha_{m}-\alpha_{n}
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On Variations of statistical ward continuity
arXiv: Functional Analysis, 2017Co-Authors: Huseyin CakalliAbstract:In this paper, we introduce a concept of statistically $p$-quasi-Cauchyness of a real Sequence in the sense that a Sequence $(\alpha_{k})$ is statistically $p$-quasi-Cauchy if $\lim_{n\rightarrow\infty}\frac{1}{n}|\{k\leq n: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called statistically $p$-ward continuous on a subset $A$ of the set of real umbers $\mathbb{R}$ if it preserves statistically $p$-quasi-Cauchy Sequences, i.e. the Sequence $f(\textbf{x})=(f(\alpha_{n}))$ is statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ 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 $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves statistically $p$-quasi-Cauchy Sequences of points in $A$.
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A new study on the strongly lacunary quasi Cauchyness
arXiv: Functional Analysis, 2017Co-Authors: Huseyin Kaplan, Huseyin CakalliAbstract:In this paper, the concept of an $N_{\theta}^{2}$ quasi-Cauchy Sequence is introduced. We proved interesting theorems related to $N_{\theta}^{2}$-quasi-Cauchy Sequences. A real valued function $f$ defined on a subset $A$ of $\mathbb{R}$, the set of real numbers, is $N_{\theta}^{2}$ ward continuous on $A$ if it preserves $N_{\theta}^{2}$ quasi-Cauchy Sequences of points in $A$, i.e. $(f( \alpha_{k}))$ is an $N_{\theta}^{2}$ quasi-Cauchy Sequence whenever $(\alpha_{k})$ is an $N_{\theta}^{2}$ quasi-Cauchy Sequences of points in $A$, where a Sequence $(\alpha_{k})$ is called $N_{\theta}^{2}$ quasi-Cauchy if $(\Delta^{2} \alpha_{k})$ is an $N_{\theta}$ quasi-Cauchy Sequence where $\Delta^{2}\alpha_{k}=\alpha_{k+2}-2\alpha_{k+1}+\alpha_{k}$ for each positive integer $k$.
Huseyin Kaplan - One of the best experts on this subject based on the ideXlab platform.
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A new study on the strongly lacunary quasi Cauchy Sequences
2018Co-Authors: Huseyin Cakalli, Huseyin KaplanAbstract:In this paper, the concept of a strongly lacunary δ2 quasi-Cauchy Sequence is introduced. We proved interesting theorems related to strongly lacunary δ2-quasi-Cauchy Sequences. A real valued function f defined on a subset A of the set of real numbers, is strongly lacunary δ2 ward continuous on A if it preserves strongly lacunary δ2 quasi-Cauchy Sequences of points in A, i.e. (f(αk)) is a strongly lacunary δ2 quasi-Cauchy Sequence whenever (αk) is a strongly lacunary δ2 quasi-Cauchy Sequences of points in A, where a Sequence (αk) is called strongly lacunary δ2 quasi-Cauchy if (Δ2αk) is a strongly lacunary δ2 quasi-Cauchy Sequence where Δ2αk = αk+2 − 2αk+1 + αk for each positive integer k.
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A new study on the strongly lacunary quasi Cauchyness
arXiv: Functional Analysis, 2017Co-Authors: Huseyin Kaplan, Huseyin CakalliAbstract:In this paper, the concept of an $N_{\theta}^{2}$ quasi-Cauchy Sequence is introduced. We proved interesting theorems related to $N_{\theta}^{2}$-quasi-Cauchy Sequences. A real valued function $f$ defined on a subset $A$ of $\mathbb{R}$, the set of real numbers, is $N_{\theta}^{2}$ ward continuous on $A$ if it preserves $N_{\theta}^{2}$ quasi-Cauchy Sequences of points in $A$, i.e. $(f( \alpha_{k}))$ is an $N_{\theta}^{2}$ quasi-Cauchy Sequence whenever $(\alpha_{k})$ is an $N_{\theta}^{2}$ quasi-Cauchy Sequences of points in $A$, where a Sequence $(\alpha_{k})$ is called $N_{\theta}^{2}$ quasi-Cauchy if $(\Delta^{2} \alpha_{k})$ is an $N_{\theta}$ quasi-Cauchy Sequence where $\Delta^{2}\alpha_{k}=\alpha_{k+2}-2\alpha_{k+1}+\alpha_{k}$ for each positive integer $k$.
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A VARIATION ON LACUNARY STATISTICAL QUASI Cauchy SequenceS
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2017Co-Authors: Huseyin Cakalli, Huseyin KaplanAbstract:In this paper, the concept of a lacunary statistically -quasi-CauchySequence is investigated. In this investigation, we proved interesting theoremsrelated to lacunary statisticallycontinuities. A real valued function f de…ned on a subset A of R, the set ofreal numbers, is called lacunary statisticallyserves lacunary statistically delta quasi-Cauchy Sequences of points in A, i.e.(f (k))is a lacunary statistically delta quasi-Cauchy Sequence whenever (k)is a lacunary statistically delta quasi-Cauchy Sequence of points in A, wherea Sequence (k)is called lacunary statistically delta quasi-Cauchy if (a lacunary statistically quasi-Cauchy Sequence. It turns out that the set oflacunary statisticallyof continuous functions
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Variations on strongly lacunary quasi Cauchy Sequences
2016Co-Authors: Huseyin Kaplan, Huseyin CakalliAbstract:We introduce a new function space, namely the space of Nθ (p)-ward continuous functions, which turns out to be a closed subspace of the space of continuous functions for each positive integer p. Nθα(p)-ward continuity is also introduced and investigated for any fixed 0 < α ≤ 1, and for any fixed positive integer p. A real valued function f defined on a subset A of R, the set of real numbers is Nθα(p)-ward continuous if it preserves Nθα(p)-quasi-Cauchy Sequences, i.e. (f (xn)) is an Nθα(p)-quasi-Cauchy Sequence whenever (xn) is Nθα(p)-quasi-Cauchy Sequence of points in A, where a Sequence (xk) of points in R is called Nθα(p)-quasi-Cauchy if limr→∞1hrα∑k∈Ir|Δxk|p=0, where Δxk = xk+1−xk for each positive integer k, p is a fixed positive integer, α is fixed in ]0, 1], Ir = (kr−1, kr], and θ = (kr) is a lacunary Sequence, i.e. an increasing Sequence of positive integers such that k0 ≠ 0, and hr: kr−kr−1 →∞.
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Strongly lacunary delta ward continuity
2015Co-Authors: Huseyin Cakalli, Huseyin KaplanAbstract:In this paper, the concepts of a lacunary statistically δ-quasi-Cauchy Sequence and a strongly lacunary δ-quasi-Cauchy Sequence are introduced, and investigated. In this investigation, we proved interesting theorems related to some newly defined continuities here, mainly, lacunary statistically δ-ward continuity, and strongly lacunary δ-ward continuity. A real valued function f defined on a subset A of R, the set of real numbers, is called lacunary statistically delta ward continuous on A if it preserves lacunary statistically delta quasi-Cauchy Sequences of points in A, i.e. (f (αk)) is a lacunary statistically quasi-Cauchy Sequence whenever (αk) is a lacunary statistically quasi-Cauchy Sequences of points in A, and a real valued function f defined on a subset A of R is called strongly lacunary delta ward continuous on A if it preserves strongly lacunary delta quasi-Cauchy Sequences of points in A, i.e. (f (αk)) is a strongly lacunary quasi-Cauchy Sequence whenever (αk) is a strongly lacunary quasi-Cauch...
Sebnem Yildiz - One of the best experts on this subject based on the ideXlab platform.
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Lacunary Statistical $p$-Quasi Cauchy Sequences
2019Co-Authors: Sebnem YildizAbstract:In this paper, we introduce a concept of lacunary statistically $p$-quasi-Cauchyness of a real Sequence in the sense that a Sequence $(\alpha_{k})$ is lacunary statistically $p$-quasi-Cauchy if $\lim_{r\rightarrow\infty}\frac{1}{h_{r}}|\{k\in I_{r}: |\alpha_{k+p}-\alpha_{k}|\geq{\varepsilon}\}|=0$ for each $\varepsilon>0$. A function $f$ is called lacunary statistically $p$-ward continuous on a subset $A$ of the set of real numbers $\mathbb{R}$ if it preserves lacunary statistically $p$-quasi-Cauchy Sequences, i.e. the Sequence $(f(\alpha_{n}))$ is lacunary statistically $p$-quasi-Cauchy whenever $\boldsymbol\alpha=(\alpha_{n})$ is a lacunary 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 $\mathbb{R}$ if there exists a positive integer $p$ such that $f$ preserves lacunary statistically $p$-quasi-Cauchy Sequences of points in $A$.
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A new variation on lacunary statistical quasi Cauchy Sequences
2018Co-Authors: Sebnem YildizAbstract:In this paper, the concept of an Sθ-δ2-quasi-Cauchy Sequence is investigated. In this investigation, we proved interesting theorems related to Sθ-δ2-ward continuity, and some other kinds of continuities. A real valued function f defined on a subset A of R, the set of real numbers, is called Sθ-δ2-ward continuous on A if it preserves Sθ-δ2-quasi-Cauchy Sequences of points in A, i.e. (f(αk)) is an Sθ-δ2-quasi-Cauchy Sequence whenever (αk) is an Sθ-δ2-quasi-Cauchy Sequence of points in A, where a Sequence (αk) is called Sθ-δ2-quasi-Cauchy if (Δ2αk) is an Sθ-quasi-Cauchy Sequence. It turns out that the set of Sθ-δ2-ward continuous functions is a closed subset of the set of continuous functions.In this paper, the concept of an Sθ-δ2-quasi-Cauchy Sequence is investigated. In this investigation, we proved interesting theorems related to Sθ-δ2-ward continuity, and some other kinds of continuities. A real valued function f defined on a subset A of R, the set of real numbers, is called Sθ-δ2-ward continuous on A if it preserves Sθ-δ2-quasi-Cauchy Sequences of points in A, i.e. (f(αk)) is an Sθ-δ2-quasi-Cauchy Sequence whenever (αk) is an Sθ-δ2-quasi-Cauchy Sequence of points in A, where a Sequence (αk) is called Sθ-δ2-quasi-Cauchy if (Δ2αk) is an Sθ-quasi-Cauchy Sequence. It turns out that the set of Sθ-δ2-ward continuous functions is a closed subset of the set of continuous functions.
Sibel Ersan - One of the best experts on this subject based on the ideXlab platform.
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A variation on strongly lacunary delta ward continuity in 2-normed spaces
Boletim da Sociedade Paranaense de Matemática, 2019Co-Authors: Sibel ErsanAbstract:A Sequence $(x_{k})$ of points in a subset E of a 2-normed space $X$ is called strongly lacunary $\delta$-quasi-Cauchy, or $N_\theta$-$\delta$-quasi-Cauchy if $(\Delta x_k)$ is $N_\theta$-convergent to 0, that is $\lim_{r\rightarrow\infty}\frac{1}{h_r}\sum_{k\in I_r}||\Delta^2 x_k, z||=0$ for every fixed $z\in X$. A function defined on a subset $E$ of $X$ is called strongly lacunary $\delta$-ward continuous if it preserves $N_{\theta}$-$\delta$-quasi-Cauchy Sequences, i.e. $(f(x_{k}))$ is an $N_{\theta}$-$\delta$-quasi-Cauchy Sequence whenever $(x_{k})$ is. In this study we obtain some theorems related to strongly lacunary $\delta$-quasi-Cauchy Sequences.
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Strongly lacunary ward continuity in 2-normed spaces.
The Scientific World Journal, 2014Co-Authors: Huseyin Cakalli, Sibel ErsanAbstract: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(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.
Hacer Sengul - One of the best experts on this subject based on the ideXlab platform.
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A variation on lacunary quasi Cauchy Sequences
2016Co-Authors: Huseyin Cakalli, Mikail Et, Hacer SengulAbstract:In the present paper, we introduce a concept of ideal lacunary statistical quasi-Cauchy Sequence of order α of real numbers in the sense that a Sequence (xk) of points in R is called I−lacunary statistically quasi-Cauchy of order α, if {r∈N:1hrα|{k∈Ir:|Δxk|≥e}|≥δ}∈I for each e > 0 and for each δ > 0, where an ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. The main purpose of this paper is to investigate ideal lacunary statistical ward continuity of order α, where a function f is called I− lacunary statistically ward continuous of order α if it preserves I-lacunary statistically quasi-Cauchy Sequences of order α, i.e. (f (xn)) is a Sθα(I)-quasi-Cauchy Sequence whenever (xn) is.