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Huseyin Cakalli

Publications and source records attributed to Huseyin Cakalli.

26 records · Page 2Linked to original sources
arXiv2011

On quasi-Cauchy sequences

This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling.

math.GN↗
arXiv2011

On Boundedness

This paper is withdrawn because the results in the paper are included in a paper to be published in Mathematical and Computer Modelling.

math.CA↗
arXiv2011

Strongly Cesaro type quasi-Cauchy sequences

In this paper we call a real-valued function $N_θ$-ward continuous if it preserves $N_θ$-quasi-Cauchy sequences where a sequence $\boldsymbolα=(α_{k})$ is defined to be $N_θ$-quasi-Cauchy when the sequence $Δ\boldsymbolα$ is in $N^{0}_θ$. We prove not only inclusion and compactness type theorems, but also continuity type theorems.

math.FA↗
arXiv2011

Abel Continuity

A sequence $\textbf{p}=(p_{n})$ of real numbers is called Abel convergent to $\ell$ if the series $Σ_{k=0}^{\infty}p_{k}x^{k}$ is convergent for $0\leq x<1$ and \[\lim_{x \to 1^{-}}(1-x) \sum_{k=0}^{\infty}p_{k}x^{k}=\ell.\] We introduce a concept of Abel continuity in the sense that a function $f$ defined on a subset of $\Re$, the set of real numbers, is Abel continuous if it preserves Abel convergent sequences, i.e. $(f(p_{n}))$ is an Abel convergent sequence whenever $(p_{n})$ is. A new type of compactness, namely Abel sequential compactness is also introduced, and interesting theorems related to this kind of compactnes, Abel continuity, statistical continuity, lacunary statistical continuity, ordinary continuity, and uniform continuity are obtained.

math.CA↗
arXiv2010

On G-Continuity

A function $f$ on a topological space is sequentially continuous at a point $u$ if, given a sequence $(x_{n})$, $\lim x_{n}=u$ implies that $\lim f(x_{n})=f(u)$. This definition was modified by Connor and Grosse-Erdmann for real functions by replacing $lim$ with an arbitrary linear functional $G$ defined on a linear subspace of the vector space of all real sequences. In this paper, we extend this definition to a topological group $X$ by replacing $G$ a linear functional with an arbitrary additive function defined on a subgroup of the group of all $X$-valued sequences and not only give new theorems in this generalized setting but also obtain theorems which are not appeared even for real functions so far.

math.GN↗
arXiv2010

$delta$-Quasi Cauchy Sequences

Recently, a concept of forward continuity and a concept of forward compactness are introduced in the senses that a function $f$ is forward continuous if $\lim_{n\to\infty} Δf(x_{n})=0$ whenever $\lim_{n\to\infty} Δx_{n}=0$,\; and a subset $E$ of $\textbf{R}$ is forward compact if any sequence $\textbf{x}=(x_{n})$ of points in $E$ has a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of the sequence $\textbf{x}$ such that $\lim_{k\to \infty} Δz_{k}=0$ where $Δz_{k}=z_{k+1}-z_{k}$. These concepts suggest us to introduce a concept of second forward continuity in the sense that a function $f$ is second forward continuous if $\lim_{n\to\infty}Δ^{2}f(x_{n})=0$ whenever $\lim_{n\to\infty}Δ^{2}x_{n}=0$, and a subset $E$ of $\textbf{R}$ is second forward compact if whenever $\textbf{x}=(x_{n})$ is a sequence of points in $E$ there is a subsequence $\textbf{z}=(z_{k})=(x_{n_{k}})$ of $\textbf{x}$ with $\lim_{k\to \infty} Δ^{2}z_{k}=0$ where $Δ^{2} y_{n}=y_{n+2}-2y_{n+1}+y_{n}$. We investigate the impact of changing the definition of convergence of sequences on the structure of forward continuity in the sense of second forward continuity, and compactness of sets in the sense of second forward compactness, and prove related theorems.

math.FA↗