Huseyin Cakalli

Slowly Oscillating Continuity

A function 𝑓 is continuous if and only if, for each point 𝑥0 in the domain, lim𝑛→∞𝑓(𝑥𝑛)=𝑓(𝑥0), whenever lim𝑛→∞𝑥𝑛=𝑥0. This is equivalent to the statement that (𝑓(𝑥𝑛)) is a convergent sequence whenever (𝑥𝑛) is convergent. The concept of slowly oscillating continuity is defined in the sense that a function 𝑓 is slowly oscillating continuous if it transforms slowly oscillating sequences to slowly oscillating sequences, that is, (𝑓(𝑥𝑛)) is slowly oscillating whenever (𝑥𝑛) is slowly oscillating. A sequence (𝑥𝑛) of points in 𝐑 is slowly oscillating if lim𝜆→1

Sequential definitions of compactness

Abstract A subset F of a topological space is sequentially compact if any sequence x = ( x n ) of points in F has a convergent subsequence whose limit is in F . We say that a subset F of a topological group X is G -sequentially compact if any sequence x = ( x n ) of points in F has a convergent subsequence y such that G ( y ) ∈ F where G is an additive function from a subgroup of the group of all sequences of points in X . We investigate the impact of changing the definition of convergence of sequences on the structure of sequentially compactness of sets in the sense of G -sequential compactness. Sequential compactness is a special case of this generalization when G = lim .

IDEAL QUASI-CAUCHY SEQUENCES

An ideal I is a family of subsets of positive integers N which is closed under taking finite unions and subsets of its elements. A sequence (xn) of real numbers is said to be I-convergent to a real number L if for each ε>0, the set {n:|xn−L|≥ε} belongs to I. We introduce I-ward compactness of a subset of R, the set of real numbers, and I-ward continuity of a real function in the senses that a subset E of R is I-ward compact if any sequence (xn) of points in E has an I-quasi-Cauchy subsequence, and a real function is I-ward continuous if it preserves I-quasi-Cauchy sequences where a sequence (xn) is called to be I-quasi-Cauchy when (Δxn) is I-convergent to 0. We obtain results related to I-ward continuity, I-ward compactness, ward continuity, ward compactness, ordinary compactness, ordinary continuity, δ-ward continuity, and slowly oscillating continuity.MSC: 40A35, 40A05, 40G15, 26A15.

Statistical ward continuity

Abstract Recently, it has been proved that a real-valued function defined on an interval A of R, the set of real numbers, is uniformly continuous on A if and only if it is defined on A and preserves quasi-Cauchy sequences of points in A . In this paper we call a real-valued function statistically ward continuous if it preserves statistical quasi-Cauchy sequences where a sequence ( α k ) is defined to be statistically quasi-Cauchy if the sequence ( Δ α k ) is statistically convergent to 0. It turns out that any statistically ward continuous function on a statistically ward compact subset A of R is uniformly continuous on A . We prove theorems related to statistical ward compactness, statistical compactness, continuity, statistical continuity, ward continuity, and uniform continuity.

Summability in topological spaces

Abstract The main purpose of the paper is to introduce the notion of summability in abstract Hausdorff topological spaces. We give a characterization of such summability methods when the space allows a countable base. We also provide several Tauberian theorems in topological structures. Some open problems are discussed.

Cone normed spaces and weighted means

In this paper, we study the main properties of cone normed spaces, and prove some theorems of weighted means in cone normed spaces.

Sequential definitions of connectedness

Abstract We investigate the impact of changing the definition of convergence of sequences on the structure of the set of connected subsets of a topological group, X . A non-empty subset A of X is called G -sequentially connected if there are no non-empty and disjoint G -sequentially closed subsets U and V , both meeting A, such that A ⊆ U ⋃ V . Sequential connectedness in a topological group is a special case of this generalization when G = lim .

Δ-quasi-slowly oscillating continuity

In this paper, a new concept of Δ-quasi-slowly oscillating continuity is introduced. Furthermore, it is shown that this kind of continuity implies ordinary continuity. A new type of compactness is also defined and some new results related to compactness are proved.

I{CONVERGENCE ON CONE METRIC SPACES

The concept of I{convergence is an important generaliza- tion of statistical convergence which depends on the notion of an ideal I of subsets of the set N of positive integers. In this paper we introduce the ideas of I{Cauchy and I {Cauchy sequences in cone metric spaces and study their properties. We also investigate the relation between this new Cauchy type condition and the property of completeness.

Quasi Cauchy double sequences

Abstract A double sequence {xk,l} is quasi-Cauchy if given an Ɛ > 0 there exists an N ∈ N such that We study continuity type properties of factorable double functions defined on a double subset A x A of R2 into R, and obtain interesting results related to uniform continuity, sequential continuity, continuity, and a newly introduced type of continuity of factorable double functions defined on a double subset A x A of R2 into R.