×

Lacunary ideal quasi Cauchy sequences. (English) Zbl 1438.40016

Summary: A real function is lacunary ideal ward continuous if it preserves lacunary ideal quasi Cauchy sequences where a sequence \((x_n)\) is said to be lacunary ideal quasi Cauchy (or \(I_\theta\)-quasi Cauchy) when \((\Delta x_n)=(x_{n+1}-x_n)\) is lacunary ideal convergent to 0, i.e. a sequence \((x_n)\) of points in \(\mathbb{R}\) is called lacunary ideal quasi Cauchy (or \(I_\theta\)-quasi Cauchy) for every \(\varepsilon>0\) if \[ \left\{r\in\mathbb{N} :\frac{1}{h_r}\sum_{n\in J_r}\vert x_{n+1}-x_n\vert\ge\varepsilon\right\}\in I. \] Also, we introduce the concept of lacunary ideal ward compactness and obtain results related to lacunary ideal ward continuity, lacunary ideal ward compactness, ward continuity, ward compactness, ordinary compactness, uniform continuity, ordinary continuity, \(\delta\)-ward continuity, and slowly oscillating continuity. Finally, we introduce the concept of ideal Cauchy continuous function in metric space and prove some results related to this notion.

MSC:

40A35 Ideal and statistical convergence
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
40G15 Summability methods using statistical convergence
PDFBibTeX XMLCite