Nonlinear Second Order Impulsive Difference Equations and their Oscillation Properties

In this work, we state and discuss the sufficient conditions for oscillation and nonoscillation of a class of nonlinear second order neutral impulsive difference equations with fixed moments of impulsive effect for various ranges of the neutral coefficient p(n).

σ-Continuous Functions and Related Cardinal Characteristics of the Continuum

A function f : X → Y between topological spaces is called σ-continuous (resp. ̄σ-continuous) if there exists a (closed) cover {Xn}n∈ω of X such that for every n ∈ ω the restriction f ↾ Xn is continuous. By 𝔠 σ (resp. 𝔠¯σ)we denote the largest cardinal κ ≤ 𝔠 such that every function f : X → ℝ defined on a subset X ⊂ ℝ of cardinality |X| <κ is σ-continuous (resp. ¯σ-continuous). It is clear that ω1 ≤ 𝔠¯σ ≤ 𝔠 σ ≤ 𝔠.We prove that 𝔭 ≤ 𝔮0 = 𝔠¯σ =min{𝔠 σ, 𝔟, 𝔮 }≤ 𝔠 σ ≤ min{non(ℳ), non(𝒩)}.

Some Remark on Oscillation of Second Order Impulsive Delay Dynamic Equations on Time Scales

This article deals with the oscillation criteria for a very extensively studied second order impulsive delay dynamic equations on time scale by using the Riccati transformation technique. Some examples are given to show the effect of impulse and to illustrate our main results.

A Note on Expansions of Rational Numbers by Certain Series

This paper deals with representations of rational numbers defined in terms of numeral systems that are certain generalizations of the classical q-ary numeral system.

On J(r, n)-Jacobsthal Hybrid Numbers

In this paper we introduce a generalization of Jacobsthal hybrid numbers – J(r, n)-Jacobsthal hybrid numbers. We give some of their properties: character, Binet’s formula, a summation formula and a generating function.

Explicit Evaluation of Some Quadratic Euler-Type Sums Containing Double-Index Harmonic Numbers

In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers H2n are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers Hn. As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung \sum\limits_{n = 1}^\infty {{{\left( {{{{H_n}} \over n}} \right)}^2} = {{17{\pi ^4}} \over {360}}} together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.

Compositions of ϱ-Lower Continuous Functions

In the paper some properties of compositions of ϱ-lower continuous functions are presented. We will show conditions for a function and for a homeomorphism f : I → I under which g ◦ f is ϱ-lower continuous for each ϱ-lower continuous function g : I → 𝕉. Some relevant properties will be discussed.

The Absolutely Strongly Star-Hurewicz Property with Respect to an Ideal

Aspace X is said to have the absolutely strongly star -𝒤-Hurewicz (ASS𝒤H) property if for each sequence (𝒰n : n ∈ 𝕅)of opencovers of X and each dense subset Y of X, there is a sequence (Fn : n ∈ 𝕅) of finite subsets of Y such that for each x ∈ X, {n ∈ 𝕅 : x ∉ St(Fn, 𝒰n)}∈ 𝒤, where 𝒤 is the proper admissible ideal of 𝕅. In this paper, we investigate the relationship between the ASS𝒤H property and other related properties and study the topological properties of the ASS𝒤H property. This paper generalizes several results of Song [25] to the larger class of spaces having the ASS𝒤H properties.

On Ideals Generated by Partitions into Meager and Null Sets

We examine the σ-ideal generated by complements of sums of sets from partitions into meager and null sets. We prove some characterizations of this σ-ideal.

Solution of the Fractional Bratu-Type Equation Via Fractional Residual Power Series Method

In this paper, we present numerical solution for the fractional Bratu-type equation via fractional residual power series method (FRPSM). The fractional derivatives are described in Caputo sense. The main advantage of the FRPSM in comparison with the existing methods is that the method solves the nonlinear problems without using linearization, discretization, perturbation or any other restriction. Three numerical examples are given and the results are numerically and graphically compared with the exact solutions. The solutions obtained by the proposed method are in complete agreement with the solutions available in the literature. The results reveal that the FRPSM is a very effective, simple and efficient technique to handle a wide range of fractional differential equations.