Oscillation of second-order nonlinear difference equations with sublinear neutral term

Mathematica Moravica, Vol. **23**, No. **1** (2019), 1–10.

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Abstract and keywords

**Abstract.** We establish some new criteria for the oscillation of
second-order nonlinear difference equations with a sublinear neutral term.
This is accomplished by reducing the involved nonlinear equation to a linear inequality.

**Keywords.** Neutral term, nonlinear difference equation, oscillation.

Behavior of solutions of a second order rational difference equation

Mathematica Moravica, Vol. **23**, No. **1** (2019), 11–25.

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Abstract and keywords

**Abstract.** In this paper, we solve the difference equation
\[
x_{n+1}=\frac{\alpha}{x_nx_{n-1}-1},
\quad n=0,1,\dots,
\]
where $\alpha>0$ and the initial values $x_{-1}$, $x_{0}$ are real numbers. We
find some invariant sets and discuss the global behavior of the solutions of that equation. We show that when $\alpha>\frac{2}{3\sqrt3}$,
under certain conditions there exist solutions, that are either periodic or converging to periodic solutions. We show also the existence of dense solutions in the real line. Finally, we show that when $\alpha<\frac{2}{3\sqrt3}$, one of the negative equilibrium points attracts all orbits with initials outside a set of Lebesgue measure zero.

**Keywords.** Difference equation, forbidden set, periodic solution, unbounded solution.

Approximation by Zygmund means in variable exponent Lebesque Spaces

Mathematica Moravica, Vol. **23**, No. **1** (2019), 27–39.

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Abstract and keywords

**Abstract.** In the present work we investigate the approximation of the functions by the
Zygmund means in variable exponent Lebesgue spaces. Here the estimate which
is obtained depends on sequence of the best approximation in Lebesgue spaces
with variable exponent\textit{. }Also, these results were applied to
estimates of approximations of Zygmund sums in Smirnov classes with variable
exponent defined on simply connected domains of the complex plane.

**Keywords.** Lebesgue spaces with variable exponent, best approximation by
trigonometric polynomials, Zygmund means, modulus of smoothness.

Transcendental Picard-Mann Hybrid Julia and Mandelbrot Sets

Mathematica Moravica, Vol. **23**, No. **1** (2019), 41–49.

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Abstract and keywords

**Abstract.** In this paper, the fascinating Julia and Mandelbrot sets for the complex-valued transcendental functions
$z\rightarrow\sin(z^m)+c$, $(m\geq 2)\in\mathbb{N}$ have been obtained in Picard, Ishikawa and Noor orbits.
The purpose of the paper is to visualize transcendental Julia and Mandelbrot sets in Picard-Mann hybrid orbit.

**Keywords.** Transcendental functions, Picard-Mann hybrid iterates, Picard-Mann hybrid Julia sets, Picard-Mann hybrid Mandelbrot sets.

On Location in a Half-plane of Zeros of Perturbed First Order Entire Functions

Mathematica Moravica, Vol. **23**, No. **1** (2019), 51–61.

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Abstract and keywords

**Abstract.** We consider the entire functions
\[
h(z)=\sum_{k=0}^\infty \frac{a_{k}z^{k}}{k!}
\quad\mbox{and}\quad
\tilde h(z)=\sum_{k=0}^\infty \frac{\tilde a_kz^{k}}{k!}
\]
$( a_0=\tilde a_0=1; z, a_k, \tilde a_k\in {\bf C}, k=1, 2, \dots )$, provided
\[
\sum_{k=0}^\infty |a_{k}|^2<\infty, \sum_{k=0}^\infty |\tilde a_{k}|^2<\infty
\]
and all the zeros of $h(z)$ are in a half-plane.
We investigate the following problem: how small should be the quantity
$q:=(\sum_{k=1}^\infty |a_k-\tilde a_k|^2)^{1/2}$
in order to all the zeros of $\tilde h(z)$ lie in the same half-plane?

**Keywords.** Entire functions, zeros, perturbations.

Strong Commutativity Preserving Derivations on Lie ideals of Prime $\Gamma$-Rings

Mathematica Moravica, Vol. **23**, No. **1** (2019), 63–73.

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Abstract and keywords

**Abstract.** Let $M$ be a $\Gamma$-ring and $S\subseteq M$.
A mapping $f:M\rightarrow M$ is called \textit{strong commutativity preserving} on
$S$ if $[f(x),f(y)]_{\alpha}=[x,y]_{\alpha}$, for all $x,y\in S$, $\alpha\in\Gamma$.
In the present paper, we investigate the commutativity of the prime $\Gamma$-ring $M$ of characteristic
not $2$ with center $Z(M)\neq (0)$ admitting a derivation which is strong commutativity preserving
on a nonzero square closed Lie ideal $U$ of $M$. Moreover, we also obtain a related result when a mapping $d$
is assumed to be a derivation on $U$ satisfying the condition $d(u)\circ_{\alpha}d(v)=u\circ_{\alpha}v$, for all $u,v\in U$, $\alpha\in \Gamma$.

**Keywords.** Prime gamma rings, Lie ideals, derivations, strong commutativity preserving maps.

Property of growth determined by the spectrum of operator associated to Timoshenko system with memory

Mathematica Moravica, Vol. **23**, No. **1** (2019), 75–96.

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Abstract and keywords

**Abstract.** In this manuscript we prove the property of growth determined
by spectrum of the linear operator associated with the Timoshenko system with two histories.

**Keywords.** Timoshenko, $C_0$-semigroup, property of growth, memory.

On almost topological groups

Mathematica Moravica, Vol. **23**, No. **1** (2019), 97–106.

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Abstract and keywords

**Abstract.** We introduce and study the almost topological groups which are
fundamentally a generalization of topological groups. Almost
topological groups are defined by using almost continuous mappings
in the sense of Singal and Singal. We investigate some permanence
properties of almost topological groups. It is proved that
translation of a regularly open (resp. regularly closed) set in an
almost topological group is regularly open (resp. regularly closed).
And this fact gives us a lot of important and useful results of
almost topological groups.

**Keywords.** Regularly open sets, regularly closed sets, almost topological groups.