arXiv subjects

Zuzana Došlá

Publications and source records attributed to Zuzana Došlá.

4 recordsLinked to original sources

arXiv2020

A fixed-point approach for decaying solutions of difference equations

A boundary value problem associated to the difference equation with advanced argument \begin{equation} \label{*}Δ\bigl (a_{n}Φ(Δx_{n})\bigr)+b_{n}Φ(x_{n+p} )=0,\ \ n\geq1 \tag{$*$} \end{equation} is presented, where $Φ(u)=|u|^α$sgn $u,$ $α>0,p$ is a positive integer and the sequences $a,b,$ are positive. We deal with a particular type of decaying solutions of (\ref{*}), that is the so-called intermediate solutions (see below for the definition) . In particular, we prove the existence of these type of solutions for (\ref{*}) by reducing it to a suitable boundary value problem associated to a difference equation without deviating argument. Our approach is based on a fixed point result for difference equations, which originates from existing ones stated in the continuous case. Some examples and suggestions for future researches complete the paper.

math.CA↗

arXiv2020

Decaying positive global solutions of second order difference equations with mean curvature operator

A boundary value problem on an unbounded domain, associated to difference equations with the Euclidean mean curvature operator is considered. The existence of solutions which are positive on the whole domain and decaying at infinity is examined by proving new Sturm comparison theorems for linear difference equations and using a fixed point approach based on a linearization device. %The process from the continuous problem to discrete one is examined, too. The process of discretization of the boundary value problem on the unbounded domain is examined, and some discrepancies between the discrete and the continuous case are pointed out, too.

math.CA↗

arXiv2020

Riemann-Liouville derivative over the space of integrable distributions

In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, $D_{HK}$. We obtain new fundamental properties of the fractional derivative and integral, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.

math.FA↗

arXiv2015

A Dirichlet problem on the half-line for nonlinear equations with indefinite weight

We study the existence of positive solutions on the half-line $[0,\infty)$ for the nonlinear second order differential equation \[ \bigl(a(t)x^{\prime}\bigr)^{\prime}+b(t)F(x)=0,\quad t\geq0, \] satisfying Dirichlet type conditions, say $x(0)=0$, $\lim_{t\rightarrow\infty}x(t)=0$. The function $b$ is allowed to change sign and the nonlinearity $F$ is assumed to be asymptotically linear in a neighborhood of zero and infinity. Our results cover also the cases in which $b$ is a periodic function for large $t$ or it is unbounded from below.

math.CA↗