arXiv subjects

Rafał Kapica

Publications and source records attributed to Rafał Kapica.

4 recordsLinked to original sources

arXiv2026

On the Existence of Geometrically Attracting Measures for Iterated Function Systems with Varying Sets of Transformations

In this paper, we focus on the asymptotic behavior of random iterated functions systems, in which both the family of transformations and the distribution of selecting them vary at each step. We consider two types of dynamics: the time-inhomogeneous Markov chain arising from forward iterates and the non-Markovian process generated by backward iterates. For both settings, we provide certain criteria for the existence of a probability measure that is geometrically attracting in the bounded Lipschitz distance, independently of the initial distribution. Finally, we illustrate our results through applications to specific models involving affine transformations.

math.DS↗

arXiv2022

Conditions for asymptotic stability of first order scalar differential-difference equation with complex coefficients

We investigate a scalar characteristic exponential polynomial with complex coefficients associated with a first order scalar differential-difference equation. Our analysis provides necessary and sufficient conditions for allocation of the roots in the complex open left half-plane what guarantees asymptotic stability of the differential-difference equation. The conditions are expressed explicitly in terms of complex coefficients of the characteristic exponential polynomial, what makes them easy to use in applications. We show examples including those for retarded PDEs in an abstract formulation.

math.DS↗

arXiv2015

Integrable solutions of inhomogeneous refinement type equations on intervals

Given a probability measure $P$ on a $σ$-algebra of subsets of a set $Ω$, an interval $I\subset\mathbb R$, $g\in L^1(I)$, and a function $φ\colon I\timesΩ\to I$ fulfilling some conditions we obtain results on the existence of solutions $f\in L^1(I)$ of the inhomogeneous refinement type equation $$ f(x)=\int_Ω\big|φ'_x(x,ω)\big|f(φ(x,ω))dP(ω)+g(x). $$

math.CA↗

arXiv2015

Inhomogeneous refinement equations with random affine maps

Given a probability space $(Ω,{\mathcal A},P)$, random variables $L,M\colonΩ\to\mathbb R$ and $g\in L^1(\mathbb R)$ we obtain two characterizations of these $f\in L^1(\mathbb R)$ which are solutions of the inhomogeneous refinement equation with a random affine map of the form $f(x)=\int_Ω|L(ω)|f(L(ω)x-M(ω))P(dω)+g(x)$.

math.CA↗