The recurrent iterated function systems (RIFS) were first introduced by Barnsley and Demko and generalized the usual iterated function systems (IFS). This new method allowed the construction of more general sets, which do not have to exhibit the strict self similarity of the IFS case and, in particular, the construction of recurrent fractal interpolation functions (RFIF). Given a data set $\{(x_n,y_n)\in I\times \mathbb{R},n=0,1,\dots ,N\}$ where $I=[x_0,x_N]$, we ensured that attractors of RIFS constructed using Geraghty contractions were graphs of some continuous functions which interpolated the given data. Our approach goes beyond the classical framework and provided a wide variety of systems for different approximations problems and, thus, gives more flexibility and applicability of the fractal interpolation method. As an application, we studied the error rates of time series related to the vaccination of COVID-19 using RFIF, and we compared them with the obtained results on the FIF.

In this paper we investigate an iterated function system that defines a fractal interpolation function, where ordinate scaling, that is Lipschitz constant in Banach contraction principle is substituted by real-valued control function. In such a manner, fractal interpolation functions associated with Matkowski contractions are obtained and provide a new framework of approximating experimental data. Furthermore, given a data generating function $f$, we study a new class of fractal interpolation functions which converge to $f$.

Multifractal analysis is typically used to describe objects possessing some type of scale invariance. During the last few decades, multifractal analysis has shown results of outstanding significance in theory and applications. In particular, it is widely used to characterize the geometry of the singularity of a measure $\mu$ or to study the time series, which has become an important tool for the study of several natural phenomena. In this paper, we investigate a more general level set studied in multifractal analysis. We use functions defined on balls in a metric space and that are Banach valued which is more general than measures used in the classical multifractal analysis. This is done by investigating Peyrière's multifractal Hausdorff and packing measures to study a relative vectorial multifractal formalism. This leads to results on the simultaneous behavior of possibly many branching random walks or many local Hölder exponents. As an application, we study the relative multifractal binomial measure in symbolic space $\mathrm{\partial }\mathcal{A}$.

The properties of various fractal and multifractal measures and dimensions have been under extensive study in the real-line and higher-dimensional Euclidean spaces. In non-Euclidean spaces, it is often impossible to construct non-trivial self-similar or self-conformal sets, etc. We consider in the present paper the proper way to phrase the definitions for use in general metric spaces. We investigate the relative Hausdorff measures ${\mathcal{H}}_{\mathit{\mu }}^{q,t}$ and the relative packing measures ${\mathcal{P}}_{\mathit{\mu }}^{q,t}$ defined in a separable metric space. We give some product inequalities which are a consequence of a new version of density theorems for these measures. Moreover, we prove that ${\mathcal{H}}_{\mathit{\mu }}^{q,t}$ and ${\mathcal{P}}_{\mathit{\mu }}^{q,t}$ can be expressed as Henstock-Thomson variation measures. The question of the weak-Vitali property arises in this context.

This paper presents a detailed procedure for deriving a Binet's type formula for the Tribonacci sequence $\{{\mathsf{T}}_n\}$. We examine the limiting distribution of a Markov chain that encapsulates the entire sequence $\{{\mathsf{T}}_n\}$, offering insights into its asymptotic behavior. An approximation of ${\mathsf{T}}_n$ is provided using two distinct probabilistic approaches. Furthermore, we study random sequences of the form $\{{\mathsf{Z}}_0,{\mathsf{Z}}_1,{\mathsf{Z}}_2,{\mathsf{Z}}_n={\mathsf{Z}}_{n-3}+{\mathsf{Z}}_{n-2}+{\mathsf{Z}}_{n-1},n=3,\dots \}$, referred to as the Tribonacci sequence of Random Variables. These sequences, fully defined by their initial random variables, are analyzed in terms of their distributional and limiting properties.

Let $\mu$ and $\nu$ be two compactly supported Borel probability measures on ${\mathbb{R}}^d$ and ${\mathbb{R}}^l$, respectively, and let $q\in \mathbb{R}$ and $h,g$ be two Hausdorff functions. In this paper, we are concerned with evaluation of the lower and upper Hewitt-Stromberg measure of Cartesian product sets, denoted, respectively, by ${\mathsf{H}}_{\mu }^{q,g}$ and ${\mathsf{P}}_{\nu }^{q,h}$, by means of the measure of their components. This is done by the construction of new multifractal measures in a similar manner to Hewitt-Stomberg measures but using the class of all (semi-) half-open binary cubes of covering sets in the definition rather than the class of all balls. Our derived product formula excludes the $0$– $\mathrm{\infty }$ case, and our approach is uniquely applied within an Euclidean space, distinguishing it from those previously utilized in metric spaces. Furthermore, by examining the measures of symmetric generalized Cantor sets, we establish that the exclusion of the $0$– $\mathrm{\infty }$ condition is essential and cannot be omitted.

This paper presents a detailed procedure for determining the probability of return for random walks on $\mathbb{Z}$, whose increment is given by a generalization of a well-known Fibonacci sequence, namely the $k$-Fibonacci-like sequence $(G_{k,n}{)}_n$. Also, we study the size of the set of these walks that return to the origin an infinite number of times, in term of fractal dimension. In addition, we investigate the limiting distribution of an adequate Markov chain that encapsulates the entire Tribonacci sequence $({\mathsf{T}}_n)$ to provide the limiting behavior of this sequence.

In this work, we investigate a class of generalized affine fractal interpolation functions (FIF) with variable parameters, where ordinate scaling is substituted by a real-valued control function. Let $\mathcal{S}$ be an iterated function system (IFS) with the attractor $G_{\mathrm{\Delta }}$, where $\mathrm{\Delta }$ is a given data set. We consider an affine transformation $\omega (\mathrm{\Delta })$ of $\mathrm{\Delta }$, and we define the IFS $\hat{\mathcal{S}}$ with the attractor $G_{\omega (\mathrm{\Delta })}$. We give a sufficient condition so that $G_{\omega (\mathrm{\Delta })}=\omega (G_{\mathrm{\Delta }})$. In addition, we compare the definite integrals of the corresponding FIF and study the additivity property. Some examples will be given, highlighting the effectiveness of our results.

This paper investigated the interplay of generalized fractal measures, specifically packing and Hewitt-Stromberg measures, on Cartesian products of symmetric generalized Cantor sets. By developing a unified framework for analyzing product sets in separable metric spaces, we uncovered how these measures interact in both typical and critical scenarios. Through explicit construction of multidimensional Cantor sets, we demonstrated instances where classical inequalities break down, particularly in cases involving vanishing and divergent measures. Our results enhance the theoretical foundation for multifractal analysis, offering new tools to address complexity in product geometries.

This paper focuses on the existence and analytical formulation of closed-form solutions for a three-dimensional system of nonlinear difference equations. The proposed system possesses a mathematical architecture that encapsulates complex nonlinear interactions among three mutually dependent variables. Through the application of systematic analytical transformations, the original system was reduced to a set of solvable recurrence relations, thereby allowing the derivation of explicit closed-form expressions with a high degree of analytical precision. Furthermore, numerical examples revealed that even minute perturbations in the system parameters or initial conditions can induce significant variations in oscillatory patterns, highlighting the system's structural sensitivity and rich dynamical diversity. This paper, therefore, constitutes a natural and essential extension of previously studied two-dimensional frameworks toward more sophisticated three-dimensional models, which provide a more faithful representation of interdependent relationships within discrete nonlinear systems.