Generalized multivariate Fractal Interpolation Function and $α$-Fractal Function

Abstract: In this paper, we introduce a new approach for constructing multivariate fractal interpolation functions and $α$-fractal functions associated with multivariate functions. Unlike the existing methods that rely on the Banach contraction principle, here the construction is based on Matkowski and Rakotch contractions. While numerous methods for constructing multivariate fractal interpolation functions have been explored in the literature, the approach given in this paper is distinct in the sense that it generalizes all previously known techniques and provides a broader framework for such constructions. We propose a technique to develop nonlinear iterated function systems using the generalized contractions and establish that the attractors of such systems are the graphs of continuous multivariate functions interpolating theoretical data points. Furthermore, for the Rakotch contractions, the existence of an invariant Borel probability measure supported on the graph of the associated multivariate fractal interpolation function is explored.

• The paper establishes existence and uniqueness of continuous multivariate fractal interpolation functions as fixed points and graph attractors of nonlinear iterated function systems under Matkowski or Rakotch contractions.
• The paper extends multivariate α-fractal functions beyond constant scaling by supporting variable and nonlinear perturbations, including forms such as α(x)y and y/(1+y).
• The paper proves that Rakotch-type systems possess a unique invariant Borel probability measure supported on the interpolation graph, while leaving the Matkowski measure problem and fractal-dimension analysis open.

Overview

This paper extends the construction of fractal interpolation functions (FIFs) and $\alpha$-fractal functions to the multivariate setting using generalized fixed point theory. Whereas the classical Barnsley framework [barnsley1986fractal] and its multivariate descendants rely on the Banach contraction principle, the authors build nonlinear iterated function systems (IFSs) whose maps satisfy Matkowski or Rakotch contractive conditions — both of which strictly weaken the Banach hypothesis, since a Banach contraction is a special case of each. The paper's central claim is that this construction subsumes all previously published techniques for multivariate FIF construction; this claim rests on the fact that every prior method invokes Banach-type contractions, which are contained in the present framework.

The contributions are threefold: (i) existence and uniqueness of continuous multivariate FIFs as attractors of IFSs satisfying Matkowski and Rakotch conditions; (ii) an analogous construction of multivariate $\alpha$-fractal functions associated with a base function $b \in \mathcal{C}(I^q)$; and (iii) existence and uniqueness of an invariant Borel probability measure supported on the graph of the multivariate FIF for Rakotch-type IFSs.

Preliminaries

A self-map $f$ on a metric space is a $\phi$-contraction if $d(f(x),f(y)) \leq \phi(d(x,y))$ for some $\phi:\mathbb{R}_+\to\mathbb{R}_+$. It is a Matkowski contraction when $\phi$ is non-decreasing with $\phi^n(t)\to 0$ for all $t>0$, and a Rakotch contraction when $\alpha$0 and $\alpha$1 is non-increasing. Every Rakotch contraction is a Matkowski contraction, and by Jachymski–Jóźwik's characterization, Rakotch contractions are precisely the $\alpha$2-contractions with $\alpha$3 strictly increasing and concave. The attractor theorem of Strobin guarantees that IFSs composed of Matkowski contractions possess unique compact attractors in the Hausdorff metric, which is the key structural input for the Rakotch case.

Multivariate FIFs under generalized contractions

Interpolation data is prescribed on a rectangular grid $\alpha$4 over $\alpha$5, with contractive homeomorphisms $\alpha$6 satisfying Lipschitz bounds $\alpha$7 in each coordinate. Maps $\alpha$8 define an IFS on $\alpha$9 via $b \in \mathcal{C}(I^q)$0, and a Read–Bajraktarević-type operator $b \in \mathcal{C}(I^q)$1 acts on the closed subspace $b \in \mathcal{C}(I^q)$2 of continuous functions matching boundary data.

Matkowski case. If each $b \in \mathcal{C}(I^q)$3 is a Matkowski contraction in the last variable with common $b \in \mathcal{C}(I^q)$4, then $b \in \mathcal{C}(I^q)$5 is itself a Matkowski contraction on $b \in \mathcal{C}(I^q)$6. A lemma establishes that $b \in \mathcal{C}(I^q)$7 maps into the subspace interpolating the full grid, so the unique fixed point $b \in \mathcal{C}(I^q)$8 interpolates the data at every node of $b \in \mathcal{C}(I^q)$9, and its graph satisfies the invariance relation $f$0. The proof hinges on monotonicity of $f$1 combined with the coordinatewise inverse maps $f$2 to dominate the sup-norm difference by $f$3.

Rakotch case. Here $f$4 must additionally be Lipschitz in the first $f$5 variables with constants $f$6. The authors introduce the equivalent metric $f$7 with $f$8, chosen so that the horizontal components become strict contractions. Under this metric each $f$9 is a Rakotch contraction with comparison function $\phi$0, which is non-increasing into $\phi$1. The Strobin attractor theorem then yields a unique compact invariant set, which coincides with the graph of the fixed point from the Matkowski analysis. Metric equivalence ensures convergence of $\phi$2 holds under the Euclidean Hausdorff metric as well.

An implication worth noting: because the Rakotch condition permits $\phi$3 as $\phi$4, these results cover vertical maps whose local contraction ratios degenerate near the diagonal — a class excluded by uniform Banach bounds.

Generalized multivariate $\phi$5-fractal functions

For a base function $\phi$6 agreeing with $\phi$7 on the boundary nodes, the perturbation maps take the form

$\phi$8

where $\phi$9 is respectively a Matkowski or Rakotch contraction in $d(f(x),f(y)) \leq \phi(d(x,y))$0. The induced operator $d(f(x),f(y)) \leq \phi(d(x,y))$1 on $d(f(x),f(y)) \leq \phi(d(x,y))$2 is shown to be a Matkowski contraction (with $d(f(x),f(y)) \leq \phi(d(x,y))$3) in the first case, and a Rakotch contraction under $d(f(x),f(y)) \leq \phi(d(x,y))$4 in the second. Consequently there exists a unique continuous function $d(f(x),f(y)) \leq \phi(d(x,y))$5 interpolating $d(f(x),f(y)) \leq \phi(d(x,y))$6 on $d(f(x),f(y)) \leq \phi(d(x,y))$7, satisfying the self-referential equation

$d(f(x),f(y)) \leq \phi(d(x,y))$8

whose graph is the attractor of the associated IFS.

A remark demonstrates the generality of the scheme: choosing $d(f(x),f(y)) \leq \phi(d(x,y))$9 recovers the classical multivariate $\phi:\mathbb{R}_+\to\mathbb{R}_+$0-fractal function; choosing $\phi:\mathbb{R}_+\to\mathbb{R}_+$1 yields variable scaling; and choices such as $\phi:\mathbb{R}_+\to\mathbb{R}_+$2 or $\phi:\mathbb{R}_+\to\mathbb{R}_+$3 produce new families of multivariate fractal functions not previously available. This substantiates the paper's claim that the framework generalizes prior constructions rather than merely paralleling them.

Invariant measures for Rakotch IFSs

Let a probability vector $\phi:\mathbb{R}_+\to\mathbb{R}_+$4 weight the IFS. The Markov operator $\phi:\mathbb{R}_+\to\mathbb{R}_+$5 is analyzed on the space $\phi:\mathbb{R}_+\to\mathbb{R}_+$6 of compactly supported Borel probability measures equipped with the Monge–Kantorovich metric $\phi:\mathbb{R}_+\to\mathbb{R}_+$7. Using the Kantorovich–Rubinstein duality representation of $\phi:\mathbb{R}_+\to\mathbb{R}_+$8 as an infimum over couplings, the authors construct a coupling $\phi:\mathbb{R}_+\to\mathbb{R}_+$9 of $\phi$0 and $\phi$1 from an optimal coupling $\phi$2 of $\phi$3, and exploit concavity of $\phi$4 via Jensen's inequality to obtain

$\phi$5

Since $\phi$6 inherits strict increase and concavity from $\phi$7, $\phi$8 is a Rakotch contraction, so it has a unique fixed point $\phi$9 satisfying the invariance equation $\phi^n(t)\to 0$0. The support argument shows both inclusions $\phi^n(t)\to 0$1 and conversely, forcing $\phi^n(t)\to 0$2 to be the unique attractor $\phi^n(t)\to 0$3, i.e., the graph of the multivariate FIF.

This result extends the univariate/bivariate measure-theoretic work of Verma and Priyadarshi to arbitrary dimension within the Rakotch setting. The concavity of $\phi^n(t)\to 0$4 is essential: it enters through Jensen's inequality in bounding $\phi^n(t)\to 0$5, and no analogous argument is available without it.

Limitations and open questions

Two restrictions are explicit in the paper. First, the invariant measure result requires Rakotch contractions specifically; the authors state plainly that whether existence and uniqueness of an invariant probability measure holds for IFSs generated by general Matkowski contractions remains open, since the concavity characterization used in the Markov-operator argument fails outside the Rakotch class. Second, the Rakotch construction imposes Lipschitz conditions on $\phi^n(t)\to 0$6 in the spatial variables with explicit constants $\phi^n(t)\to 0$7, which constrains the admissible horizontal geometry beyond what the pure Matkowski theorem demands. The paper also leaves quantitative aspects untouched: fractal dimension estimates (Hausdorff, box, packing) for the newly constructed FIFs and $\phi^n(t)\to 0$8-fractal functions are not derived, nor is any calculus (fractional or classical) developed for them, although the conclusion identifies both as natural follow-up problems given the existing dimensional and fractional-calculus literature for Barnsley-type FIFs.

Conclusion

The paper delivers a unified, strictly more general machinery for multivariate fractal interpolation: Matkowski and Rakotch contractions replace the Banach principle in constructing nonlinear multivariate IFSs whose attractors are graphs of continuous interpolants, the same contractions yield multivariate $\phi^n(t)\to 0$9-fractal functions recovering classical and variable-scaling constructions as special cases, and invariant Borel probability measures supported on the FIF graphs are established for the Rakotch class. The main open problem left by the work is the extension of the invariant measure theory to Matkowski-type IFSs, together with the development of dimension theory and fractional calculus for the new function classes.