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

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.