Local fractal functions on Orlicz-Sobolev spaces
DOI:
https://doi.org/10.24193/subbmath.2025.4.03Keywords:
Fractal, attractor, IFS, Orlicz-Sobolev space, Read-Bajraktarevic operator, contractive mapAbstract
In these notes we consider a class of iterated function systems whose attractors are the graphs of local fractal functions which belong to Orlicz and to Orlicz-Sobolev spaces. We prove that these maps are in correspondence with the fixed points of the Read-Bajraktarevi\'c operator. Our procedure extends a number of known theorems on the existence of local fractal functions of the Lebesgue and Sobolev classes, into more general function spaces where the role of the norm is now played by a Young function. The existence of local fractal functions of the Orlicz and of the Orlicz-Sobolev classes is demonstrated through an intermediary result. The realization of an IFS in the (previously untreated) multidimensional case is obtained via a stronger version of the finite increments theorem. Our results show that it would be natural to extend the Read-Bajraktarevi\'c operator to other function spaces on subdomains of differentiable and real analytic manifolds. Other questions, such as the existence of fixed points in higher-order Orlicz-Sobolev spaces etc., remain open as well. Our generalizations may be useful in applications.References
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