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Calculus on fractal subsets of real line. I: Formulation. (English) Zbl 1173.28005

That is a very laborious paper which contain a new calculus based upon fractal subsets of the real line. The \(F^\alpha\) \((0<\alpha\leq 1)\) integral is defined, which is suitable to integrate functions with fractal support of dimension \(\alpha\). Further, it is introduced the \(F^\alpha\) derivative, which enables to differentiate functions. The \(F^\alpha\) calculus retains much of the simplicity of ordinary calculus. Several results including analogous of fundamental theorems of calculus are proved. In this formulation, the integral staircase function, which is a generalization of the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, namely the \(\gamma\)-dimension.
The spaces of \(F^\alpha\)-differentiable and \(F^\alpha\)-integrable functions are analyzed and \(F^\alpha\)-differentiability is generalized on \(F\) using Sobolev-like construction.
The \(F^\alpha\)-differential equations can be used to model sublinear dynamical systems and fractal time process and it is given an important example.
The paper is very well written and the subject is very actual.

MSC:

28A80 Fractals
28A78 Hausdorff and packing measures
26A30 Singular functions, Cantor functions, functions with other special properties
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