On a new measure on fractals
Journal of Inequalities and Applications, vol.2013, 2013 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 2013
- Publication Date: 2013
- Doi Number: 10.1186/1029-242x-2013-522
- Journal Name: Journal of Inequalities and Applications
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
- Keywords: Fractal calculus, Fractal curve, Fractal measure
- Open Archive Collection: AVESIS Open Access Collection
- Van Yüzüncü Yıl University Affiliated: Yes
Abstract
Fractals are sets whose Hausdorff dimension strictly exceeds their topological dimension. The algorithmic Riemannian-like method, Fα-calculus, has been suggested very recently. Henstock-Kurzweil integral is the generalized Riemann integral method by using the gauge function. In this paper we generalize the Fα-calculus as a fractional local calculus that is more suitable to describe some physical process. We introduce the new measure using the gauge function on fractal sets that gives a finer dimension in comparison with the Hausdorff and box dimension. Hilbert Fα-spaces are defined. We suggest the self-adjoint Fα-differential operator so that it can be applied in the fractal quantum mechanics and on the fractal curves. ©2013 Golmankhaneh and Baleanu; licensee Springer.