Fractals in extended b-metric space.

Ledia Subashi; Nertila Gjini

Ledia Subashi Department of Mathematics, University of Tirana,Albania
Nertila Gjini Deparment of Mathematics, University of New York Tirana.

Keywords: Iterated function system; fractal; extended b-metric space

Abstract

Iterated function systems are method of constructing fractals, which are based on the mathematical foundations laid by Hutchinson[1] and Barnsley[2]. Formally an Iterated function systems is a finite set of ‘contraction mappings’, on a complete metric space X. In this paper we construct a fractal set of Iterated function systems, which in our case are a collection of mappings defined in an extended b-metric space, of compact subsets of the space. We will prove that the Hutchinson operator defined with the help of a finite family of ‘generalized F-contraction mappings’ on a complete extended b- metric space is itself a generalized F- contraction mapping on a family of compact subsets of X. Then by successive application of a generalized F-Hutchinson operator we obtain a final fractal in an extended b- metric space

Downloads

Download data is not yet available.

References

J. E. Hutchinson (1981) “Fractals and self-similarity,” Indiana University Mathematics Journal, vol. 30, no. 5, pp. 713–74. doi:10.1512/iumj.1981.30.30055

M. F. Barnsley (1993). Fractals Everywhere, Academic Press, San Diego, Calif, USA, 2nd edition.

Czerwik, S. (1993). Contraction mappings in b-metric spaces. (pp 5–11). Acta Math. Inform. Univ. Ostra. 1.

T. Kamran, M. Samreen, Q. UL Ain (2017). A generalization of b-metric space and some fixed point theorems. MDPI Mathematics Journal, 5 ,19, 2017 doi:10.3390/math5020019

Subashi, L. (2017). Some topological properties of extended b-metric space, Proceedings of The 5th International Virtual Conference on Advanced Scientific Results (SCIECONF-2017), Vol. 5, pg 164-167. Zilina, Slovakia. doi: 10.18638/scieconf.2017.5.1.451

Nazir,T. Silvestrov,S. Abbas,M. (2016). Fractals of generalized F-hutchinson operator. Waves, Wavelets and Fractals—Advanced Analysis, vol. 2, pp. 29–40,.

D. Pompeiu (1905). Sur la continuite' des fonctions de variables complexes (These). Gauthier- Villars, Paris, ;Ann.Fac.Sci.deToulouse 7:264-315.

F. Hausdorff (1914). Grunclzuege der Mengenlehre. Viet, Leipzig.

Subashi, L (2017).Some results on extended b-metric spaces and Pompeiu-Hausdorff metric. Preprint

D. Wardowski, (2012). Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory and Applications, vol. 2012, article 94, 6 pages.