Iterated function system in ∅- Metric Spaces
Abstract
Fractals have gained great attention from researchers due to their wide applications in engineering and applied sciences. Especially, in several topics of applied sciences, the iterated function systems theory has important roles. As is well known, examples of fractals are derived from the fixed point theory for suitable operators in spaces with complete or compact structures. In this article, a new generalization of Hausdorff distance on , is a class of all nonempty compact subsets of the metric space ( , ). Completeness and compactness of are analogously obtained from its counterparts of ( , ). Furthermore, a fractal is presented under a finite set of generalized -contraction mappings. Also, other special cases are presented.
Downloads
References
L. M. Alain, "Fractal Geometries Theory and Applications", Hermes, Paris, (1990).
M. F. Barnesly, "Fractal Everywhere", Second edition, Academic Press, (1988).
N. M. G. Al-Saidi, Sh. S. Al-Bundi, N. J. Al-Jawari, "A hybrid of fractal image coding and fractal dimension for an efficient retrieval method", Computational and Applied Mathematics, Vol. 37, pp 996-1011, (2018). https://doi.org/10.1007/s40314-016-0378-9
S. S. Al-Bundi, N. M. G. Al-Saidi, Al- N. J., Al-Jawari, "Crowding Optimization Method to Improve Fractal Image Compressions Based Iterated Function Systems", (IJACSA) International Journal of Advanced Computer Science and Applications, Vol. 7, No. 7, (2016). https://doi.org/10.14569/IJACSA.2016.070755
N. M. G. Al-Saidi. S. S. Al-Bundi N. J., Al-Jawari, "An Improved Harmony Search Algorithm For Reducing Computational Time of Fractal Image Coding", Vol.95. No 8, pp. 1669-1679, (2017).
J. E. Hutchinson, "Fractals and Self-Similarity", Indiana University journal of mathematics, Vol. 30, No. 5, pp. 713-747, (1981). https://doi.org/10.1512/iumj.1981.30.30055
M. F. Barnsley, S. Demko, "Iterated Function Systems and The Global Construction of Fractals", In Proceedings of the Royal Society of London A399, pp. 243- 275, (1985). https://doi.org/10.1098/rspa.1985.0057
M. F. Barnsley, V. Ervin, D. Hardin, J. Lancaster, "Solution of an Inverse Problem of Fractals and Other Sets", Proceeding of the National Academy of Science, Vol. 83, pp. 1975-1977, (1986). https://doi.org/10.1073/pnas.83.7.1975
S. Czerwik, contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav. 1,5-11, (1993).
A. Branciari, "a fixed point theorem of Banach-Caccioppoli type on a class of generalized metric spaces", publ. Math. Debrcen ,57:1-2, 31-37, (2000).
Z. Mustafa, B. Sims, "a new approach to generalized metric spaces", J. Nonlinear Convex Anal. 7, 289-297, (2006).
S. S. Abed, "Fixed Point Principles in General b-Metric Spaces and b-Menger Probabilistic spaces", Journal of Al-Qadisiyah for computer science and mathematics, 10(2), pp 42-53, (2018). https://doi.org/10.29304/jqcm.2018.10.2.366
S. S. Abed, K. E. Abdul Sada, "Common Fixed Points in Modular Spaces", Ibn Al-Haitham Journal for Pure and Applied science, (2018). https://doi.org/10.30526/2017.IHSCICONF.1822
S. S. Abed, A. Abdullah, "Fixed Point Theorem for Uncommuting Mappings", Ibn Al-Haitham Journal for Pure and Applied science, 26(1), pp. 312-319, (2013).
T. Nazir, S. Silvestrov, X. Qi, "Fractals of generalized F− Hutchinson operator in b-metric spaces", Waves Wavelets Fractals Adv. Anal.,(2),pp.29-40, (2016). https://doi.org/10.1515/wwfaa-2016-0006
M. Abbas, T. Nazir, "Attractor of the Generalized Contractive Iterated Function System", In Mathematical Analysis and Applications, pp. 401-420, (2018). https://doi.org/10.1002/9781119414421.ch11
S. S. Abed, A. N. Faraj., "Fixed Point Theorems and Iterative Function System in G-Metric Spaces", Journal of the University of Babylon for pure and applied sciences, Vol.27, No.2, pp. 329-340, 2019. https://doi.org/10.29196/jubpas.v27i2.2228
M. Samreen, T. Kamran, M. Postolache, "Extended b- Metric Space, Extended b-Comparison Function and Nonlinear Contractions", Journal Scientific Bulletin Series A, Applied Mathematics and Physics, Vol. 80, Iss. 4, (2018).
J. R. Roshan, N. Shobkolaei, S. Sedghi, and M. Abbas, "Common fixed point of four maps in b−metric spaces", Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics Vol. 43 (4), pp. 613-624, (2014).
Wardowski, D. ,Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory and Applications, 2012, 94, (2012). https://doi.org/10.1186/1687-1812-2012-94
Copyright (c) 2022 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
