Multi Fractals of Generalized Multivalued Iterated Function Systems in <i>b</i>-Metric Spaces with Applications

• Main Authors: Sudesh Kumari, Renu Chugh, Jinde Cao, Chuangxia Huang
• Format: Article
• Language: English
• Published: MDPI AG 2019-10-01
• Series: Mathematics
• Subjects: generalized multivalued <i>g</i>—contraction; generalized multivalued iterated function systems; hausdorff <i>b</i> metric space; fractal space; multifractal space; fixed point
• Online Access: https://www.mdpi.com/2227-7390/7/10/967

In this paper, we obtain multifractals (attractors) in the framework of Hausdorff <i>b</i>-metric spaces. Fractals and multifractals are defined to be the fixed points of associated fractal operators, which are known as attractors in the literature of fractals. We extend the results obtained by Chifu et al. (2014) and N.A. Secelean (2015) and generalize the results of Nazir et al. (2016) by using the assumptions imposed by Dung et al. (2017) to the case of ciric type generalized multi-iterated function system (CGMIFS) composed of ciric type generalized multivalued <i>G</i>-contractions defined on multifractal space <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">C</mi> <mo>(</mo> <mi mathvariant="script">U</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> in the framework of a Hausdorff <i>b</i>-metric space, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">U</mi> <mo>=</mo> <msub> <mi>U</mi> <mn>1</mn> </msub> <mo>&#215;</mo> <msub> <mi>U</mi> <mn>2</mn> </msub> <mo>&#215;</mo> <mo>⋯</mo> <mo>&#215;</mo> <msub> <mi>U</mi> <mi>N</mi> </msub> </mrow> </semantics> </math> </inline-formula>, <i>N</i> being a finite natural number. As an application of our study, we derive collage theorem which can be used to construct general fractals and to solve inverse problem in Hausdorff <i>b</i>-metric spaces which are more general spaces than Hausdorff metric spaces.