Multi Fractals of Generalized Multivalued Iterated Function Systems in b-Metric Spaces with Applications

In this paper, we obtain multifractals (attractors) in the framework of Hausdorff b-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 G-contractions defined on multifractal space C ( U ) in the framework of a Hausdorff b-metric space, where U = U 1 × U 2 × ⋯ × U N , N 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 b-metric spaces which are more general spaces than Hausdorff metric spaces.

ITERATED FUNCTION SYSTEMS ON FUNCTIONS OF BOUNDED VARIATION

We show that under certain hypotheses, an iterated function system on mappings (IFSM) is a contraction on the complete space of functions of bounded variation (BV). It then possesses a unique attractor of BV. Some BV-based inverse problems based on the Collage Theorem for contraction maps are considered.

IMPLEMENTASI TEOREMA COLLAGE UNTUK MENDESAIN SISTEM FUNGSI ITERASI (SFI): STUDI AWAL MENDESAIN SFI MOTIF SONGKET LOMBOK

Abstrak: Geometri fraktal merupakan cabang matematika yang memfokuskan kajiannya pada objek-objek fraktal. SFI yaitu teknik yang dapat digunakan untuk memodelkan objek fraktal. Tulisan ini memaparkan bagaimana mengaplikasikan teorema Collage untuk mendesain SFI suatu himpunan K yang memiliki sifat self-similarity. Mendesain SFI suatu himpunan K berarti mencari sejumlah berhingga pemetaan kontraktif berupa transformasi affine: dengan sedemikian sehingga untuk n=1,2,…,N. Keenam parameter pada persamaan di atas disebut sebagai kode SFI. Penelitian ini bertujuan untuk merancang suatu algoritma berdasarkan ide dari teorema Collage dalam menentukan kode SFI yang akan digunakan untuk memvisualisasikan atraktor dari objek fraktal menggunakan bahasa pemrograman. Algoritma yang telah disusun selanjutnya diterapkan untuk memperoleh SFI motif songket Lombok dan diperoleh hasil sebagai berikut:dengan faktor kontraktivitas s = 0.70434.Kata Kunci: Teorema Collage; Sifat Self-Similarity; Transformasi Affine; Algoritma; SFI; Atraktor. Abstract: Fractal geometry is the branch of mathematics that focus its studies on fractals. Iterated Function Systems (IFS) acts as a technique to generate fractal models. This article presents how to implement the Collage Theorem to design IFS of K which hold self-similarity property. Designing IFS of K means that finding the finite contractive mapping i.e. affine transformation: where such that for n=1,2,…,N. The six parameters on the equation above are called IFS codes. The aim of the study is constructing the algorithm based on the Collage theorem to determine the IFS codes which are used to visualize the attractor of the fractal objects through programming language. The Algorithm is implemented to obtain the IFS of Songket’s texture of Lombok and the result is given below:with a contractivity factor s = 0.70434.Keywords: Collage Theorem; Self-Similarity; Affine Transformation; Algorithm; IFS; Attractor.

Inverse Problems via the “Generalized Collage Theorem” for Vector-Valued Lax-Milgram-Based Variational Problems

We present an extended version of the Generalized Collage Theorem to deal with inverse problems for vector-valued Lax-Milgram systems. Numerical examples show how the method works in practical cases.