Generalized rational contractions in semi metric spaces via iterated function system
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Generalized rational contractions in semi metric spaces via iterated function system Marwan Amin Kutbi1 · Abdul Latif1 · Talat Nazir2 Received: 27 December 2019 / Accepted: 3 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We present the iterated function systems of generalized rational contractive mappings in Hausdorff semi metric spaces. We also study the well-posedness of attractors based problems of generalized rational contractive operator in the framework of semi metric spaces. An example is presented to support the results proved therein. These results extend, improve and generalize many results in the existing literature. Keywords Hutchinson operator · Iterated function system · Fractal · Attractor · Fixed point · Generalized rational contractive mapping Mathematics Subject Classification 47H10 · 54H25 · 54E50
1 Introduction and preliminaries Metric fixed point theory is a powerful tool for solving variety of scientific problems. It has many applications in the different branches of mathematics such as, for existence of solutions of differential equations, integral equations, optimization problems, equilibrium problems and split feasibility problems. Recently, various interesting and useful results of iterated methods as well as generalization of metric spaces are studied in [8, 10, 18, 20, 25–28, 31, 37, 38]. There are many generalizations and extensions of distance structure presented in the current literature. The concept of distance between two elements plays important role in defining the nature of the topology of an underlying set. Frechet [16] studied the concept of metric space defined on a nonempty set X that induces the Hausdorff topology on X . Consequently, there have been several generalizations of a metric function. Among which, one of the important generalization is the concept of a semi metric or symmetric metric space which gives rise to a topology which is not a Hausdorff. * Abdul Latif [email protected] 1
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
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Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus, Abbottabad 22060, Pakistan
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Cicchese [11] studied the contractive mappings in semi metric space and proved some useful fixed point results. Further useful results in this direction were proved in [1, 2, 7, 12, 21–24, 29, 32, 39, 40, 42]. On the other hand, Hutchinson [19] introduced the iterated function systems for constructing fractals from contractive self-mappings in the setup of metric spaces. He defined Hutchinson operator from the finite family of contractions and then obtained the attractor of iterated function system which is the fixed point of Hutchinson operator (see also [9]). Recently Dung and Petrusel [14] obtained some results of iterated function systems consisting of Kannan, Reich and Chatterjea type maps. Georgescu et al. [17] presented the Hardy–Rogers type iterated function systems. Then several imp
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