Generation of fractals via iterated function system of Kannan contractions in controlled metric space

Abstract

The fixed point theory is one of the most essential techniques of applicable mathematics for solving many realistic problems to get a unique solution by using the well known Banach contraction principle. It has paved the ways for numerous extensions, generalization and development of the theory of fixed points in very diverse settings. Our intention in the present paper is to study the Kannan contraction maps defined on a controlled metric space. The generalization of the fixed point theorem for Kannan contraction on controlled metric space is investigated in this paper. We construct an iterated function system called Controlled Kannan Iterated Function System (CK-IFS) with Kannan contraction maps in a controlled metric space and use it to develop a new kind of invariant set, which is called a Controlled Kannan Attractor or Controlled Kannan Fractal (CK-Fractal). Subsequently, the collage theorem for controlled Kannan fractal is also proved. The multivalued fractals are also constructed in the controlled metric space using Kannan and Reich-type contraction maps. The newly developing iterated function system and fractal set in the controlled metric space can provide a novel direction in the fractal theory.