Self Similarity Sets via Fixed Point Theory with lack of convexity

Abstract

A well-known theorem of fractal geometry, presented by J. Hutchinson (\cite{9}), says that there exists a unique compact self similar set with respect to any finite set of contractions on a complete metric space. Motivated by this result, in this paper, we prove fixed set theoretical theorems in order to obtain useful variations of this important result for Meir-Keeler operators and using the technique of measure of weak-noncompactness for operators acting in Banach spaces and Banach algebras. 

ref: J E. Hutchinson, Fractals and self semiliraty . Indians Univ.Math.J,30, 1981.