Remarkable results obtained when studying a question concerning the invariant set of an IFS.

Some years ago, a question was raised about whether an iterated function system consisting of a finite number of Reich (or Kannan) contractions (generally discontinuous contractive mappings) has a unique compact invariant set. In this article, we give remarkable results obtained by studying previous research articles that tried to answer this question positively and previous counterexamples that demonstrate that this question is negatively answered. Our results strongly encourage the development of fractal theory because the existence, uniqueness, compactness and boundedness of invariant sets with respect to functional families are one of the main research topics in the theory of fractals. • We provide counterexamples demonstrating that the question of whether an IFS consisting of a finite number of Kannan (or Reich) contractions (generally a discontinuous contractive mappings) with unique fixed points has a unique compact attractor is negatively solved. • We establish new theorems under weaker assumptions. • Explicit examples are given to illustrate the main results. [ABSTRACT FROM AUTHOR]