Iterated function systems and well-posedness

Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems in several topics of applied sciences [see for example: El Naschie MS. Iterated function systems and the two-slit experiment of quantum mechanics. Chaos, Solitons & Fractals 1994;4:1965–8; Iovane G. Cantorian spacetime and Hilbert space: Part I-Foundations. Chaos, Solitons & Fractals 2006;28:857–78; Iovane G. Cantorian space-time and Hilbert space: Part II-Relevant consequences. Chaos, Solitons & Fractals 2006;29:1–22; Fedeli A. On chaotic set-valued discrete dynamical systems. Chaos, Solitons & Fractals 2005;23:13814; Shi Y, Chen G. Chaos of discrete dynamical systems in complete metric spaces. Chaos, Solitons & Fractals 2004;22:55571]. The purpose of this paper is twofold. First, some existence and uniqueness results for the self-similar sets of a mixed iterated function systems are given. Then, using the concept of well-posed fixed point problem, the well-posedness of the self-similarity problem for some classes of iterated multifunction systems is also studied. Well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals.