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IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem. (English) Zbl 1171.28002
Authors’ abstract: We develop the Hutchinson-Barnsley theory for finite families of mappings on a metric space endowed with a directed graph. In particular, our results subsume a classical theorem of J. E. Hutchinson [Indiana Univ. Math. J. 30, 713–747 (1981; Zbl 0598.28011)] on the existence of an invariant set for an iterated function system of Banach contractions, and a theorem of L. Máté [Period. Math. Hung. 27, No. 1, 21–33 (1993; Zbl 0936.28004)] concerning finite families of locally uniformly contractions introduced by Edelstein. Also, they generalize recent fixed point theorems due to A. C. M. Ran and M. C. B. Reurings [Proc. Am. Math. Soc. 132, No. 5, 1435–1443 (2004; Zbl 1060.47056)], J. J. Nieto and R. Rodríguez-López [Order 22, No. 3, 223–239 (2005; Zbl 1095.47013); Acta Math. Sin., Engl. Ser. 23, No. 12, 2205–2212 (2007; Zbl 1140.47045)], and A. Petrusel and I. A. Rus [Proc. Am. Math. Soc. 134, No. 2, 411–418 (2006; Zbl 1086.47026)] for contractive mappings on an ordered metric space. As an application, we obtain a theorem on the convergence of infinite products of linear operators on an arbitrary Banach space. This result yields new generalizations of the Kelisky-Rivlin theorem on iterates of the Bernstein operators on the space $C$[0,1] as well as its extensions given recently by H. Oruc and N. Tuncer [J. Approximation Theory 117, No. 2, 301–313 (2002; Zbl 1015.33012)], and H. Gonska and P. Pitul [Commentat. Math. Univ. Carol. 46, No. 4, 645–652 (2005; Zbl 1121.41013)].
##### MSC:
 28A80 Fractals 47-99 Operator theory (MSC2000)