IFS on a metric space with a graph structure and extensions of the Kelisky–Rivlin theorem

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 [J.E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981) 713–747] on the existence of an invariant set for an iterated function system of Banach contractions, and a theorem of L. Máté [L. Máté, The Hutchinson–Barnsley theory for certain non-contraction mappings, Period. Math. Hungar. 27 (1993) 21–33] concerning finite families of locally uniformly contractions introduced by Edelstein. Also, they generalize recent fixed point theorems of A.C.M. Ran and M.C.B. Reurings [A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435–1443], J.J. Nieto and R. Rodríguez-López [J.J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223–239; J.J. Nieto, R. Rodríguez-López, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205–2212], and A. Petruşel and I.A. Rus [A. Petruşel, I.A. Rus, Fixed point theorems in ordered L-spaces, Proc. Amer. Math. Soc. 134 (2006) 411–418] 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. Oruç and N. Tuncer [H. Oruç, N. Tuncer, On the convergence and iterates of q-Bernstein polynomials, J. Approx. Theory 117 (2002) 301–313], and H. Gonska and P. Piţul [H. Gonska, P. Piţul, Remarks on an article of J.P. King, Comment. Math. Univ. Carolin. 46 (2005) 645–652].