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

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:356

Article

Gwozdz-Lukawska, Gertruda²  Jachymski, Jacek¹ 

[1] Tech Univ Lodz, Inst Math, PL-93005 Lodz, Poland

[2] Tech Univ Lodz, Ctr Math & Phys, PL-90924 Lodz, Poland

关键词: Iterated function system;    Invariant set;    Hutchinson system;    Contractive mapping;    Reflexive graph;    Connected graph;    Edge-preserving mapping;    Bernstein operator;    q-Bernstein operator;    Kelisky-Rivlin theorem;    Infinite products of operators;   

DOI  :  10.1016/j.jmaa.2009.03.023

来源: Elsevier

【 摘 要 】

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. Mate [L Mate, 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. Rodriguez-Lopez [J.J. Nieto, R. Rodriguez-Lopez. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22 (2005) 223-239; J.J. Nieto, R. Rodriguez-Lopez, 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. Petrusel and I.A. Rus [A. Petru el, 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. Oruc and N. Tuncer [H. Oruc, N. Tuncer, On the convergence and iterates of q-Bernstein polynomials, J. Approx. Theory 117 (2002) 301-313], and H. Gonska and P. Pitul [H. Gonska, R Pitul, Remarks on an article of J.P. King, Comment. Math. Univ. Carolin. 46 (2005) 645-652). (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free