zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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)].
47-99Operator theory (MSC2000)
[1]Abel, U.; Arends, G.: The powers of the bivariate Bernstein operators, Extracta math. 11, 389-404 (1996) · Zbl 0883.41018
[2]Adell, J. A.; Badía, F. G.; De La Cal, J.: On the iterates of some Bernstein-type operators, J. math. Anal. appl. 209, 529-541 (1997) · Zbl 0872.41009 · doi:10.1006/jmaa.1997.5371
[3]Agarwal, R. P.; El-Gebeily, M. A.; O’regan, D.: Generalized contractions in partially ordered metric spaces, Appl. anal. 87, 109-116 (2008) · Zbl 1140.47042 · doi:10.1080/00036810701556151
[4]Agratini, O.; Rus, I. A.: Iterates of a class of discrete linear operators via contraction principle, Comment. math. Univ. carolin. 44, 555-563 (2003) · Zbl 1096.41015 · doi:emis:journals/CMUC/cmuc0303/cmuc0303.htm
[5]Barnsley, M. F.: Fractals everywhere, (1988)
[6]Chen, F. L.; Feng, Y. Y.: Limit of iterates for Bernstein polynomials defined on a triangle, Appl. math. J. chinese univ. Ser. B 8, 45-53 (1993) · Zbl 0783.41017 · doi:10.1007/BF02661990
[7]Edelstein, M.: An extension of Banach’s contraction principle, Proc. amer. Math. soc. 12, 7-10 (1961) · Zbl 0096.17101 · doi:10.2307/2034113
[8]Edgar, G. A.: Measure, topology, and fractal geometry, Undergrad. texts math. (1990) · Zbl 0727.28003
[9]Edgar, G. A.; Golds, J.: A fractal dimension estimate for a graph-directed iterated function system of non-similarities, Indiana univ. Math. J. 48, 429-447 (1999) · Zbl 0944.28008 · doi:10.1512/iumj.1999.48.1641 · doi:http://www.iumj.indiana.edu/TOC/992.htm
[10]Espínola, R.; Kirk, W. A.: Fixed point theorems in R-trees with applications to graph theory, Topology appl. 153, 1046-1055 (2006) · Zbl 1095.54012 · doi:10.1016/j.topol.2005.03.001
[11]Gonska, H.; Piţul, P.: Remarks on an article of J.P. King, Comment. math. Univ. carolin. 46, 645-652 (2005) · Zbl 1121.41013 · doi:emis:journals/CMUC/cmuc0504/cmuc0504.htm
[12]Gwóźdź-łukawska, G.; Jachymski, J.: The Hutchinson – barnsley theory for infinite iterated function systems, Bull. austral. Math. soc. 72, 441-454 (2005)
[13]Hutchinson, J. E.: Fractals and self-similarity, Indiana univ. Math. J. 30, 713-747 (1981) · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[14]Jachymski, J.: The contraction principle for mappings on a metric space with a graph, Proc. amer. Math. soc. 136, 1359-1373 (2008) · Zbl 1139.47040 · doi:10.1090/S0002-9939-07-09110-1
[15]Kelisky, R. P.; Rivlin, T. J.: Iterates of Bernstein polynomials, Pacific J. Math. 21, 511-520 (1967) · Zbl 0177.31302
[16]King, J. P.: Positive linear operators which preserve x2, Acta math. Hungar. 99, 203-208 (2003) · Zbl 1027.41028 · doi:10.1023/A:1024571126455
[17]Máté, L.: The Hutchinson – barnsley theory for certain non-contraction mappings, Period. math. Hungar. 27, 21-33 (1993) · Zbl 0936.28004 · doi:10.1007/BF01877158
[18]Matkowski, J.: Integrable solutions of functional equations, Dissertationes math. (Rozprawy mat.) 127 (1975) · Zbl 0318.39005
[19]Mauldin, R. D.; Williams, S. C.: Hausdorff dimension in graph directed constructions, Trans. amer. Math. soc. 309, 811-829 (1988) · Zbl 0706.28007 · doi:10.2307/2000940
[20]Nieto, J. J.; Pouso, R. L.; Rodríguez-López, R.: Fixed point theorems in ordered abstract spaces, Proc. amer. Math. soc. 135, 2505-2517 (2007) · Zbl 1126.47045 · doi:10.1090/S0002-9939-07-08729-1
[21]Nieto, J. J.; Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5
[22]Nieto, J. J.; Rodríguez-López, R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta math. Sin. (Engl. Ser.) 23, 2205-2212 (2007) · Zbl 1140.47045 · doi:10.1007/s10114-005-0769-0
[23]Nowakowski, R.; Rival, I.: Fixed-edge theorem for graphs with loops, J. graph theory 3, 339-350 (1979) · Zbl 0432.05030 · doi:10.1002/jgt.3190030404
[24]O’regan, D.; Petruşel, A.: Fixed point theorems in ordered metric spaces, J. math. Anal. appl. 341, 1241-1252 (2008)
[25]Oruç, H.; Tuncer, N.: On the convergence and iterates of q-Bernstein polynomials, J. approx. Theory 117, 301-313 (2002) · Zbl 1015.33012 · doi:10.1006/jath.2002.3703
[26]Ostrovska, S.: Q-Bernstein polynomials and their iterates, J. approx. Theory 123, 232-255 (2003) · Zbl 1093.41013 · doi:10.1016/S0021-9045(03)00104-7
[27]Petruşel, A.; Rus, I. A.: Fixed point theorems in ordered L-spaces, Proc. amer. Math. soc. 134, 411-418 (2006) · Zbl 1086.47026 · doi:10.1090/S0002-9939-05-07982-7
[28]Phillips, G. M.: Bernstein polynomials based on the q-integers, Ann. numer. Math. 4, 511-518 (1997) · Zbl 0881.41008
[29]Ran, A. C. M.; Reurings, M. C. B.: A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. amer. Math. soc. 132, 1435-1443 (2004) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4
[30]Rus, I. A.: Iterates of Bernstein operators, via contraction principle, J. math. Anal. appl. 292, 259-261 (2004) · Zbl 1056.41004 · doi:10.1016/j.jmaa.2003.11.056
[31]Wenz, H. -J.: On the limits of (linear combinations of) iterates of linear operators, J. approx. Theory 89, 219-237 (1997) · Zbl 0871.41014 · doi:10.1006/jath.1996.3039