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)
A general class of iterative equations on the unit circle. (English) Zbl 1174.39005
Summary: A class of functional equations with nonlinear iterates is discussed on the unit circle 𝕋 1 . By lifting maps on 𝕋 1 and maps on the torus 𝕋 n to Euclidean spaces and extending their restrictions to a compact interval or cube, we prove existence, uniqueness and stability for their continuous solutions.
39B12Iterative and composite functional equations
39B32Functional equations for complex functions
39B82Stability, separation, extension, and related topics
37E05Maps of the interval (piecewise continuous, continuous, smooth)
[1]M. Bajger: On the structure of some flows on the unit circle. Aequationes Math. 55 (1998), 106–121. · Zbl 0891.39017 · doi:10.1007/s000100050023
[2]K. Baron and W. Jarczyk: Recent results on functional equations in a single variable, perspectives and open problems. Aequationes Math. 61 (2001), 1–48. · Zbl 0972.39011 · doi:10.1007/s000100050159
[3]K. Ciepliński: On the embeddability of a homeomorphism of the unit circle in disjoint iteration groups. Publ. Math. Debrecen 55 (1999), 363–383.
[4]K. Ciepliński: On properties of monotone mappings of the circle. J. Anal. Appl. 4 (2006), 169–178.
[5]I. P. Cornfeld, S. V. Fomin and Y. G. Sinai: Ergodic Theory, Grundlehren 245, Springer Verlag, Berlin-Heidelberg-New York. 1982.
[6]W. Jarczyk: On an equation of linear iteration. Aequationes Math. 51 (1996), 303–310. · Zbl 0872.39010 · doi:10.1007/BF01833285
[7]W. Jarczyk: Babbage equation on the circle. Publ. Math. Debrecen 63 (2003), 389–400.
[8]M. Kuczma, B. Choczewski and R. Ger: Iterative Functional Equations. Encycl. Math. Appl. 32, Cambridge Univ. Press, Cambridge, 1990.
[9]M. Kulczycki and J. Tabor: Iterative functional equations in the class of Lipschitz functions. Aequationes Math. 64 (2002), 24–33. · Zbl 1009.39021 · doi:10.1007/s00010-002-8028-2
[10]J. Mai: Conditions of existence for N-th iterative roots of homeomorphisms on the circle, in Chinese. Acta Math. Sinica 30 (1987), 280–283.
[11]J. Mai and X. Liu: Existence, uniqueness and stability of C m solutions of iterative functional equations. Science in China A43 (2000), 897–913. · Zbl 0999.39020 · doi:10.1007/BF02879796
[12]J. Matkowski and W. Zhang: On the polynomial-like iterative functional equation. Functional Equations & Inequalities, Math.& Its Appl. Vol. 518, ed. T.M. Rassias, Kluwer Academic, Dordrecht, 2000, pp. 145–170.
[13]A. Mukherjea and J. S. Ratti: On a functional equation involving iterates of a bijection on the unit interval. Nonlinear Anal. 7 (1983), 899–908; Nonlinear Anal. 31 (1998), 459–464. · Zbl 0518.39005 · doi:10.1016/0362-546X(83)90065-2
[14]J. Palis and W. Melo: Geometric Theory of Dynamical Systems, An Introduction. Springer-Verlag, New York, 1982.
[15]J. Si: Continuous solutions of iterative equation G(f(x), f n 2 (x),..., f nk (x)) = F(x). J. Math. Res. Exp. 15 (1995), 149–150. (In Chinese.)
[16]P. Solarz: On some iterative roots on the circle. Publ. Math. Debrecen 63 (2003), 677–692.
[17]J. Tabor and J. Tabor: On a linear iterative equation. Results in Math. 27 (1995), 412–421.
[18]C. T. C. Wall: A Geometric Introduction to Topology. Addison-Wesley, Reading, 1972.
[19]D. Yang and W. Zhang: Characteristic solutions of polynomial-like iterative equations. Aequationes Math. 67 (2004), 80–105. · Zbl 1060.39019 · doi:10.1007/s00010-003-2708-4
[20]M.C. Zdun: On iterative roots of homeomorphisms of the circle. Bull. Polish Acad. Sci. Math. 48 (2000), 203–213.
[21]J. Zhang, L. Yang and W. Zhang: Some advances on functional equations. Adv. Math. (Chin.) 24 (1995), 385–405.
[22]W. Zhang: Discussion on the solutions of the iterated equation i=1 n λ i f i (x)=F(x) . Chin. Sci. Bul. 32 (1987), 1444–1451.
[23]W. Zhang: Discussion on the differentiable solutions of the iterated equation i=1 n λ i f i (x)=F(x) . Nonlinear Anal. 15 (1990), 387–398. · Zbl 0717.39005 · doi:10.1016/0362-546X(90)90147-9
[24]W. Zhang and J. A. Baker: Continuous solutions of a polynomial-like iterative equation with variable coefficients. Ann. Polon. Math. 73 (2000), 29–36.
[25]W. Zhang: Solutions of equivariance for a polynomial-like iterative equation. Proc. Royal Soc. Edinburgh 130A (2000), 1153–1163. · Zbl 0983.39010 · doi:10.1017/S0308210500000615
[26]Zhu-Sheng Zhang: Relations between embedding flows and transformation groups of self-mappings on the circle. Acta Math. Sinica 24 (1981), 953–957. (In Chinese.)