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)
Convex solutions to polynomial-like iterative equations on open intervals. (English) Zbl 1223.39014

The first part of this article deals with nondecreasing convex solutions to the polynomial-like iterative functional equation

λ 1 f(x)+λ 2 f 2 (x)++λ n f n (x)=F(x),(1)

(f(·):SS (S), f 0 (x)=x, f j (x)=f(f j-1 (x))). Under assumptions λ 1 >0 λ 2 ,,λ n 0 and the existence of continuous nondecreasing convex functions g,hS satisfying inequalities

λ 1 g(x)+λ 2 g 2 (x)++λ n g n (x)F(x),
λ 1 h(x)+λ 2 h 2 (x)++λ n h n (x)F(x),

there exists at least one continuous nondecreasing convex solution f(x) such that g(x)f(x)h(x). The second part of the article presents the analoguous result for p-convex (p-1) solutions of (1) (a function f:S is called convex of order p or p-convex, if

[x 0 ,x 1 ,,x p+1 ;f]0,x 0 <x 1 <<x p+1 S,

([x 0 ,x 1 ,,x p+1 ;f] denotes the divided difference of f at the points x 0 ,x 1 ,,x p+1 ).

39B12Iterative and composite functional equations
39B22Functional equations for real functions
26A18Iteration of functions of one real variable
47H10Fixed point theorems for nonlinear operators on topological linear spaces
[1]Agarwal R.P., Meehan M., O’Regan D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001)
[2]Baron K., Jarczyk W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequationes Math. 61, 1–48 (2001) · Zbl 0972.39011 · doi:10.1007/s000100050159
[3]Breckner W.W.: Equicontinuous families of generalized convex mappings. Math. Rev. Anal. Numér. Théor. Approx., Sér. Math. 26(49), 9–20 (1984)
[4]Breckner W.W., Trif T.: On the singularities of certain families of nonlinear mappings. Pure Math. Appl. 6, 121–137 (1995)
[5]Breckner W.W., Trif T.: Convex Functions and Related Functional Equations, Selected Topics. Cluj University Press, Cluj-Napoca (2008)
[6]Chen J., Zhang W.: Leading coefficient problem for polynomial-like iterative equations. J. Math. Anal. Appl. 349, 413–419 (2009) · Zbl 1152.39017 · doi:10.1016/j.jmaa.2008.09.015
[7]Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series and Products. Academic Press, New York (1980)
[8]Jarczyk W.: On an equation of linear iteration. Aequationes Math. 51, 303–310 (1996) · Zbl 0872.39010 · doi:10.1007/BF01833285
[9]Kuczma, M.: An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality. Second edn. Edited and with a preface by Attila Gilányi. Birkhäuser, Basel (2009)
[10]Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Encyclopedia Math. Appl. 32, Cambridge University Press, Cambridge (1990)
[11]Neumann M.M.: Uniform boundedness and closed graph theorems for convex operators. Math. Nachr. 120, 113–125 (1985) · Zbl 0582.46006 · doi:10.1002/mana.19851200111
[12]Rassias Th.M., Trif T.: Log-convex solutions of the second order to the functional equation f(x + 1) = g(x)f(x). J. Math. Anal. Appl. 331, 1440–1451 (2007) · Zbl 1122.39017 · doi:10.1016/j.jmaa.2006.09.060
[13]Roberts A.W., Varberg D.E.: Convex Functions. Academic Press, New York (1973)
[14]Tabor J., Tabor J.: On a linear iterative equation. Results Math. 27, 412–421 (1995)
[15]Xu B., Zhang W.: Construction of continuous solutions and stability for the polynomial-like iterative equation. J. Math. Anal. Appl. 325, 1160–1170 (2007) · Zbl 1111.39020 · doi:10.1016/j.jmaa.2006.02.065
[16]Xu B., Zhang W.: Decreasing solutions and convex solutions of the polynomial-like iterative equation. J. Math. Anal. Appl. 329, 483–497 (2007) · Zbl 1114.39007 · doi:10.1016/j.jmaa.2006.06.087
[17]Zhang W.: Discussion on the iterated equation i=1 n λ i f i (x)=F(x) . Chinese Sci. Bull. 32, 1444–1451 (1987)
[18]Zhang W.: Discussion on the differentiable solutions of the iterated equation i=1 n λ i f i (x)=F(x) . Nonlinear Anal. 15, 387–398 (1990) · Zbl 0717.39005 · doi:10.1016/0362-546X(90)90147-9
[19]Zhang W., Nikodem K., Xu B.: Convex solutions of polynomial-like iterative equations. J. Math. Anal. Appl. 315, 29–40 (2006) · Zbl 1090.39012 · doi:10.1016/j.jmaa.2005.10.006