×

zbMATH — the first resource for mathematics

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
\[ \lambda_1f(x) + \lambda_2f^2(x) + \dots +\lambda_n f^n(x) = F(x),\tag{1} \]
(\(f(\cdot):\;S \to S\) (\(S \subset {\mathbb R}\)), \(f^0(x) = x\), \(f^j(x) = f(f^{j-1}(x))\)). Under assumptions \(\lambda_1 > 0\) \(\lambda_2, \dots, \lambda_n \leq 0\) and the existence of continuous nondecreasing convex functions \(g,h \in S\) satisfying inequalities
\[ \lambda_1g(x) + \lambda_2g^2(x) + \dots + \lambda_n g^n(x) \leq F(x), \]
\[ \lambda_1h(x) + \lambda_2h^2(x) + \dots +\lambda_n h^n(x) \geq F(x), \]
there exists at least one continuous nondecreasing convex solution \(f(x)\) such that \(g(x) \leq f(x) \leq h(x)\). The second part of the article presents the analoguous result for \(p\)-convex (\(p \geq -1\)) solutions of (1) (a function \(f:\;S \to {\mathbb R}\) is called convex of order \(p\) or \(p\)-convex, if
\[ [x_0,x_1,\dots,x_{p+1};f] \geq 0, \qquad x_0 < x_1 < \dots < x_{p+1} \in S, \]
(\([x_0,x_1,\dots,x_{p+1};f]\) denotes the divided difference of \(f\) at the points \(x_0,x_1,\dots,x_{p+1}\)).

MSC:
39B12 Iteration theory, iterative and composite equations
39B22 Functional equations for real functions
26A18 Iteration of real functions in one variable
47H10 Fixed-point theorems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agarwal R.P., Meehan M., O’Regan D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001) · Zbl 0960.54027
[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) · Zbl 0852.47034
[5] Breckner W.W., Trif T.: Convex Functions and Related Functional Equations, Selected Topics. Cluj University Press, Cluj-Napoca (2008) · Zbl 1202.26001
[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) · Zbl 0521.33001
[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) · Zbl 1221.39041
[10] Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations. Encyclopedia Math. Appl. 32, Cambridge University Press, Cambridge (1990) · Zbl 0703.39005
[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) · Zbl 0271.26009
[14] Tabor J., Tabor J.: On a linear iterative equation. Results Math. 27, 412–421 (1995) · Zbl 0831.39006
[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 \({\sum_{i=1}^n\lambda_i f^i(x)=F(x)}\) . Chinese Sci. Bull. 32, 1444–1451 (1987) · Zbl 0639.39006
[18] Zhang W.: Discussion on the differentiable solutions of the iterated equation \({\sum_{i=1}^n\lambda_i f^i(x)=F(x)}\) . 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.