zbMATH — the first resource for mathematics

Convex solutions of polynomial-like iterative equations. (English) Zbl 1090.39012
The authors study the following polynomial-like iterative functional equation $\lambda_1 f(x)+\lambda_2 f^2(x)+\cdots+\lambda_n f^n(x)=F(x),$ where $$F:I=[a,b] \to I$$ is given, $$f$$ is the unknown function and $$\lambda_1>0$$, $$\sum_{i=1}^n \lambda_i=1$$. Under suitable hypothesis for $$\lambda_i$$’s and Lipschitzianity of $$F$$, they obtain existence theorems for continuous and for convex solutions.

MSC:
 39B12 Iteration theory, iterative and composite equations 39B22 Functional equations for real functions
Full Text:
References:
 [1] 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 [2] Collet, P.; Eckmann, J.P.; Lanford, O.E., Universal properties of maps on an interval, Comm. math. phys., 76, 211-254, (1980) · Zbl 0455.58024 [3] Dhombres, J.G., Itération linéaire d’ordre deux, Publ. math. debrecen, 24, 177-187, (1977) · Zbl 0398.39006 [4] Dubbey, J.M., The mathematical work of charles babbage, (1978), Cambridge Univ. Press Cambridge · Zbl 0376.01002 [5] Jarczyk, W., On an equation of linear iteration, Aequationes math., 51, 303-310, (1996) · Zbl 0872.39010 [6] Kuczma, M., An introduction to the theory of functional equations and inequalities, (1985), PWN/Uniwersytet Śla̧ski Warszawa [7] Kuczma, M.; Choczewski, B.; Ger, R., Iterative functional equations, Encyclopedia math. appl., vol. 32, (1990), Cambridge Univ. Press Cambridge · Zbl 0703.39005 [8] Kulczycki, M.; Tabor, J., Iterative functional equations in the class of Lipschitz functions, Aequationes math., 64, 24-33, (2002) · Zbl 1009.39021 [9] Matkowski, J.; Zhang, W., On the polynomial-like iterative functional equation, (), 145-170 · Zbl 0976.39014 [10] Matkowski, J.; Zhang, W., On linear dependence of iterates, J. appl. anal., 6, 149-157, (2000) · Zbl 0972.39012 [11] Mukherjea, A.; Ratti, J.S., On a functional equation involving iterates of a bijection on the unit interval, Nonlinear anal., 7, 899-908, (1983) · Zbl 0518.39005 [12] Mukherjea, A.; Ratti, J.S., A functional equation involving iterates of a bijection on the unit interval II, Nonlinear anal., 31, 459-464, (1998) · Zbl 0899.39005 [13] Nabeya, S., On the functional equation $$f(p + q x + r f(x)) = a + b x + c f(x)$$, Aequationes math., 11, 199-211, (1974) · Zbl 0289.39003 [14] Ng, C.T.; Zhang, W., Invariant curves for planar mappings, J. difference equ. appl., 3, 147-168, (1997) · Zbl 0891.34070 [15] Rice, R.E.; Schweizer, B.; Sklar, A., When is $$f(f(z)) = a z^2 + b z + c$$, Amer. math. monthly, 87, 252-263, (1980) · Zbl 0441.30033 [16] Roberts, A.W.; Varberg, D.E., Convex functions, (1973), Academic Press New York · Zbl 0289.26012 [17] Tabor, J.; Tabor, J., On a linear iterative equation, Results math., 27, 412-421, (1995) · Zbl 0831.39006 [18] Yang, D.; Zhang, W., Characteristic solutions of polynomial-like iterative equations, Aequationes math., 67, 80-105, (2004) · Zbl 1060.39019 [19] 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 [20] 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 [21] Zhang, W., Solutions of equivariance for a polynomial-like iterative equation, Proc. roy. soc. Edinburgh sect. A, 130, 1153-1163, (2000) · Zbl 0983.39010 [22] Zhao, L., Theorems on existence and uniqueness for the functional equation $$\lambda_1 f(x) + \lambda_2 f^2(x) = F(x)$$, J. chin. univ. sci. tech., Math. Issue, 21-27, (1983), (in Chinese)
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.