Abstract
We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive iterates of the unknown function at a given point. As an application we solve a generalized problem of Zoltán Boros posed during the 50th ISFE.
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References
Baron K., Jarczyk W.: Recent results on functional equations in a single variable, perspectives and open problems. Aequ. Math. 61, 1–48 (2001)
Boros, Z.: Talk given during the 50th international symposium on functional equations. Aequ. Math. 86, 293 (2013)
Draga S., Morawiec J.: On a Zoltán Boros’ problem connected with polynomial-like iterative equations. Nonlinear Anal. Real 26, 56–63 (2015)
Jarczyk W.: On an equation of linear iteration. Aequ. Math. 51, 303–310 (1996)
Jerri A.J.: Linear Difference Equations with Discrete Transform Methods. Springer, Berlin (1996)
Li L., Zhang W.: Continuously decreasing solutions for polynomial-like iterative equations. Sci. China Math. 56, 1051–1058 (2013)
Matkowski, J.: Remark done during the twenty-sixth international symposium on functional equations. Aequ. Math. 37, 119 (1989)
Matkowski J., Zhang W.: Method of characteristic for functional equations in polynomial form. Acta Math. Sin. 13, 421–432 (1997)
Matkowski J., Zhang W.: Characteristic analysis for a polynomial-like iterative equation. Chin. Sci. Bull. 43, 192–196 (1998)
Morawiec J.: On a functional equation involving iterates and powers. Adv. Differ. Equ. 2014, 271 (2014)
Mukherjea A., Ratti J.S.: On a functional equation involving iterates of a bijection on the unit interval. Nonlinear Anal. 7, 899–908 (1983)
Mukherjea A., Ratti J.S.: On a functional equation involving iterates of a bijection on the unit interval, II. Nonlinear Anal. 31, 459–464 (1998)
Nabeya S.: On the functional equation f(p + qx + rf(x)) = a + bx + cf(x). Aequ. Math. 11, 199–211 (1974)
Ratti J.S., Lin Y.F.: A functional equation involving f and f −1. Colloq. Math. 60/61, 519–523 (1990)
Tabor J., Tabor J.: On a linear iterative equation. Results Math. 27, 412–421 (1995)
Yang D., Zhang W.: Characteristic solutions of polynomial-like iterative equations. Aequ. Math. 67, 80–105 (2004)
Zhang W., Baker A.J.: Continuous solutions of a polynomial-like iterative equation with variable coefficients. Ann. Pol. Math. 73, 29–36 (2000)
Zhang, P., Gong, X.: Continuous solutions of 3-order iterative equation of linear dependence. Adv. Differ. Equ. 2014, 318 (2014)
Zhang W., Xu B., Zhang W.: Global solutions for leading coefficient problem of polynomial-like iterative equations. Results Math. 63, 79–93 (2013)
Zhang J., Yang L., Zhang W.: Some advances on functional equations. Adv. Math. (China) 24, 385–405 (1995)
Zhang W., Zhang W.: On continuous solutions of n-th order polynomial-like iterative equations. Publ. Math. Debr. 76, 117–134 (2010)
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Draga, S., Morawiec, J. Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros’ problem. Aequat. Math. 90, 935–950 (2016). https://doi.org/10.1007/s00010-016-0420-4
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DOI: https://doi.org/10.1007/s00010-016-0420-4