Smooth solutions for the $p$-order functional equation \(f(\varphi(x))=\varphi^p(f(x))\)
-
1666
Downloads
-
3197
Views
Authors
Min Zhang
- College of Science, China University of Petroleum, Qingdao, Shandong 266580, P. R. China.
Jie Rui
- College of Science, China University of Petroleum, Qingdao, Shandong 266580, P. R. China.
Abstract
This paper deals with the \(p\)-order functional equation
\[\left\{
\begin{array}{ll}
f(\varphi(x))=\varphi^p(f(x)),\\
\varphi(0)=1, \quad -1\leq \varphi(x)\leq1 , \quad x\in[-1,1],
\end{array}
\right.
\]
where \(p\geq 2\) is an integer, \(\varphi^p\) is the \(p\)-fold iteration of
\(\varphi\), and \(f(x)\) is smooth odd function on
\([-1,1]\) and satisfies \(f(0)=0, -1<f^{'}(x)<0, (x\in[-1,1]).\)
Using constructive method, the existence of
unimodal-even-smooth solutions of the above equation on \([-1,1]\) can be proved.
Share and Cite
ISRP Style
Min Zhang, Jie Rui, Smooth solutions for the $p$-order functional equation \(f(\varphi(x))=\varphi^p(f(x))\), Journal of Nonlinear Sciences and Applications, 10 (2017), no. 8, 4418--4429
AMA Style
Zhang Min, Rui Jie, Smooth solutions for the $p$-order functional equation \(f(\varphi(x))=\varphi^p(f(x))\). J. Nonlinear Sci. Appl. (2017); 10(8):4418--4429
Chicago/Turabian Style
Zhang, Min, Rui, Jie. "Smooth solutions for the $p$-order functional equation \(f(\varphi(x))=\varphi^p(f(x))\)." Journal of Nonlinear Sciences and Applications, 10, no. 8 (2017): 4418--4429
Keywords
- Functional equation
- constructive method
- unimodal-even-smooth solution.
MSC
References
-
[1]
F. Y. Chen, Solutions of the three order renormalization group equation, J. Appl. Math. of Chinese Univ., 4 (1989), 108–116.
-
[2]
P. Coullet, C. Tresser, Iterations d’endomorphismes et groupe de renormalisation, J. Phys. Colloque., 39 (1978), 5–25.
-
[3]
J. P. Eckmann, H. Epstein, P. Wittwer, Fixed points of Feigenbaum’s type for the equation \(f^p(\lambda x) = \lambda f(x)\), Commun. Math. Phys., 93 (1984), 495–516.
-
[4]
J. P. Eckmann, P. Wittwer, A complete proof of the Feigenbaum conjectures, J. Stat. Phys., 46 (1987), 455–475.
-
[5]
H. Epstein, Fixed point of composition operators II, Nonlinearity, 2 (1989), 305–310.
-
[6]
M. J. Feigenbaum, Quantitative universality for a class of non-linear transformations, J. Stat. Phys., 19 (1978), 25–52.
-
[7]
M. J. Feigenbaum, The universal metric properties of non-linear transformations, J. Stat. Phys., 21 (1979), 669–706.
-
[8]
G. F. Liao, On the Feigenbaum’s functional equation \(f^p(\lambda x) = \lambda f(x)\), Chinese Ann. Math., 15 (1994), 81–88.
-
[9]
P. J. McCarthy, The general exact bijective continuous solution of Feigenbaum’s functional equation, Commun. Math. Phys., 91 (1983), 431–443.
-
[10]
D. Sullivan, Boubds quadratic differentials and renormalization conjectures, Amer. Math. Soc., 2 (1992), 417–466.
-
[11]
L. Yang, J. Z. Zhang, The second type of Feigenbaum’s functional equation, Sci. Sinica Ser. A, 29 (1986), 1252–1262.
-
[12]
M. Zhang, J. G. Si, Solutions for the p-order Feigenbaum’s functional equation \(h(g(x)) = g^p(h(x))\), Annales Polonici Mathematici, 111 (2014), 183–195.
