IJPAM: Volume 46, No. 1 (2008)

ON THE FIXED POINTS OF A FUNCTION AND
THE FIXED POINTS OF ITS COMPOSITE FUNCTIONS

Mohammad K. Azarian
Department of Mathematics
University of Evansville
1800 Lincoln Avenue, Evansville, IN 47722, USA
e-mail: azarian@evansville.edu


Abstract.In this paper we study the relationship between the fixed point(s) of a function $f(x)$ (if they exist), and the fixed point(s) of its composite functions $f^{n}(x)$ for $n\geq2$. We start by showing that if $x_{p}$ is a fixed point of the function $f(x)$, then $x_{p}$ is a fixed point of the composite function $f^{n}(x)$ $(n\geq2)$, and if $f(x)$ is fixed point free, then $f^{n}(x)$ $(n\geq2)$ is fixed point free as well. The functions under our discussion will include polynomials and natural trigonometric functions as well as hyperbolic functions. Finally, we pose ten questions for the reader.

Received: November 11, 2007

AMS Subject Classification: 65-01

Key Words and Phrases: fixed point, fixed point free, attractive fixed point, elliptic fixed point, group fixed point, hyperbolic fixed point, parabolic fixed point, fixed point of a composite function, idempotent function, natural trigonometric function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2008
Volume: 46
Issue: 1