Fixed Points of Functions below the Line y = x

Author(s)

Grace Fryling, Baldwin Wallace University
Harrison Rouse, Baldwin Wallace University

Document Type

Article

Publication Date

2022

Publication Source

Proceedings of the 2021 Undergraduate Mathematics Day

Volume

7

Inclusive pages

20-26

Abstract

This paper concerns fixed points of functions whose graphs lie on or below the line y = x. Using the Monotone Convergence Theorem, we show that positive fixed points of such functions are “attracting on the right” so long as we include a couple of further assumptions about these functions near their fixed points. As an illustrative example, we confirm that this is the case for the function y = x sin x; the positive fixed points of this function “attract on the right” and “repel on the left.” Further, we generalize by showing that differentiability is in fact not needed to conclude that a fixed point is attracting on the right. Continuing in this direction, we identify a class of discontinuous functions whose fixed points are attracting on the right.

Keywords

Fixed points of real valued functions, attracting fixed point, repelling fixed point

Disciplines

Mathematics

Comments

Presented at University of Dayton Undergraduate Mathematics Day Nov. 6, 2021.

eCommons Citation

Fryling, Grace and Rouse, Harrison, "Fixed Points of Functions below the Line y = x" (2022). Undergraduate Mathematics Day: Past Content. 44.
https://ecommons.udayton.edu/mth_epumd/44