Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Faith Grice - Texas Southern University
Co-Author(s): Willie E. Taylor, Texas Southern University, Houston, TX
The equation x_(n+1)=(f(x_n))/x_(n-1) (E1) has been studied by many authors, but few of them focus on specific functions for it. To address this issue, this research investigates the solutions of (E1), along with its linearized form y_(n+1)=F(y_n )-y_(n-1) (E2) specifically, where f and F is constructed using a maximum operator. Various conditions are given on f and F so that the solutions of (E1) and (E2) have periods 5, 7, and 9. While both (E1) and (E2) are second order difference equations, (E1) will be nonlinear, but (E2) will involve only linear terms and will be called a maxilinear difference equation. The conditions studied on f include f(x)= max{1,x}, f(x)=max{1,1/x}, and f(x)=max{x,1/x}, while the conditions studied on F include F(y)=max{0,y_n}, F(y)=max{0,-y_n}, and F(y)=max{y_n,-y_n}. For each of these conditions, three specific cases were examined. Assuming that x_(-1)=a and x_0=b, the following cases were tested for f: 0 Funder Acknowledgement(s): Texas Southern University College of Science, Engineering and Technology.
National Science Foundation.
Louis Stokes Alliances for Minority Participation.
L.L. Clarkson.
Faculty Advisor: Willie Taylor,
taylor_we@tsu.edu
Role: I conducted the research. I was given the second-order difference equation to study along with the various functions used, and I calculated and recorded the general solutions and solution behaviors. I also chose the cases and sub-cases used to cover all solution behaviors.