# Global behavior of some nonlinear difference equations

#### Abstract

Our goal is to study the global properties of some nonlinear rational difference equations. ^ Consider the nonlinear rational difference equation: yn+1=pyn-1+yn-2 q+yn-2,n=0,1,&ldots; 0.1 with positive parameters and non-negative initial conditions. We investigate the global behavior of solutions of Eq.(0.1). We obtain a precise description of the regions of parameters where the equilibrium is globally asymptotically stable, the regions where there exist unbounded solutions, and the regions where all solutions converge to period two solutions. ^ Consider the delay difference equation: yn+1=p+yn-2k+1 1+yn-2l,n =0,1,&ldots; 0.2 where *k* and *l* are non-negative integers, *p* is a positive parameter, and the initial conditions are non-negative real numbers. We prove that every positive solution converges to a periodic solution with period 2*d* where *d* is the greatest common divisor of *k* + 1 and 2*l* + 1. ^ The extensions of Eq.(0.1) to equations with higher delays play an important role in the development of the basic theory of nonlinear difference equations. ^ We investigate the global stability, the periodic character and the boundedness nature of solutions of the difference equation: yn+1=a+gyn- 2k+1+dyn-2l A+yn-2l,n=0,1,&ldots; 0.3 where *k* and *l* are non-negative integers, the parameters a,g , δ, *A* are non-negative real numbers with a+g + δ > 0, and the initial conditions are non-negative real numbers. We show that the solutions exhibit a trichotomy character depending upon how γ compares with δ + *A*. ^

#### Subject Area

Mathematics

#### Recommended Citation

Mihaela Predescu,
"Global behavior of some nonlinear difference equations"
(2003).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI3103720.

http://digitalcommons.uri.edu/dissertations/AAI3103720