Lothar Berg

On the Difference Equation\newline $\boldsymbol{x_{n+1}=(\beta x_n+\gamma x_{n-1})/(\gamma x_n+\beta x_{n-1})}$

The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 61, 3 - 11 (2006)

MSC: 39A11   Stability and asymptotics of difference equations; oscillatory and periodic solutions, etc.
  41A60   Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Abstract:   The difference equation in the title is solved by means of functions, which can be represented as composed functions of the exponential function and a function being odd with respect to one or two arbitrary parameters. In the case $\beta=1/4$, $\gamma=3/4$ there is given a conjecture concerning a solution of a new type. A second conjecture concerns the existence of asymptotically 3-periodic solutions. Though the difference equation is of second order, we point out singular cases where three initial values can be prescribed.

Keywords:   Nonlinear difference equations, odd functions, asymptotic behaviour, 3-periodic solutions, three initial values, conjectures.

Seite generiert am 28.09.2006,   11:57   Uhr