On the global attractivity and the periodic character of some difference equations

The global attractivity of the rational difference equation $y_{n}=1+\frac{y_{n-k}}{y_{n-m}}$

“…First, note that (as in [5]), if g = gcd(m, k) > 1, then {y i } can be separated into g different equations of the form…”

On the oscillation and periodic character of a third order rational difference equation

“…Here we prove this conjecture using a general convergence result established in [3]. We also prove that every nontrivial solution of (1.2) oscillates.…”

“…For rational difference equations of order greater than two, several results have been obtained in [1,2,3,4,5]. Recently, in [7], G. Ladas conjectured that every positive solution of the third order rational difference equation…”

“…For the proof of our main convergence result in Section 4, we will need the following global convergence theorem established in [3] (see also [2] The proof of this theorem is based on the method of full limiting sequences developed by G. Karakostas [4], and it may be found in [3]. It was applied there to investigate the periodic character of the difference equation…”

Qualitative Properties for a Fourth Order Rational Difference Equation

“…The long term behavior of the solutions of nonlinear difference equations of order greater than one has been extensively studied during the last decade. For example, various results about boundedness, stability, and periodic character of the solutions of the second-order nonlinear difference equation see [17,18,21]. Many researchers have investigated the behavior of the solution of difference equations for example: Elabbasy et al [9,12,13] investigated the global stability, boundedness, periodicity character and gave the solution of some special cases of the difference equations…”