Global stability of yn + 1 = A + ynyn−k

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

“…More generally, it follows from Theorem 5.3 in [7] that if k is even and m is odd, then every positive solution of (1) converges to a nonnegative periodic solution with period 2 gcd(m, k). For a discussion of related equations, see also [1], [2], [3], [4], [6] and [11]. Here we prove the following complimentary result which answers the question when y = 2 is a global attractor.…”

“…In this section we combine a recent theorem of Grove and Ladas with the result in Theorem 1 to determine the periodic character of equation (1).…”

Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

“…In Figure 21 trajectories of solutions are shown for p 2, q 1 the point E in Figure 16 with the initial conditions x −1 1.5, x 0 1 and x −1 x 0 −0.78. One can see that the equilibrium point x 1 1.281 red trajectories is stable and the equilibrium point x 2 −0.781 green trajectories is unstable. In Figure 22 trajectories of solutions are shown for p 7, q 2 the point F in Figure 16 with the initial conditions x −1 1.5, x 0 1.9 and x −1 −1.4, x 0 −1.3.…”

The global attractivity of the rational difference equation 𝑦_{𝑛}=𝐴+(\frac{𝑦_{𝑛-𝑘}}𝑦_{𝑛-𝑚})^{𝑝}

“…There is a relatively long history in studying equation (1). For example, for p = 1, the case k = 2, m = 1 was studied in [2] by Amleh et al, the case k ∈ N, m = 1 was studied by DeVault et al in [11], and the case A > 1, k = 1, m ∈ N was studied by Stević in [20].…”

“…The existence of monotone solutions, for the case p > 0 and A > −1 was shown in [5] by developing the technique from [6,7,8,9,10,24] and [25]. Equations in papers [4] and [12] were investigated by transforming them into some special cases of equation (1).…”