In the last twenty years, many papers appeared focusing on the investigation of the qualitative analysis of solutions of difference equations (see [2, 3, 4, 7, 8, 9, 14, 15, 17, 22] and the references cited therein). Applications of difference equations have appeared in many areas such as population dynamics, ecology, economics, probability theory, genetics, psychology, physics, engineering, sociology, statistical problems, stochastic time series, number theory, electrical networks, neural networks, queuing problems and so on. Namely, the theory of difference equations gets a central position in applicable analysis. Hence, it is very valuable to study the dynamical behavior of solutions of non-linear rational difference equations.

In our opinion, it is of a great importance to investigate not only non-linear difference equations, but also those equations which contain powers of arbitrary positive numbers (see [3, 4, 6, 7, 11, 13, 21, 23]).

The purpose of this paper is to study the local asymptotic stability of equilibria, the periodic nature and the global behavior of solutions of the following fractional difference equation

where the parameters α, β, γ, p, q are non-negative numbers and the initial values x₋₂, x₋₁, x₀ are positive numbers such that the denominator is always positive.

In [7], El-Owaidy et al. investigated the global behavior of the following rational recursive sequence

with non-negative parameters and non-negative initial values.

By generalizing the results of El-Owaidy et al. [7], Chen et al. [6] studied the dynamical behavior of the following rational difference equation

where k, l ∈ ℕ, the parameters are positive real numbers and the initial values x_−max{k,l}, …,x₋₁, x₀ ∈ (0,∞).

Ahmed in [3, 4] investigated the global asymptotic behavior and the periodic character of the difference equations

and

where the parameters are non-negative real numbers and the initial values are non-negative real numbers.

In [10], Erdogan et al. investigated the dynamical behavior of positive solutions of the following higher order difference equation

where the parameters are non-negative real numbers and the initial values are non-negative real numbers.

In [16], Karatas investigated the global behavior of the equilibria of the following difference equation

where the parameters are non-negative real numbers and the initial values are non-negative real numbers.

If some parameters of Eq.(1.1) are zero, then special cases emerge. If α = 0, we have the trivial case. If β = 0, Eq.(1.1) is reduced to a linear difference equation by the change of variables x_n= e^y_n. If γ = 0, Eq.(1.1) is reduced to a linear first order difference equation.

Note that Eq.(1.1) can be reduced to the following fractional difference equation

by the change of variables xn=(βγ)1p+qyn with r=αβ. So, we shall study Eq.(1.2).

For the sake of completeness and the readers convenience, we are including some basic results (one can see [1, 5, 12, 18, 19, 20] and the references cited therein).

Let I be an interval of real numbers and let f: I × I × I → I be a continuously differentiable function. Then for any condition x₋₂, x₋₁, x₀ ∈ I, the difference equation

has a unique positive solution {xn}n=-2∞.

Definition 2.1

An equilibrium point of Eq.(2.1) is a point χ̄ that satisfies

The point χ̄ is also said to a fixed point of the function f.

Definition 2.2

Let χ̄ be a positive equilibrium of (2.1).

• χ̄ is stable if for every ɛ > 0, there is δ > 0 such that for every positive solution {xn}n=-2∞ of (2.1) with ∑i=-20∣xi-x¯∣<δ, |x_n− x̄| < ɛ, holds for n ∈ ℕ.

• χ̄ is locally asymptotically stable if x̄ is stable and there is γ > 0 such that lim x_n= x̄ holds for every positive solution {xn}n=-2∞ of (2.1) with ∑i=-20∣xi-x¯∣<γ.

• χ̄ is a global attractor if lim x_n= χ̄ holds for every positive solution {xn}n=-2∞ of (2.1).

• x̄ is globally asymptotically stable if x̄ is both stable and global attractor.

Definition 2.3

The linearized equation of (2.1) about the equilibrium point χ̄ is

where

The characteristic equation of (2.2) is

The following result, known as the Linearized Stability Theorem, is very useful in determining the local stability character of the equilibrium point χ̄ of equation (2.1).

Theorem 2.4. (The Linearized Stability Theorem)

Assume that the function F is a continuously differentiable function defined on some open neighborhood of an equilibrium point χ̄. Then, the following statements are true:

• If all roots of (2.3) have absolute value less than one, then the equilibrium point χ̄ of (2.1) is locally asymptotically stable.

• If at least one of the roots of (2.3) has absolute value greater than one, then the equilibrium point χ̄ of (2.1) is unstable. Also, the equilibrium point χ̄of (2.1) is called a saddle point if (2.3) has roots both inside and outside the unit disk.

Theorem 2.5

Assume that α₂, α₁, and α₀are real numbers. Then, a necessary and sufficient condition for all roots of the equation

to lie inside the unit disk is

In this section we prove our main results.

Theorem 3.1

We have the following cases for the equilibrium points of Eq.(1.2).

• ȳ₀ = 0 is always the equilibrium point of Eq.(1.2).

• Ifr > 1, then Eq.(1.2)has the positive equilibrium y¯1=(r-1)1p+q.

• Ifr < 1 and 1p+qis an even positive integer, then Eq.(1.2)has the positive equilibrium y¯2=(r-1)1p+qwhich is always in the interval (0, 1).

Theorem 3.2

For Eq.(1.2), we have the following results.

• Ifr < 1, then the zero equilibrium point is locally asymptotically stable.

• Ifr > 1, then the zero equilibrium point is locally unstable.

• Ifr = 1, then the zero equilibrium point is non-hyperbolic point.

• Assume thatr > 1 and let q<rr-1(-12+125-4p(r-1r)). Then the positive equilibrium point y¯1=(r-1)1p+qis locally asymptotically stable if either

  (3.1)q≤p

  or

  (3.2)p<q<p+2rr-1.

• Assume thatr ∈ (0, 1) such that 1p+qis an even positive integer. Then the positive equilibrium point y¯2=(r-1)1p+qis unstable.

Theorem 3.3

Every solution of Eq.(1.2) is bounded.

Theorem 3.4

Assume that r > 1. Then the following statements are true:

• Let {yn}n=-2∞be a solution of Eq.(1.2)such that for somen₀ ∈ ℕ, either

  yn>y¯1=r-1p+q         for         n>n0

  or

  yn<y¯1=r-1p+q         for         n>n0.

  Then, forn ≥ n₀ + 2, the sequence {y_n} is monotonic and

  limn→∞yn=y¯1.

• Let {yn}n=-2∞be a non-oscillatory solution of Eq.(1.2)and consider the positive equilibrium ȳ₁. Then, the extreme in each semicycle about ȳ₁occurs at either the second term or the third.

Theorem 3.5

Assume that r < 1, then the zero equilibrium point of Eq.(1.2)is globally asymptotically stable.

Theorem 3.6

If p + 2 ≥ q, then Eq.(1.2)has no prime period-2 solutions. If q > p + 2 and r>q-pq-p-2, then Eq.(1.2)has prime period-2 solutions.