Abstract
The main objective of this paper is to study the behavior of the rational difference equation of the fifth-order , where , and are real numbers and the initial conditions and are positive real numbers such that . Also, we obtain the solution of some special cases of this equation.
1. Introduction
In recent years, there has been a great interest in studying the rational difference equations. These equations describe real life situations in stochastic time series, combinatorial analysis, electrical network, number theory, biology, genetics, probability theory, physics, ecology, statistical problems, and economics, for example [1–5]. It is so important to investigate the asymptotic behavior of solutions of a nonlinear difference equations and to discuss the boundedness, periodicity, and stability (local and global) of their equilibrium points; see [6–36] and references therein.
Kalabušić et al. [6] investigated the periodic nature, the boundedness character, and the global asymptotic stability of solutions of the difference equation
In [7], Elabbasy et al. got the solution and the periodicity character of the recursive sequence
Cinar [8] found the solution of the difference equation
In [9] Obaid et al. studied the global stability, boundedness, and the periodicity of solutions of the rational difference equation
Elsayed et al. [10] studied the dynamical analysis of rational difference equation
Elabbasy et al. [11] obtained the global behavior of the solutions of the difference equation
Aloqeili [12] investigated the dynamics of the difference equation
The aim of this paper is to study some qualitative behavior of the positive solutions of the difference equationwhere , and are real numbers and the initial conditions , and are positive real numbers such that .
2. Preliminaries
Let be some interval of real numbers and let be a continuously differentiable function. Then for every set of initial conditions , the difference equationhas a unique solution .
Definition 1 (equilibrium point). A point is called an equilibrium point of the difference equation (10) if That is, for is a solution of the difference equation (10) or, equivalently, is a fixed point of
Definition 2 (stability). Let be an equilibrium point of the difference equation (10). Then, we have the following:
(i) The equilibrium point of the difference equation (10) is called locally stable if for every ,there exists such that for all with we have (ii) The equilibrium point of the difference equation (10) is called locally asymptotically stable if is locally stable solution of (10) and there exists , such that, for all with we have (iii) The equilibrium point of the difference equation (10) is called global attractor if for all , we have (iv) The equilibrium point of the difference equation (10) is called globally asymptotically stable if is locally stable, and is also a global attractor of the difference equation (10).
(v) The equilibrium point of the difference equation (10) is called unstable if is not locally stable.
Definition 3 (periodicity). A sequence is said to be periodic with period if for all A sequence is said to be periodic with prime period if is the smallest positive integer having this property.
Definition 4 (Fibonacci sequence). The sequence , i.e., , is called Fibonacci sequence.
Definition 5. The linearized equation of the difference equation (10) about the equilibrium is the linear difference equationNow, assume that the characteristic equation associated with (17) iswhere
Theorem 6 (see [1]). Assume that , , and is nonnegative integer. Then is a sufficient condition for the asymptotic stability of the difference equation
Theorem 7 (see [2]). Let be a continuous function and be an interval of real numbers. Consider the difference equationAssume that satisfies the following conditions:
(i) is nondecreasing in and in for all and nonincreasing in for all and in
(2) If is a solution of the system then Then (22) has a unique equilibrium point and every solution of (22) converges to .
3. Dynamics of (8)
In this section, we study the local stability, global stability of the solutions, and the boundedness of
where , and are positive real numbers.
3.1. Local Stability of the Equilibrium Point
In this subsection, we study the local stability of the equilibrium point of (8).
Equation (8) has a unique equilibrium point and is given by or If , then the only equilibrium point is .
Theorem 8. LetThen the equilibrium point of (8) is locally asymptotically stable.
Proof. Let be a continuous function defined by Therefore, it follows that Thenand the linearized equation of (8) about is It follows by Theorem 6 that (8) is asymptotically stable if and only if and so and, thus, The proof is complete.
Example 9. Figure 1 shows that the zero solution of the difference Equation (8) is local stability if , , , and and the initial conditions , , , , and .
3.2. Global Stability of the Equilibrium Point
In this subsection we study the global stability of the positive solutions of (8).
Theorem 10. The equilibrium point of (8) is global stability if
Proof. Let and be nonnegative real numbers and assume that is a function defined by Then we can see that the function is increasing in and and decreasing .
Assume that is a solution of the system Then from (8), we see thatand then Subtracting these two equations, we obtain and if , then we see that
According to Theorem 7 the equilibrium point is a global attractor of (8). The proof is complete.
Example 11. The zero solution of the difference Equation (8) is global stability if , , , and and the initial conditions , , , , and (See Figure 2).
3.3. Existence of Boundedness and Unboundedness Solutions
Here we look at the boundedness and unboundedness solutions of (8).
Theorem 12. Every solution of (8) is bounded if
Proof. Let be a solution of (8). It follows from (8) that Then Then the sequence is decreasing and so is bounded from the above by .
Theorem 13. Every solution of (8) is unbounded if
Proof. Let be a solution of (8). It follows from (8) that We see that the right hand side can be written as follows: and this equation is unstable because and Then by using ratio test is unbounded from above.
Example 14. Figure 3 shows that (8) is bounded when , , , and and the initial conditions , , , , and .
Example 15. Figure 4 shows that (8) is unbounded when , , , and and the initial conditions , , , , and .
4. Special Cases of (8)
In this section we investigate the following special case:
where , , , and are integrals numbers.
4.1. First Case When
Theorem 16. Suppose that is a solution of rational difference equationThen, for , we see that where , , , , , , and
Proof. For the result holds. Now, suppose that and that our assumption holds for . That is,From (48), we see that Also, we see from this that Also, This completes the proof.
4.2. Second Case When and
Theorem 17. Let be a solution of rational difference equationThen, for , we see that where are arbitrary nonzero real numbers and , .
Proof. For the result holds. Now, suppose that and that our assumption holds for . That is, Now, it follows from (55) thatAlso, we see that Also, andThis completes the proof.
Example 18. Figure 5 illustrates the solutions of (55) when the initial conditions , , , , and .
4.3. Third Case When ,
Theorem 19. Assume that is solutions of rational difference equationand then for , we see that where , , , , and are arbitrary nonzero real numbers with , , , and
Proof. The proof of Theorem 16 will be omitted.
4.4. Fourth Case When and
Theorem 20. Suppose that is solutions of rational difference equationand then for , we see that where , and are arbitrary nonzero real numbers with .
Proof. For the result holds. Now, suppose that and that our assumption holds for . That is,Now, it follows from (64) that This completes the proof.
Theorem 21. Every solution of rational difference equation (64) is periodic with period 12.
Proof. The proof follows from the previous theorem and will be omitted.
Example 22. Figure 6 shows that the solution of (64) has a periodic solution with period 12 when the initial conditions , , , and .
4.5. Fifth Case When and
Theorem 23. Suppose that is solutions of rational difference equationThen every solution of (68) is periodic with period 12. Moreover takes the form where , , , , and are arbitrary nonzero real numbers with , , , and .
Proof. The proof is left to the reader.
Example 24. Figure 7 shows that the solution of (68) has a periodic solution with period 12 when the initial conditions , , , , and .
4.6. Sixth Case When and
Theorem 25. Suppose that is solutions of rational difference equationand then for , we see that where , , , , , , , , and
4.7. Seventh Case When and
Theorem 26. Suppose that is solutions of rational difference equationand then for , we see that where , and are arbitrary nonzero real numbers and ,
4.8. Eighth Case When and
Theorem 27. Suppose that is solutions of rational difference equationand then, for , we see that where , , , and .
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.