Abstract
In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, global behavior of equilibrium points, boundedness and periodicity of the rational recursive sequence , where for . With initial values are positive real numbers. Some numerical examples are given to verify our theoretical results.
1. Introduction
It is very amusing to explore the nature of the solutions of a higher-order rational difference equation and to explain the local asymptotic stability of its equilibrium points. The inspection of some properties associated with these equations is a very enormous activity. Discrete dynamical systems or difference equations are diverse fields because various biological systems and models naturally lead to their study by means of a discrete variable. Applications of discrete dynamical systems and difference equations have appeared recently in many fields of science and technology. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Rational difference equations are a special form of nonlinear difference equations. Delay difference equations have rich dynamics to study. Due to adequate computational outcomes, discrete dynamical systems are awful lot better than allied structures in differential equations. Specifically, in case of nonoverlapping generations, difference equations are greater apposite to take a look at the behavior of population models [1, 2]. Also, an epidemiological approach to insurgent population modeling is mentioned in [3, 4]. The research of delay difference equations is a divergent field that involves most aspects of mathematics, including both applied and pure. Recently, there has been a symbolic development in the applications of difference equations. It is very interesting to investigate the behavior of solutions of a system of nonlinear difference equations and to discuss the local asymptotic stability of their equilibrium points. There are many papers in which systems and behavior of rational difference equations have been studied see [5–20]. Rational difference equations have been studied by several authors. There has been a great interest exclusively in the study of the attractiveness of the solution of such equations.
Here, we recall some basics and preliminaries that will be useful in our results and investigation.
2. Preliminaries and Definitions
Definition 1. Let be some interval of real numbers, and letbe a continuously differentiable function. Then, for every set of initial conditions , the difference equationhas a unique solution .
Definition 2. A point is called an equilibrium point of equation (2) ifThat is, , for all is a solution of equation (2), or equivalently, is a fixed point of .
Definition 3. (periodicity). A sequence is said to be periodic with period if for all .
Definition 4. (Fibonacci sequence). The sequence is called Fibonacci sequence.
Definition 5. Equation (2) is called permanent and bounded if there exist numbers and with such that for any initial values , there exists a positive integer which depends on these initial values such that for all .
Definition 6. (stability)(i)The equilibrium point of equation (2) is locally stable if for every there exists such that for all with we have(ii)The equilibrium point of equation (2) is locally asymptotically stable if is a locally stable solution of equation (2) and there exists , such that for all , with we have(iii)The equilibrium point of equation (2) is global attractor if for all , we have(iv)The equilibrium point of equation (2) is globally asymptotically stable if is locally stable and is also a global attractor of equation (2).(v)The equilibrium point of equation (2) is unstable if is not locally stable.
Definition 7. The linearized equation of equation (2) about the equilibrium is defined by the following equation:where
Lemma 1 (see[21]). Assume that and . Then,is a sufficient condition for the asymptotic stability of the following difference equation:
The following theorem will be useful for the proof of our results in this paper.
Theorem 1. Let be an interval of real numbers and assume that is a continuous function, and consider the following equation:satisfying the following conditions:(a) is nondecreasing in for each fixed , and is nonincreasing in for each fixed (b)If is a solution of the systems then and equation (2) has a unique equilibrium and every solution of equation (2) converges to .Some published works include the following:
Ibrahim and El-Moneam [22] investigate the local and global stability of the following recursive sequencewhere for , and are real numbers.
Aboutaleb et al. [23] studied the global attractiveness of the positive equilibrium of the following rational recursive equation:where the coefficients are the nonnegative real numbers.
Several other researchers have studied the behavior of the solution of difference equations; for example, Devault et al. [24] investigated the global behavior of all positive solutions of the following equation:Elabbasy et al. [25] studied the boundedness, global stability, and periodicity character and gave the solution of some special cases of the following difference equation:Elabbasy [26] gave the solution of the following difference equation:El-Moneam and Zayed [27] investigated the global stability and periodicity character and gave the solution of some special cases of the following difference equation:Zayed and El-Moneam [28] studied the global behavior of the following rational recursive sequence:Motivated by the above study, the main focus of this article is to discuss some qualitative behaviors of the solutions of the following rational recursive sequence:where for With initial values are the positive real numbers. Some numerical examples are given to verify our theoretical results.
3. Local Stability of Equation (22)
This section deals with the local stability character of the solutions of equation (22). Equation (22) has a unique equilibrium point, and it is given fromif
Then, the unique equilibrium point is .
Let be a function defined by
Therefore, it follows thatWe see that
The linearized equation of equation (22) about is
The characteristic equation of equation (28) is
Theorem 2. Assume thatThen, the equilibrium point of equation (22) is locally asymptotically stable.
Proof. It follows by Lemma 7 that equation (28) is asymptotically stable ifand soHence,
4. Boundedness of the Solutions
In this section, we show the boundedness of the positive solutions of equation (22).
Theorem 3. Every solution of equation (22) is bounded from above byif
Proof. Let be a solution of equation (22). It follows from equation (22) thatThen, for all Then, all the subsequences , are decreasing and so are bounded from above by
Theorem 4. If , then each solution of equation (22) will be unbounded.
Proof. Let be a solution of equation (22). Then, from equation (22), we see thatFrom above, the right-hand side can be written as follows:and this equation is unstable because and .
Then, by using the ratio test, is unbounded from above.
5. Periodic Solutions of Equation (22)
In this section, we satisfy the periodic solutions of equation (22).
Theorem 5. Equation (22) has no positive solutions of prime period two solution in the following case, if p is even and r is odd.
Proof. Assume that there exists two distinct positive real numbers such that , , is prime period two solution of equation (22). We haveBy simplifying equations (41) and (42), we obtainBy subtracting, we deduce thatThis implies that equation (22) has no positive solutions of prime period two.
Theorem 6. Equation (22) has positive period two solution in the following cases:(i)(ii)(iii)
Proof. We will prove case (i) and the other cases (ii) and (iii) are obtained by the same way. Assume that there exist two distinct positive real numbers such that , , is prime period two solution of equation (22):From equation (45) and equation (46), we obtainBy solving these equations, we obtainSince are nonzero positive real numbers, and
This implies that
6. Global Stability
We will study the global asymptotic stability of the positive solutions of equation (22) in this section.
Theorem 7. The equilibrium point of equation (22) is a global attractor if
Proof. Let and are real numbers and assume that be function defined by . Thus, we see that the function is increasing in and is decreasing in . Let is a solution of the system and . Then, from (22), we see thatThen,By subtracting these equations, we obtainThus,It follows that the equilibrium point of equation (22) is a global attractor.
7. Applications
In this section, we will discuss the solution of some special cases of equation (22).
Case 1. When . In this case, we have the following special type of difference equation:
Theorem 8. The general solution of equation (56) iswhere
Proof. Proof is obtained by induction, and it is easy to do.
Case 2. When . In this case, we have the following special case of difference equation:where , and are the arbitrary real numbers.
Theorem 9. Let be a solution of equation (59). Then, for ,where , , and .
Proof. For , the result holds. Now, suppose that and that our assumption holds for :Now, it follows from (59) thatThus, we haveHence, the proof is completed.
Similarly, we can prove other relations.
Case 3. When In this case, we have the following special case of difference equation:The following theorem gives the solution of equation (64).
Theorem 10. Let be a solution of equation (64). Then, the solution for equation (64) is given bywhere and initial values are the arbitrary positive real numbers.
Proof. For Hence, the relation holds true. Now, suppose that relation (64) holds for .
That is,We now want to show that relation (64) holds true for :
Case 4. When In this case, we have the following special case of difference equation:where , and are the arbitrary real numbers.
Theorem 11. Let be a solution of equation (69). Then, for ,where , , and .
Proof. For , the result holds. Now, suppose that and that our assumption holds for :Now, it follows from (69) thatHence, the proof is complete.
By using the same way, we can prove other relations.
8. Numerical Examples
This section discusses some numerical results of our previous results.
Example 1. Figure 1 shows the behavior of the solution bounded of equation (56) when we take
Example 2. Figure 2 shows the behavior of solution of equation (59) when we take
Example 3. Figure 3 shows that the solution of equation (64) is globally asymptotically stable when we take
Example 4. Figure 4 shows the solution of equation (69) has no prime period two solution if we take
9. Concluding Remarks
In the literature, several articles are related to qualitative behavior of rational difference equations. It is a very interesting mathematical problem to study the dynamics of such equations because these are closely related to models in population dynamics and biological sciences. We have investigated the existence and uniqueness of positive equilibrium points, and boundedness and persistence of positive solutions are proved. Moreover, we have shown that they are locally as well as globally asymptotically stable. The main objective of dynamical systems theory is to predict the global behavior of a system based on the knowledge of its present state. We have taken some special cases as applications of equation (22) in Section 7 and found the closed form of the solutions. Finally, some illustrative examples are provided to support our theoretical discussion.
Data Availability
All the data utilized are included in this article and their sources are cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest.