Abstract
In this paper, we explore local stability, attractor, periodicity character, and boundedness solutions of the second-order nonlinear difference equation. Finally, obtained results are verified numerically.
1. Introduction
For decades, the qualitative analysis of difference equations has been steadily increasing. This is due to the fact that difference equations appear as mathematical models in statistical problems, queuing theory, combinatorial analysis, electrical networks, genetics in biology, probability theory, economics, psychology, stochastic time series, sociology, geometry, number theory, etc. Precisely, there is an increasing interest in the qualitative analysis of difference equations. For instance, Devault et al. [1] have explored boundedness, existence of unbounded solutions, persistence, and global attractivity results for following nonautonomous difference equation:where is a positive bounded sequence and initial conditions are positive. Amleh et al. [2] have explored global stability, periodic nature, and boundedness character of the following difference equation:where and are positive constants. DeVault et al. [3] have explored boundedness, periodic character, and global stability of the following difference equation:where , is positive, and initial conditions are arbitrary positive numbers. Berenhaut and Stević [4] have explored the behaviour of the following difference equation:where and . Stević [5] has explored the behavior of the following difference equation:where is a negative number. For more results on the behavior of the difference equation, we refer the reader to recent published articles [6–10] and books [11–13]. Motivated from aforementioned studies, we explore the behavior of the following difference equation:where is a nonnegative real number. Moreover, initial conditions are positive real numbers, and is a nonnegative periodic sequence withwhere .
2. Dynamics of Solutions of Equation (6)
In this study, we consider the following three cases of the function .
2.1. Case 1:
In this case, (6) becomes
It is easy to see that is the only positive fixed point of (8).
Now, the function is defined by
Therefore,
Now,
So, the linearized equation of (8) about is
Theorem 1. (i)If , then of (8) is locally asymptotically stable, and so it is also called a sink(ii)If , then of (8) is unstable and is called a repeller(iii)If , then of (8) is unstable and is called a nonhyperbolic point
Proof. (i)We set and . Now, and also, which is valid if So, by Theorem 1.1.1 (a) and (c) of [13], one can obtain that is locally asymptotically stable when .(ii)Again, and then, . Thus, by Theorem 1.1.1 (d) of [13], is unstable (repeller point) when .(iii)Note that Thus, by Theorem 1.1.1 (e) of [13], is unstable (repeller point) when .
Theorem 2. Positive solution of (8) is bounded and persists if .
Proof. We obtain from (8) thatHence, persists. Then, again, from (8), it follows thatNow consideringIf the solution of (20) with is , thenNow, we have to show that is bounded. Letand then,Therefore, is nondecreasing and concave. Therefore, one gets as the unique fixed point of . Moreover, also satisfiesBy Theorem 2.6.2 of [14], is a global attractor for all positive solutions of (20), and hence, it is bounded. So, from (8), is also bounded.
Theorem 3. Let , and then, (8) has unbounded solutions.
Proof. It is noted that following holds:for every solution of (8). Let . Then, it follows from (25) thatNow, roots ofare given bySince , we have that andTherefore, both roots of are positive if . Moreover, (26) can also written asThen, we see thatIt follows thatChoose and so thatIt follows from this and (32) thatand consequently,It follows by letting in (35) that as , and hence, it follows from this result.
Theorem 4. Let and ; then, of (8) is globally asymptotically stable.
Proof. By Theorem 1 (i), is a sink. Hence, it is enough to prove further that of (8) tends to . Recall that of (8) is bounded by Theorem 2. Thus,Then, from (8), we getNow, claiming that , otherwise, . From (37), we obtainSince holds, thenor equivalentlyIt follows from (38) and (40) thatHence,which is impossible for . This is contradiction, and hence, the result follows.
Theorem 5. Every positive solution of (8) oscillates about with semicycles of length two or three, and extreme of every semicycle occurs at the first or the second term.
Proof. Let the positive solution of (8) is . First, we prove that every positive semicycle except possibly the first term has two or three terms. Assuming and for some , we obtain from (8) thatIf , then we haveOn the contrary, since , we see thatSo, Therefore,
Theorem 6. Equation (8) has no periodic solution having prime period two.
Proof. Letbe a periodic solution of period two of (8). It follows thatwhich implies thatSubstituting from (49) into (48) and after some calculation, we getFrom (50), one hasObviously, is a solution of (51). But one has to prove that this is the unique solution of (51). Now,Thus, for . This implies that, on , is strictly nondecreasing. Hence, is the unique solution of (51), and consequently, is the unique solution of (48) completing the theorem’s proof.
2.2. Case 2: be a Function of Period Two
We will explore dynamics of equation (6) when is a periodic sequence having period two with and . Consider and . Then, we have
By separating the even-indexed and odd-indexed terms, equation (6) now becomeswhere is the unique fixed point of system (54).
Theorem 7. If , then of (54) is a sink.
Proof. We consider the map on , which is described as follows:Then,Therefore, the Jacobian matrix of evaluated at isand the auxiliary equation associated with isand then, we obtainIt follows by Corollary 1.3.1 of [14] that of (54) is locally stable ifThen, the proof is completed.
2.3. Case 3: A Positive Bounded Sequence is
We assume that is positive bounded withfor some real constants and .
Theorem 8. of (6) is bounded and persists if .
Proof. Its proof is same as proof of Theorem 2, and hence, it is omitted.
Lemma 1. Assume (61) is satisfied, and ifthen
Proof. Let for , and we getTherefore,Taking the for (65), we obtainSince is arbitrary, it follows thatSimilarly,We get from inequalities (67) and (68) thatSince holds, we getor equivalentlyIt follows from equation (69) thatSo,and one hasWe have from (67), for all ,Similarly, we obtain from (68) thatThis completes the proof.
Now, we will explore attractively of solutions of equation (6).
Let represent the arbitrary positive solution of (6). Now, one can find appropriate conditions such that attracts all positive solutions of (6), that is,Now, define :and then, equation (6) becomes
Lemma 2. Let be a positive solution of (6), and then,(i) is the positive fixed point of (79).(ii)If for some and , then . Moreover, if for some and , then .(iii)Every semicycle, except first one, of any oscillatory solution of (79) contains exactly one term.
Proof. (i)The proof of (i) is trivial.(ii)If , then and In a similar way, the case is the same when is proven.(iii)Let be an eventually oscillatory solution of (27) such that and . It follows from part (ii) that . So, the positive semicycle has exactly one term. In similar way, one can prove for the negative semicycle.
Lemma 3. Every nonoscillatory solution of (79) converges to 1.
Proof. Assuming be a nonoscillatory solution of (79). We may assume, without losing generality, that , for . Clearly, for , one has ; otherwise, there exists such that , and it follows by Lemma 1 (ii) that , that is, not possible. As is decreasing and , it converges. Assumewhere . We have to prove that . Sincefor and , the sufficiently large one hasHence,So, we obtain .
Lemma 4. If is a positive solution of (79) and suppose that there exists , s.t.,thenMoreover, ifthen
Proof. If be a solution of (79) such that (85) holds for , then we obtainand by working inductively, one can prove that (86) is satisfied.
In similar way, one can prove that if (87) holds for , then (88) is satisfied.
Theorem 9. If is a particular positive solution of (6) and is a positive solution of (79) and suppose that or , , and , then
Proof. If is a solution of (79), then it is enough to prove thatAssuming there exists such that (86) or (88) hold. We may also assume that (86) holds for , and holds. Letand also, letNow, consideringfor , then one hasThus, it can be observed that(i) is nonincreasing in and nondecreasing in if (ii) is nonincreasing in and nondecreasing in if Let . Using (79), one hasSince (85) holds, so by Lemma 4, one can obtainUsing (61) and (92)–(95) and the monotone properties of , we getorThen,Hence,So,orThus,But, from , one can obtain . This implies thatorThus, we get that . The proof is completed.
Now, suppose that , , and hold. Then, using relations (61) and (92)–(95) and , holds; we obtainorSince , it follows that . Therefore, from equation (109), we getorSince , , and hold, then from (112), we have ; so, . Then, the proof is completed.
Theorem 10. Assume that , and is a periodic sequence, s.t., for all and then, (6) has a periodic solution of the prime period two.
Proof. For (6) possesses a periodic solution having prime period two, one can find positive numbers , s.t.,Or, equivalently,Now, one has to prove that system (114) is consistent. From (114), one getsand from this, it follows thatDefineThen,where . So, has a zero, say , and in view of equation (115) and (116), one can get that (6) has a two-periodic solution. Now, assuming that , one can defineThen,Thus, has a zero, say , and in view of equations (115) and (116), one can get that (6) has a two-periodic solution.
3. Numerical Simulation
In this section, we will provide some simulation in order to verify obtained theoretical results, for these following cases are to be considered for the completeness of this section ():
Case 1: If and , then Figure 1 implies that fixed point of equation (8) is a sink. This simulation agrees with the conclusion of Theorem 1.
Case 2: If and , then Figure 2 implies that fixed point of equation (8) is a sink. This simulation again agrees with the conclusion of Theorem 1.
Case 3: If then Figure 3 implies that fixed point of system (54) is a sink. Additionally, if , then , and hence, this simulation agrees with the conclusion of Theorem 7.
Case 4: If , then Figure 4 implies that fixed point of system (54) is a sink. Additionally, if then , and hence, this simulation again agrees with the conclusion of Theorem 7.
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Data Availability
The data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.