Abstract
We give in this work the sufficient conditions on the positive solutions of the difference equation , , where α, k, and under positive initial conditions , to be bounded, α-convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the cases of Amleh et al. (1999) and of Hamza and Morsy (2009). We offer improving conditions in the case of of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.
1. Introduction
Throughout this paper and denote, respectively, the class of bounded sequences of real numbers and its subclass of sequences convergent to . Our aim in this paper is to investigate the positive solutions of the difference equationwhere is a continuously differentiable function, , , , and the initial conditions , are arbitrary positive numbers.
Remark 1. A point is an equilibrium point of (1), if and only if it is a root for the function
If we replace and in (1) by the variables and , respectively, then we have
The linearized equation related with (1) about the equilibrium point is Its characteristic equation is
Definition 2. An equilibrium point of difference equation (1) is called locally stable if for every there exists such that if with , then for all .
Definition 3. An equilibrium point of (1) is called globally asymptotically stable if it is locally stable and, for every , one has .
Definition 4. An equilibrium point of difference equation (1) is called unstable if it is not locally stable.
Definition 5. A sequence is said to be periodic with prime period if is the smallest positive integer with for all .
Theorem 6 (see [1, 2]). (i) A necessary and sufficient condition for both roots of (5) to lie in the open disk isIn this case is locally asymptotically stable.
(ii) A necessary and sufficient condition for one of the roots of (5) to lie outside the open disk and the other root inside it isIn this case is unstable and is called a saddle point.
(iii) A necessary and sufficient condition that a root of (5) has absolute value equal to 1 is orIn this case is called a nonhyperbolic point.
By applying Theorem 6 in special difference equation (1), we obtain the following results.
Remark 7. (i) is locally asymptotically stable, if and only if(ii) is unstable, if and only if(iii) is called a nonhyperbolic point, if and only ifor
Amleh et al. [3], for , obtained important results for the following difference equation:They showed in [3] that when , the equilibrium point of (14) is globally asymptotically stable. When , every positive solution of (14) converges to a period-two solution. Every positive solution of (14) is bounded if and only if . Finally, when , the equilibrium point is an unstable saddle point. Hamza and Morsy [4], for , gave some results for the difference equationthey investigated the behavior of positive solutions of (15); they proved that when , every positive solution of (15) is bounded and when , the equilibrium point of (15) is globally asymptotically stable and they showed that (15) has periodic solutions without conditions on and . These results are improved in [5] by Gümüs and Öcalan; they investigated the boundedness character of positive solutions of (15); they proved that if , then there exist unbounded solutions of (15) in the cases when and or and ; then every positive solution of (15) is unbounded. When , , and , then every positive solution of (15) is bounded. Also they show that if , , , and , then the equilibrium point of (15) is globally asymptotically stable. Finally they explained the sufficient condition for the fact that every positive solution of (15) converges to a prime two-periodic solution. In our work we prove in Sections 3 and 4 that these results are incorrect and we give the results under finer conditions and we illustrate them in some figures. Closely related equations to (1) are investigated by many authors, for example, [6–11].
In this work in Section 2 we study the global behavior of (1). In Section 3 we investigate the boundedness character of positive solutions of (1). Section 4 is dedicated to the periodic character of the positive solutions of (1). Finally we explain the sufficient condition that every positive solution of (1) converges to a prime two-periodic solution. Throughout the paper we denote by the class and the subclass of the class by The well-known inequality in [12] will be used: if and , then
2. The Difference Equation
In this section we give the sufficient conditions to determine the classification of the equilibrium points for (1) and its uniqueness.
Lemma 8. Let be the equilibrium point of (1) and for all .(i)If , then the equilibrium point of (1) is locally asymptotically stable.(ii)If , then the equilibrium point of (1) is unstable.(iii)If , then is a nonhyperbolic point.
Proof. Give an equilibrium point of (1) and for all .(i)Suppose ; then from (2) we have Hence and using Remark 7 we get that is locally asymptotically stable.(ii)Let ; then from (2) we have hence and using Remark 7 we get that is unstable.(iii)If , then we obtain . So ; by using Remark 7 we get that is a nonhyperbolic point.
Lemma 9. One has the following:(i)If , then (1) has a unique equilibrium point .(ii)If , then (1) has a unique equilibrium point .(iii)If , then (1) has a unique equilibrium point .(iv)If , then (1) has a unique equilibrium point .
Proof. (i) If , (1) gives . See [3].
(ii) If , (1) gives . See [13].
(iii) If , the function given by (2) is decreasing on and increasing on . Since and , then has a unique root .
(iv) If , the function is increasing on . Since and , then has a unique root .
3. Global Behavior of Solutions and Boundedness
In this section, we present the conditions for the boundedness of the positive solutions and its global stability for the equilibrium point of (1).
Theorem 10. If and , then .
Proof. Suppose on the contrary that . Then, we haveTherefore, from (1) we get Let ; this implies ; we obtain which contradicts , so the proof is complete.
In Figures 1 and 2 we illustrate the result of Theorem 10 for and small values of , , , , and for, respectively, and , and explain that the solutions of (1) are unbounded for every and .
Theorem 11. If , then .
Proof. For fixed let . Choose and let with the initial conditions such that hence,Moreover we haveTherefore, by induction we can show that Hence,which gives that Thus is unbounded. Now, we assume that and choose the initial conditions such thatSo, we obtainMoreover we haveBy induction we have Therefore since , we getFor fixed let ; we deduce that Since there exists unbounded solution of the difference equation , , there exists unbounded solution of difference equation (1). This completes the proof.
In Figure 3 we illustrate the result of Theorem 11 for , , and , with the initial conditions , and choosing ; in this case the subsequence of the solution of (1) is bounded with for all , while the subsequence is unbounded, so the terms of appear approximately zero in comparison with the terms of . Also Figure 4 explains the same result for , , and ; Gümüs and Öcalan in [5] showed that when and , every solution of (1) is unbounded. Figure 5 gives a counter example of this result for , , and . We correct this result in Theorem 12 and Corollary 13.
Theorem 12. (a) , if and .
(b) , if and .
Proof. (a) From (1), for . We get for and thatSince , we have the seriesTo study the convergence of the series with general term , since for and since , we obtain Since , then by using the ratio test we obtain then the series is convergent. And for and since , we obtain So by using the ratio test we obtain then the series is convergent and hence is bounded. Also, we getTo study the convergence of the series with general term , since for and since , we obtain Since , then by using the ratio test we obtain then the series is convergent. And for and since , we obtain So by using the ratio test we obtain then the series is convergent and hence is bounded.
(b) In addition, for , , and , we haveAgain, we obtainfrom which the proof follows.
Corollary 13. For fixed , if and , then .
Proof. Since and , we have . Then, we obtainHence, from (1), we getLet ; then we obtainwhich implies . So the proof is complete.
Theorem 14. The unique positive equilibrium of (1) is globally asymptotically stable, if the following conditions are satisfied:(i),(ii) and .
Proof. By Lemma 8, is locally asymptotically stable. Thus, we have to prove that, for all , there exists a unique positive equilibrium with . Let . By Theorem 12-(b), . Thus, we haveHence from (1), we getWe maintain that ; otherwise . From (55), we obtainAnd from (56),Hence, we haveFor and , we have . Consider with ; then there exists such thatFrom (58) and (59), we obtainwhich is equivalent toUsing and in (61), we getEquation (62) with the values , , and gives a contradiction. Thus, we find . Now, assume that , , and . From (58), we get This givesMoreover, since for all , we haveThen from (65), we getBy using (64) and (66), we obtainSince and , for , , and , (67) is not satisfied. This is a contradiction. Thus, we find , which implies that tends to the unique positive equilibrium, from which the proof follows.
4. Periodicity of the Solutions of
In this section we study the convergence of the positive solution of (1) to a prime two-periodic solution when , .
Lemma 15. Suppose and . Then, the following conditions are contented: (i), if and only if .(ii), if and only if .
Proof. (i) Replace by in (1); we deduceIf , then and vice versa.
(ii) Replace by in (1); we haveLet ; then and vice versa.
Theorem 16. If these conditions are satisfiedand there exists a sufficient small positive number such thatthen, (1) has a periodic solution of prime period two.
Proof. Let be a periodic solution of period two; we havethen is a solution of the systemLet . It is obvious that if (73) holds, then is a periodic solution of period two. System (74) is correspondent toso we get the equationThus for , we have Moreover, from (71) and (76) we have Therefore, the equation has a root , where , in the interval . So, we have Let us consider the functionSince , we getFrom inequality (72), we have , and hence , which implies that By taking , , then the solution with initial values , is a prime 2-periodic solution. This completes the proof.
Lemma 17. If which consists of at least two semicycles, then is oscillatory and, except possibly for the first semicycle, every semicycle is of length 1.
Proof. The proof is directly obtained from (1).
Theorem 18. For , the sequences and are ultimately monotone.
Proof. We haveIf and , we obtain from (83) and therefore . By induction we find that is decreasing and is increasing. Correspondingly if and , using induction we obtain from (83) that is increasing and is decreasing. If and and , we have from (83) that is increasing and is decreasing. Hence, we can assume that and . If further , then is decreasing and is increasing. So we may assume that and . By induction we get the result in this case. The cases and can be treated in the same way.
Theorem 19. If (70), (71), and (72) are conformed, then every positive solution of (1) converges to a prime two-periodic solution.
Proof. The sequences and are eventually monotone and bounded by Theorems 18 and 12 for every positive solution of (1). So, the sequences and are convergent. Using Lemmas 15 and 17 the proof follows.
Figure 6 illustrates the result of Theorem 19 for , , and .
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author wants to thank the reviewers for much encouragement, support, productive feedback, cautious perusing, and a helpful remark which improved the presentation and the comprehensibility of the paper.