Journal of Function Spaces

Volume 2015, Article ID 737420, 9 pages

http://dx.doi.org/10.1155/2015/737420

## Some Discussions on the Difference Equation

^{1}Department of Mathematics, Faculty of Science and Arts, University of Jeddah (UJ), P.O. Box 355, Khulais 21921, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt

Received 6 November 2014; Revised 8 April 2015; Accepted 15 April 2015

Academic Editor: Jaeyoung Chung

Copyright © 2015 Awad A. Bakery. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 is**In 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 is**In 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 **or**In 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 *

*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*

*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 .*