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a2 -α-p =0. So we get a = 1+/I+4p y + B which is a trivial solution. Now we deal with the other case such that m is odd. Now we apply Elsayed's new method for two periodic solution, see [16]. We have from Eq.(8)

a2 -α-p =0. So we get a = 1+/I+4p y + B which is a trivial solution. Now we deal with the other case such that m is odd. Now we apply Elsayed's new method for two periodic solution, see [16]. We have from Eq.(8)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.9: Properties Of Determinants
Problem 46E
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Hence we obtain the linearized equation of Eq.(8) about its unique positive equilibrium point j as follow: 2p --m = 0. (15) Zn+1 - Motivated by the above studies, we study the dynamics of following higher order difference equation In A+B (2) In+1 'n-m where A, B are positive real numbers and the initial conditions are positive numbers. Additionally, we investigate the boundedness, periodicity, oscillation behaviours, global asymptotically stability and rate of convergence of related higher order difference equations. Firstly, we take the change of the variables for Eq.(2) as follows yn From this, we obtain the following difference equation Yn Yn+1 = 1+p Yn-m (8) where p = . From now on, we handle the difference equation (8). The unique positive equilibrium point of Eq.(8) is 1+ 1+ 4p 2
Transcribed Image Text:Hence we obtain the linearized equation of Eq.(8) about its unique positive equilibrium point j as follow: 2p --m = 0. (15) Zn+1 - Motivated by the above studies, we study the dynamics of following higher order difference equation In A+B (2) In+1 'n-m where A, B are positive real numbers and the initial conditions are positive numbers. Additionally, we investigate the boundedness, periodicity, oscillation behaviours, global asymptotically stability and rate of convergence of related higher order difference equations. Firstly, we take the change of the variables for Eq.(2) as follows yn From this, we obtain the following difference equation Yn Yn+1 = 1+p Yn-m (8) where p = . From now on, we handle the difference equation (8). The unique positive equilibrium point of Eq.(8) is 1+ 1+ 4p 2
Theorem 6 Let {yn} be a positive solution of Eq. (8). Then Eq. (8) has no two periodic solution. Proof. We assume that there exist two periodic solution such that α, β, α, β, . . . where a and B are positive and distinct real numbers. We handle two cases for the proof of Theorem. Firstly we consider a case such that m is even. We have from Eq.(8) р a = 1+ = 1+ Hence we obtain that — а — р %— 0. 1+y1+4p B which is a trivial solution. Now we deal with So we get a = the other case such that m is odd. Now we apply Elsayed's new method for two periodic solution, see [16]. We have from Eq.(8) 2 pB a = 1+ pa = 1+ 4
Transcribed Image Text:Theorem 6 Let {yn} be a positive solution of Eq. (8). Then Eq. (8) has no two periodic solution. Proof. We assume that there exist two periodic solution such that α, β, α, β, . . . where a and B are positive and distinct real numbers. We handle two cases for the proof of Theorem. Firstly we consider a case such that m is even. We have from Eq.(8) р a = 1+ = 1+ Hence we obtain that — а — р %— 0. 1+y1+4p B which is a trivial solution. Now we deal with So we get a = the other case such that m is odd. Now we apply Elsayed's new method for two periodic solution, see [16]. We have from Eq.(8) 2 pB a = 1+ pa = 1+ 4
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