Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
the positive solutions of the rational difference equation aSn-q+bSn-r+cSn-s Sn+1 = Sn-pdSn-q + eSn-r+fSn-s/ where a, b, c, d, e, f € (0, ∞). The initial conditions S-p, S-p+1,...,S_q, S-q+1,...,S_r, S-r+1,...,S-s,...,S-s+1,...,S-1 and So are arbitrary positive real numbers such that p>q>r>s ≥ 0. (1)

the positive solutions of the rational difference equation aSn-q+bSn-r+cSn-s Sn+1 = Sn-pdSn-q + eSn-r+fSn-s/ where a, b, c, d, e, f € (0, ∞). The initial conditions S-p, S-p+1,...,S_q, S-q+1,...,S_r, S-r+1,...,S-s,...,S-s+1,...,S-1 and So are arbitrary positive real numbers such that p>q>r>s ≥ 0. (1)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.6: Variation
Problem 10E
See similar textbooks
icon
Related questions
Question

find the oscillation of the difference equation to the determine red in same way of the determine yellow

In this paper, the global asymptotic behavior and the periodic character of solutions of the rational difference equation axn-1 Xn+1 = n = 0,1,2,..., (1.3) ß+ yITx i=*n-2i where the parameters a, ß, y, pi, Pl+1,...pk are nonnegative real numbers, I, k are nonnegative integers such that I< k, and the initial conditions x-2k,x-2k+1,...,xo are arbitrary nonnegative real numbers such that 3. A General Oscillation Result 1/2Pi The change of variables x, = (B/y)" reduces (1.3) to the difference equation ryn-1 Pi n = 0,1,2,..., (3.1) Yn+1 = 1+ IIYn-2i a/ß > 0. Note that y, = 0 is always an equilibrium point. When r > 1, (3.1) also possesses the unique positive equilibrium ỹ, = (r – 1)'/P. where r = %3D 1/Ek Theorem B (see [8]). Assume that F E C([0, c0)2k+1 arguments, and nondecreasing in the even arguments. Let x be an equilibrium point of the difference equation [0, 00)) is nonincreasing in the odd Xn+1 = F(xп, Хр-1,, хр-2к), п%3D0,1,2,..., (3.2) and let {xn} 2k be a solution of (3.2) such that either n=-2k X-2k, X-2k+2,...,xo 2 x, X-2k+1, X-2k+3,...,X-1 < x, (3.3) or X-2k, X-2k+2,...,xo < x, X-2k+1, X_2k+3,...,x-1 2 x. (3.4) Then {x,}-2k oscillates about x with semicycles of length one. Corollary 3.1. Assume that r> 1; let {yn}-2k be a solution of (3.1) such that either Y-2k, Y-2k+2,..., Yo > y, = (r – 1)1/Pi (3.5) Y-2k+1, Y-2k+3, -..,y-1 < y, = (r – 1)/Pi or Y-2k, Y-2k+2, -.., Yo < F2 = (r – 1)'/p (3.6) Y-2k+1, Y-2k+3,.., „Y-1 2 F2 = (r – 1)/P. %3D Then {yn}-2k Oscillates about the positive equilibrium point y, = (r – 1)'/2Pi with semicycles of length one. n=-2k Proof. The proof follows immediately from Theorem B. [7] A. M. Ahmed, H. M. El-Owaidy, A. E. Hamza, and A. M. Youssef, "On the recursive sequence xp+1 = (a + bxn-1)/(A+ Bx)," Journal of Applied Mathematics & Informatics, vol. 27, no. 1-2, pp. 275-289, 2009. [8] Alaa. E. Hamza and R. Khalaf-Allah, "Global behavior of a higher order difference equation," Journal of Mathematics and Statistics, vol. 3, no. 1, pp. 17-20, 2007. [9] E. M. Elabbasy and E. M. Elsayed, "Global attractivity and periodic nature of a difference equation," World Applied Sciences Journal, vol. 12, no. 1, pp. 39-47, 2011.
Transcribed Image Text:In this paper, the global asymptotic behavior and the periodic character of solutions of the rational difference equation axn-1 Xn+1 = n = 0,1,2,..., (1.3) ß+ yITx i=*n-2i where the parameters a, ß, y, pi, Pl+1,...pk are nonnegative real numbers, I, k are nonnegative integers such that I< k, and the initial conditions x-2k,x-2k+1,...,xo are arbitrary nonnegative real numbers such that 3. A General Oscillation Result 1/2Pi The change of variables x, = (B/y)" reduces (1.3) to the difference equation ryn-1 Pi n = 0,1,2,..., (3.1) Yn+1 = 1+ IIYn-2i a/ß > 0. Note that y, = 0 is always an equilibrium point. When r > 1, (3.1) also possesses the unique positive equilibrium ỹ, = (r – 1)'/P. where r = %3D 1/Ek Theorem B (see [8]). Assume that F E C([0, c0)2k+1 arguments, and nondecreasing in the even arguments. Let x be an equilibrium point of the difference equation [0, 00)) is nonincreasing in the odd Xn+1 = F(xп, Хр-1,, хр-2к), п%3D0,1,2,..., (3.2) and let {xn} 2k be a solution of (3.2) such that either n=-2k X-2k, X-2k+2,...,xo 2 x, X-2k+1, X-2k+3,...,X-1 < x, (3.3) or X-2k, X-2k+2,...,xo < x, X-2k+1, X_2k+3,...,x-1 2 x. (3.4) Then {x,}-2k oscillates about x with semicycles of length one. Corollary 3.1. Assume that r> 1; let {yn}-2k be a solution of (3.1) such that either Y-2k, Y-2k+2,..., Yo > y, = (r – 1)1/Pi (3.5) Y-2k+1, Y-2k+3, -..,y-1 < y, = (r – 1)/Pi or Y-2k, Y-2k+2, -.., Yo < F2 = (r – 1)'/p (3.6) Y-2k+1, Y-2k+3,.., „Y-1 2 F2 = (r – 1)/P. %3D Then {yn}-2k Oscillates about the positive equilibrium point y, = (r – 1)'/2Pi with semicycles of length one. n=-2k Proof. The proof follows immediately from Theorem B. [7] A. M. Ahmed, H. M. El-Owaidy, A. E. Hamza, and A. M. Youssef, "On the recursive sequence xp+1 = (a + bxn-1)/(A+ Bx)," Journal of Applied Mathematics & Informatics, vol. 27, no. 1-2, pp. 275-289, 2009. [8] Alaa. E. Hamza and R. Khalaf-Allah, "Global behavior of a higher order difference equation," Journal of Mathematics and Statistics, vol. 3, no. 1, pp. 17-20, 2007. [9] E. M. Elabbasy and E. M. Elsayed, "Global attractivity and periodic nature of a difference equation," World Applied Sciences Journal, vol. 12, no. 1, pp. 39-47, 2011.
the positive solutions of the rational difference equation aSn-q + bSn-r + cSn-s dSn-q+ eSn-r + f Sn-s , (1) Sn+1 = Sn-p where a, b, c, d, e, ƒ € (0, ∞). The initial conditions S-p, S-p+1,...,S-q, S-q+1,...,S-r, S-r+1,...,S-s,..,S_s+1,..,S_1 and So are arbitrary positive real numbers such that p > q > r > s > 0. +b-
Transcribed Image Text:the positive solutions of the rational difference equation aSn-q + bSn-r + cSn-s dSn-q+ eSn-r + f Sn-s , (1) Sn+1 = Sn-p where a, b, c, d, e, ƒ € (0, ∞). The initial conditions S-p, S-p+1,...,S-q, S-q+1,...,S-r, S-r+1,...,S-s,..,S_s+1,..,S_1 and So are arbitrary positive real numbers such that p > q > r > s > 0. +b-
Expert Solution
steps

Step by step

Solved in 3 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Algebra: Structure And Method, Book 1
Algebra: Structure And Method, Book 1
Algebra
ISBN:
9780395977224
Author:
Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:
McDougal Littell
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Big Ideas Math A Bridge To Success Algebra 1: Stu…
Algebra
ISBN:
9781680331141
Author:
HOUGHTON MIFFLIN HARCOURT
Publisher:
Houghton Mifflin Harcourt
SEE MORE TEXTBOOKS