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Theorem 4 Every solution of Eq.(1.1) is bounded if A < 1. 00 proof Let {ym}-5 be a solution of Eq.(1.1). It follows from Eq.(1.1) that a1Ym-1 + a2Ym-2+ a3Ym-3+ a4Ym-4+a5Ym-5 B1ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4+ B5Ym-5 Ym+1 Aym + a1Ym-1 Aym + B1Ym-1 + B2Ym-2 + B3Ym-3 + B4Ym-4 + B5Ym-5

Theorem 4 Every solution of Eq.(1.1) is bounded if A < 1. 00 proof Let {ym}-5 be a solution of Eq.(1.1). It follows from Eq.(1.1) that a1Ym-1 + a2Ym-2+ a3Ym-3+ a4Ym-4+a5Ym-5 B1ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4+ B5Ym-5 Ym+1 Aym + a1Ym-1 Aym + B1Ym-1 + B2Ym-2 + B3Ym-3 + B4Ym-4 + B5Ym-5

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 30E
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Theorem 4 Every solution of Eq.(1.1) is bounded if A< 1. proof Let {ym} 5 be a solution of Eq.(1.1). It follows from Eq.(1.1) m=-5 that a1Ym-1 +a2Ym-2+ a3ym-3 + a4Ym-4 + a5Ym-5 Ут+1 Aym + B1ym–1+ B2Ym-2 + B3Ym–3 + BaYm-4+ B5Ym–5 a1Ym-1 Аут + B1ym-1 + B2Ym-2 + B3ym-3 + B4Ym-4 + B5Ym-5 a2Ym-2 B1Ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4 + B5Ym-5 azYm-3 В1Ут-1 + В2Ут-2 + Взут-3 + ВАУт-4 + В53т-5 a4Ym-4 Віут-1 + В2Ут-2 + Взут-3 + ВАУт-4 + Взут-5 a5Ym-5 B1ym–1 + B2Ym-2 + B3ym-3 + B4Ym-4 + B5Ym-5 Then a1Ym–1 a2Ym-2 a3Ym-3 + B3ym-3 a4Ym-4 a5Ym-5 + B5Ym-5 Ym+1 < Aym + Biym-1 B2Ym-2 B4Ym-4 9. ai Aym + B1 a2 a3 for all m >1. B2 B3 B4 By using a comparison, we can write the right hand side as follows a2 Aym + B1 a3 + B3 Ym+1 = B2 B4 then Ym = a"yo + constant, and this equation is locally asymptotically stable because A < 1, and con- verges to the equilibrium point a1b2B3B4B5 + a2ß1B3B4B5 + a3ß132B4B5 + a4B1B2B3ß5 + a5ß1B2B3ß4 B1B2B3B4B5 (1 – A) Therefore, a1 B2B3B4B5 + a2ß1B3B4B5 + a3ß1B2B4B5 + ¤4B1B2B3B5 + a5ß1B2ß3ß4 lim sup ym B182B3B4B5 (1 – A) | Thus, the solution of Eq.(1.1) is bounded and the proof is complete. +
Transcribed Image Text:Theorem 4 Every solution of Eq.(1.1) is bounded if A< 1. proof Let {ym} 5 be a solution of Eq.(1.1). It follows from Eq.(1.1) m=-5 that a1Ym-1 +a2Ym-2+ a3ym-3 + a4Ym-4 + a5Ym-5 Ут+1 Aym + B1ym–1+ B2Ym-2 + B3Ym–3 + BaYm-4+ B5Ym–5 a1Ym-1 Аут + B1ym-1 + B2Ym-2 + B3ym-3 + B4Ym-4 + B5Ym-5 a2Ym-2 B1Ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4 + B5Ym-5 azYm-3 В1Ут-1 + В2Ут-2 + Взут-3 + ВАУт-4 + В53т-5 a4Ym-4 Віут-1 + В2Ут-2 + Взут-3 + ВАУт-4 + Взут-5 a5Ym-5 B1ym–1 + B2Ym-2 + B3ym-3 + B4Ym-4 + B5Ym-5 Then a1Ym–1 a2Ym-2 a3Ym-3 + B3ym-3 a4Ym-4 a5Ym-5 + B5Ym-5 Ym+1 < Aym + Biym-1 B2Ym-2 B4Ym-4 9. ai Aym + B1 a2 a3 for all m >1. B2 B3 B4 By using a comparison, we can write the right hand side as follows a2 Aym + B1 a3 + B3 Ym+1 = B2 B4 then Ym = a"yo + constant, and this equation is locally asymptotically stable because A < 1, and con- verges to the equilibrium point a1b2B3B4B5 + a2ß1B3B4B5 + a3ß132B4B5 + a4B1B2B3ß5 + a5ß1B2B3ß4 B1B2B3B4B5 (1 – A) Therefore, a1 B2B3B4B5 + a2ß1B3B4B5 + a3ß1B2B4B5 + ¤4B1B2B3B5 + a5ß1B2ß3ß4 lim sup ym B182B3B4B5 (1 – A) | Thus, the solution of Eq.(1.1) is bounded and the proof is complete. +
The main focus of this article is to discuss some qualitative behavior of the solutions of the nonlinear difference equation a1Ym-1+a2Ym-2+ a3Ym-3+ a4Ym–4+ a5Ym-5 Ут+1 — Аутt т 3 0, 1, 2, ..., B1Ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4 + B3Ym-5 (1.1) where the coefficients A, a;, B; E (0, 0), i = 1, ..., 5, while the initial condi- tions y-5,y-4,Y–3,Y–2, Y–1, yo are arbitrary positive real numbers. Note that the special case of Eq.(1.1) has been discussed in [4] when az = B3 = a4 = = a5 = B5 = 0 and Eq.(1.1) has been studied in [8] in the special case B4 when a4 = B4 = a5 = B5 = 0 and Eq.(1.1) has been discussed in [5] in the special case when az = B5 = 0.
Transcribed Image Text:The main focus of this article is to discuss some qualitative behavior of the solutions of the nonlinear difference equation a1Ym-1+a2Ym-2+ a3Ym-3+ a4Ym–4+ a5Ym-5 Ут+1 — Аутt т 3 0, 1, 2, ..., B1Ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4 + B3Ym-5 (1.1) where the coefficients A, a;, B; E (0, 0), i = 1, ..., 5, while the initial condi- tions y-5,y-4,Y–3,Y–2, Y–1, yo are arbitrary positive real numbers. Note that the special case of Eq.(1.1) has been discussed in [4] when az = B3 = a4 = = a5 = B5 = 0 and Eq.(1.1) has been studied in [8] in the special case B4 when a4 = B4 = a5 = B5 = 0 and Eq.(1.1) has been discussed in [5] in the special case when az = B5 = 0.
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