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Find the bounded of the determine red In the same way as the proof of theorem 4 in the second picture

Find the bounded of the determine red In the same way as the proof of theorem 4 in the second picture

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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Find the bounded of the determine red In the same way as the proof of theorem 4 in the second picture

aSn-q + bSn-r + cSn-s Sn+1 = Sn-p dSn-q + eS n-r + f Sn–s, are investigated, where a, b, c, d, e, f e (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 ar- bitrary positive real numbers such that p > q > r > s > 0. Some numerical examples are provided to illustrate the theoretical discussion.
Transcribed Image Text:aSn-q + bSn-r + cSn-s Sn+1 = Sn-p dSn-q + eS n-r + f Sn–s, are investigated, where a, b, c, d, e, f e (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 ar- bitrary positive real numbers such that p > q > r > s > 0. Some numerical examples are provided to illustrate the theoretical discussion.
The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation bxn-1 + cxn-2+ fxn-3 + rxn-4 Xn+1 = axn + n = 0, 1, 2, .. (1) dxn-1 + exn-2 + gxn-3 + sxn-4 where the coefficients a, b, c, d, e, f, g,r, s E (0, 00), while the initial con- ditions x-4,x_3,x-2, x-1, xo are arbitrary positive real numbers. Note that the special cases of Eq.(1) has been studied discussed in [11] when f = g = 0 and Eq.(1) has been studied discussed in [35] in the special case r = s = when r = s = 0. %3D Theorem 4 Every solution of Eq.(1) is bounded if a < 1. Proof: Let {an}-4 be a solution of Eq.(1). It follows from Eq.(1) that bxn-1+ cxn-2 + fxn-3 +rxn-4 Xn+1 axn + dxn-1 + exn-2 + gxn-3 + sxn-4 bxn-1 axn + dxn-1+ exn-2 + gxn-3+ sxn-4 cXn-2 dxn-1 + exn-2+ gxn-3 + sxn-4 fxn-3 dxn-1 + exn-2 + gxn-3 + Sxn-4 r*n-4 dxn-1 + exn-2+ gxn-3 + SXn-4 Then bxn-1 føn-3 b = axn + d cxn-2 rxn-4 f Xn+1 < axn + dxn-1 exn-2 gxn-3 sxn-4 e for all n > 1. By using a comparison, we can write the right hand side as follows b Yn+1 = ayn + d e S then n Yn = a"yo + constant, and this equation is locally asymptotically stable because a < 1, and con- verges to the equilibrium point begs + cdgs + def s+rdeg = degs (1 – a) Therefore begs + cdgs + def s+ rdeg lim sup xn S degs (1 a) Thus, the solution of Eq.(1) is bounded and the proof is now completed.
Transcribed Image Text:The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation bxn-1 + cxn-2+ fxn-3 + rxn-4 Xn+1 = axn + n = 0, 1, 2, .. (1) dxn-1 + exn-2 + gxn-3 + sxn-4 where the coefficients a, b, c, d, e, f, g,r, s E (0, 00), while the initial con- ditions x-4,x_3,x-2, x-1, xo are arbitrary positive real numbers. Note that the special cases of Eq.(1) has been studied discussed in [11] when f = g = 0 and Eq.(1) has been studied discussed in [35] in the special case r = s = when r = s = 0. %3D Theorem 4 Every solution of Eq.(1) is bounded if a < 1. Proof: Let {an}-4 be a solution of Eq.(1). It follows from Eq.(1) that bxn-1+ cxn-2 + fxn-3 +rxn-4 Xn+1 axn + dxn-1 + exn-2 + gxn-3 + sxn-4 bxn-1 axn + dxn-1+ exn-2 + gxn-3+ sxn-4 cXn-2 dxn-1 + exn-2+ gxn-3 + sxn-4 fxn-3 dxn-1 + exn-2 + gxn-3 + Sxn-4 r*n-4 dxn-1 + exn-2+ gxn-3 + SXn-4 Then bxn-1 føn-3 b = axn + d cxn-2 rxn-4 f Xn+1 < axn + dxn-1 exn-2 gxn-3 sxn-4 e for all n > 1. By using a comparison, we can write the right hand side as follows b Yn+1 = ayn + d e S then n Yn = a"yo + constant, and this equation is locally asymptotically stable because a < 1, and con- verges to the equilibrium point begs + cdgs + def s+rdeg = degs (1 – a) Therefore begs + cdgs + def s+ rdeg lim sup xn S degs (1 a) Thus, the solution of Eq.(1) is bounded and the proof is now completed.
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