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Theorem 6 If (b+ f) > (c+r) and (d+ g) > (e + s), then the necessary and sufficient condition for Eq.(1) to have positive solutions of prime period two is that the inequality [(a + 1) ((d + g) – (e+ s))] [(b+ f) – (c+r)]² +4[(b+ f) – (c+ r)] [(c+r) (d + g) + a (e + s) (b + f)] > 0. (13) is valid. Proof: Suppose that there exist positive distinctive solutions of prime period two P,Q, P,Q,. of Eq.(1). From Eq.(1) we have bxn-1 + cxn-2+ fan-3+ rXn-4 Xn+1 = axn + %3D dxn-1 + exn-2+ gan-3 + sxn-4 (b+ f) P+ (c+r)Q (d + g) P + (e + s)Q' (b+ f)Q + (c+ r) P (d + g) Q + (e + s) P' P = aQ+ Q = aP+ Consequently, we obtain (d+ g) P² + (e + s) PQ = a (d + g) PQ+a(e+ s) Q² + (b + f)P+(c+r)Q, (14) %3D

Theorem 6 If (b+ f) > (c+r) and (d+ g) > (e + s), then the necessary and sufficient condition for Eq.(1) to have positive solutions of prime period two is that the inequality [(a + 1) ((d + g) – (e+ s))] [(b+ f) – (c+r)]² +4[(b+ f) – (c+ r)] [(c+r) (d + g) + a (e + s) (b + f)] > 0. (13) is valid. Proof: Suppose that there exist positive distinctive solutions of prime period two P,Q, P,Q,. of Eq.(1). From Eq.(1) we have bxn-1 + cxn-2+ fan-3+ rXn-4 Xn+1 = axn + %3D dxn-1 + exn-2+ gan-3 + sxn-4 (b+ f) P+ (c+r)Q (d + g) P + (e + s)Q' (b+ f)Q + (c+ r) P (d + g) Q + (e + s) P' P = aQ+ Q = aP+ Consequently, we obtain (d+ g) P² + (e + s) PQ = a (d + g) PQ+a(e+ s) Q² + (b + f)P+(c+r)Q, (14) %3D

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 79E
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l stc ksa 9:21 PM C@ 974%4 Theorem 6 If (b+ f) > (c+ r) and (d+ g) > (e+ s), then the necessary and sufficient condition for Eq. (1) to have positive solutions of prime period two is that the inequality [(a + 1) ((d+ g) – (e+ s))] [(b+ f) – (c +r)]? +4[(b+ f) – (c+ r)[c +r) (d + g) + a (e+ s) (b + f)] > 0. (13) is valid. Proof: Suppose that there exist positive distinctive solutions of prime period two P,Q, P, Q,. of Eq.(1). From Eq.(1) we have bxn-1 + cxn-2+ fxn-3 + rxn-4 Xn+1 = aan + dxn-1 + exn-2+ gxn-3 + sxn-4 (P = aQ+ d+9) P + (e + s) Q' (b+ f)Q + (c+ r) P (d + g) Q + (e + s) P' Q = aP+ Consequently, we obtain (d+g) P² + (e + s) PQ = a (d + g) PQ+a (e+ s) Q² + (b+ f) P+(c+r) Q, (14)
Transcribed Image Text:l stc ksa 9:21 PM C@ 974%4 Theorem 6 If (b+ f) > (c+ r) and (d+ g) > (e+ s), then the necessary and sufficient condition for Eq. (1) to have positive solutions of prime period two is that the inequality [(a + 1) ((d+ g) – (e+ s))] [(b+ f) – (c +r)]? +4[(b+ f) – (c+ r)[c +r) (d + g) + a (e+ s) (b + f)] > 0. (13) is valid. Proof: Suppose that there exist positive distinctive solutions of prime period two P,Q, P, Q,. of Eq.(1). From Eq.(1) we have bxn-1 + cxn-2+ fxn-3 + rxn-4 Xn+1 = aan + dxn-1 + exn-2+ gxn-3 + sxn-4 (P = aQ+ d+9) P + (e + s) Q' (b+ f)Q + (c+ r) P (d + g) Q + (e + s) P' Q = aP+ Consequently, we obtain (d+g) P² + (e + s) PQ = a (d + g) PQ+a (e+ s) Q² + (b+ f) P+(c+r) Q, (14)
l stc ksa 12:14 AM @ 1 60% 4 The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation bxn-1 + cæn-2+ fxn-3 + ræn-4 Xn+1 = axn + n = 0, 1, 2, . (1) dxn-1+ exn-2 + gæn-3 + sxn-4 where the coefficients a, b, c, d, e, f, g, r, s E (0, 00), while the initial con- ditions a_4,x_3,x_2, x-1, xo are arbitrary positive real numbers. Note that Cancel Actual Size (399 KB) Choose
Transcribed Image Text:l stc ksa 12:14 AM @ 1 60% 4 The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation bxn-1 + cæn-2+ fxn-3 + ræn-4 Xn+1 = axn + n = 0, 1, 2, . (1) dxn-1+ exn-2 + gæn-3 + sxn-4 where the coefficients a, b, c, d, e, f, g, r, s E (0, 00), while the initial con- ditions a_4,x_3,x_2, x-1, xo are arbitrary positive real numbers. Note that Cancel Actual Size (399 KB) Choose
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