Explain the determine yellow and the inf is here Theorem 6 If (a1+ аз + a5) > (02 + ол) аnd (Bі + Вз + Bъ) > (32 + Ba),then the necessary and sufficient condition for Eq.(1.1) to have positive so-lutions of prime period two is that the inequality[(A + 1) ((B1 + B3 + B5) – (B2 + B4))] [(@1 + a3 + a5) – (a2 + a4)]?+4[(a1 + a3 + a5) – (a2 + a4)] [(B1 + B3 + B5) (a2 + a4) + A (82 + B4) (a1 + a3 + 5)] > 0.(4.13)is valid.proof: Suppose there exist positive distinctive solutions of prime periodtwoP,Q, P, Q, ..of Eq.(1.1). From Eq.(1.1) we havea1Ym-1+a2Ym-2+ a3Ym-3+a4Ym-4+a5Ym-5Ym+1 =Aym +B1Ym-1 + B2Ym-2 + B3Ym–3 + B4Ym-4 + B5Ym–5(a1 + az + a5) P+(a2+a4) QQ =(B1 + B3 + B5) P + (82 + B4) Q ’(a1 + az + a5) Q+(a2 + a4) PР— АQ+= AP+(B1 + B3 + B5) Q+ (B2 + B4) P *(4.14)Consequently, we get(B1 + B3 + B5) P² + (B2 + B4) PQA (B1 + B3 + B5) PQ + A (B2 + B4) Q²+ (a1 + a3 + a5) P+ (a2 + a4) Q,(4.15)and(B1 + 33 + B5) Q² + (82 + B4) PQA (B1 + B3 + B5) PQ + A (ß2 + B4) P²+ (a1 + a3 + a5) Q + (a2 + a4) P.(4.16)By subtracting (4.15) from (4.16), we obtain[(81 + B3 + B5) + A (B2 + B4)] (P² –- Q²) = [(a1 + a3 + as) – (a2 + a4)] (P – Q).11 The main focus of this article is to discuss some qualitative behavior ofthe solutions of the nonlinear difference equationa1Ym-1+ a2Ym-2 + a3Ym-3 + a4Ym-4+ a5Ym-5Ym+1 =Aym+т 3D 0, 1, 2, ...,В1ут-1 + В2ут-2 + Взут-3 + Влут-4 + B5ут-5(1.1)

Name: Explain the determine yellow and the inf is here Theorem 6 If (a1+ аз
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: Explain the determine yellow and the inf is here Theorem 6 If (a1+ аз + a5) > (02 + ол) аnd (Bі + Вз + Bъ) > (32 + Ba),then the necessary and sufficient condition for Eq.(1.1) to have positive so-lutions of prime period two is that the inequality[(A