Show me the steps of determine red and the informatian is here Theorem 9.If k is even and1,0 are odd positive integers,then Eq. (1) has prime period two solution if the conditionth(1– (C+D)) (3e- d) < (e+d) (A+B),(20)valid,provided(C+D)1andise (1-(C+ D)) – d (A+B) > 0.aiProof.If k is even and 1, o are odd positive integers, thenXn = Xp-k and Xn+1 = Xn-1= Xp-g. It follows from Eq. (1)thatbQP= (A+B) Q+(C+D)P –(21)(еР — d@)"andbPQ= (A+B) P+ (C+D) Q –(22)(e Q- dP)Consequently, we gete P – dPQ = e (A+B) PQ- d (A+ B) Q +e(C+ D) P- (C+D) dPQ– bQ,(23)(2ande Q – dPQ = e (A+B) PQ- d (A+B) P +e(C+ D) Q-(C+D) dPQ- bP.(24)By subtracting (24) from (23), we getbisP+Q=(25)[e (1- (C+ D)) – d (A+B)]'Pwhere e (1- (C+ D)) – d (A+B) > 0. By adding (23)and (24), we obtainXtheb (1- (C+ D))(e+d) [K1 + (A+B)] [e K1 – d (A+ B)]² "PQ =(26)where K =(1– (C+D)), provided (C+D) < 1.Assume that P and Q are two positive distinct real rootsof the quadratic equation2 - (P+ Q) t+ PQ=0.(27)Thus, we deduce that(P+Q)² > 4PQ.(28)Substituting (25) and (26) into (28), we get the condition(20). Thus, the proof is now completed.O The objective of this article is to investigate somequalitative behavior of the solutions of the nonlineardifference equationbxn-kXn+1 = Axn+ Bx–k+Cxn-1+ Dxn-o+dxn-k– exŋ-n= 0,1,2,.....(1)where the coefficients A, B, C, D, b, d, e E (0,0), whilek, 1 and o are positive integers. The initial conditionsX_g,..., X_1,.., X_k, ….., X_1, X are arbitrary positive realnumbers such that k <1< 0. Note that the special casesof Eq.(1) have been studied in [1] when B=C= D=0,and k= 0,1= 1, b is replaced by – b and in [27] whenB=C= D=0, and k= 0, b is replaced by[33] when B = C = D = 0, 1= 0 and in [32] whenA=C= D=0, 1=0, b is replaced by – b.b and in%3|

Name: Show me the steps of determine red and the informatian is here Theorem
Uploaded: 2024-03-20T06:38:44.000Z
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Description: Show me the steps of determine red and the informatian is here Theorem 9.If k is even and1,0 are odd positive integers,then Eq. (1) has prime period two solution if the conditionth(1– (C+D)) (3e- d) < (e+d) (A+B),(20)valid,provided(C+D)1andise (1-(C+