find the oscillation of the difference equation to the determine red in same way of the determine yellow In this paper, the global asymptotic behavior and the periodic character of solutions ofthe rational difference equationaxn-1Xn+1 =n = 0,1,2,...,(1.3)ß+ yITxi=*n-2iwhere the parameters a, ß, y, pi, Pl+1,...pk are nonnegative real numbers, I, k are nonnegativeintegers such that I< k, and the initial conditions x-2k,x-2k+1,...,xo are arbitrary nonnegativereal numbers such that3. A General Oscillation Result1/2PiThe change of variables x, = (B/y)"reduces (1.3) to the difference equationryn-1Pin = 0,1,2,...,(3.1)Yn+1 =1+ IIYn-2ia/ß > 0.Note that y, = 0 is always an equilibrium point. When r > 1, (3.1) also possesses theunique positive equilibrium ỹ, = (r – 1)'/P.where r =%3D1/EkTheorem B (see [8]). Assume that F E C([0, c0)2k+1arguments, and nondecreasing in the even arguments. Let x be an equilibrium point of the differenceequation[0, 00)) is nonincreasing in the oddXn+1 =F(xп, Хр-1,, хр-2к), п%3D0,1,2,...,(3.2)and let {xn} 2k be a solution of (3.2) such that eithern=-2kX-2k, X-2k+2,...,xo 2 x,X-2k+1, X-2k+3,...,X-1 < x,(3.3)orX-2k, X-2k+2,...,xo < x,X-2k+1, X_2k+3,...,x-1 2 x.(3.4)Then {x,}-2k oscillates about x with semicycles of length one.Corollary 3.1. Assume that r> 1; let {yn}-2k be a solution of (3.1) such that eitherY-2k, Y-2k+2,..., Yo > y, = (r – 1)1/Pi(3.5)Y-2k+1, Y-2k+3, -..,y-1 < y, = (r – 1)/PiorY-2k, Y-2k+2, -.., Yo < F2 = (r – 1)'/p(3.6)Y-2k+1, Y-2k+3,..,„Y-1 2 F2 = (r – 1)/P.%3DThen {yn}-2k Oscillates about the positive equilibrium point y, = (r – 1)'/2Pi with semicycles oflength one.n=-2kProof. The proof follows immediately from Theorem B.[7] A. M. Ahmed, H. M. El-Owaidy, A. E. Hamza, and A. M. Youssef, "On the recursive sequence xp+1 =(a + bxn-1)/(A+ Bx)," Journal of Applied Mathematics & Informatics, vol. 27, no. 1-2, pp. 275-289, 2009.[8] Alaa. E. Hamza and R. Khalaf-Allah, "Global behavior of a higher order difference equation," Journalof Mathematics and Statistics, vol. 3, no. 1, pp. 17-20, 2007.[9] E. M. Elabbasy and E. M. Elsayed, "Global attractivity and periodic nature of a difference equation,"World Applied Sciences Journal, vol. 12, no. 1, pp. 39-47, 2011. the positive solutions of the rational difference equationaSn-q + bSn-r + cSn-sdSn-q+ eSn-r + f Sn-s ,(1)Sn+1 = Sn-pwhere a, b, c, d, e, ƒ € (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 arbitrary positive real numbers such thatp > q > r > s > 0.+b-

As per the question, we are given a specific form of a difference equation and are tasked with…

the positive solutions of the rational difference equation aSn-q+bSn-r+cSn-s Sn+1 = Sn-pdSn-q + eSn-r+fSn-s/ where a, b, c, d, e, f € (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 arbitrary positive real numbers such that p>q>r>s ≥ 0. (1)