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)
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)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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 of
the rational difference equation
axn-1
Xn+1 =
n = 0,1,2,...,
(1.3)
ß+ yITx
i=*n-2i
where the parameters a, ß, y, pi, Pl+1,...pk are nonnegative real numbers, I, k are nonnegative
integers such that I< k, and the initial conditions x-2k,x-2k+1,...,xo are arbitrary nonnegative
real numbers such that
3. A General Oscillation Result
1/2Pi
The change of variables x, = (B/y)"
reduces (1.3) to the difference equation
ryn-1
Pi
n = 0,1,2,...,
(3.1)
Yn+1 =
1+ IIYn-2i
a/ß > 0.
Note that y, = 0 is always an equilibrium point. When r > 1, (3.1) also possesses the
unique positive equilibrium ỹ, = (r – 1)'/P.
where r =
%3D
1/Ek
Theorem B (see [8]). Assume that F E C([0, c0)2k+1
arguments, and nondecreasing in the even arguments. Let x be an equilibrium point of the difference
equation
[0, 00)) is nonincreasing in the odd
Xn+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 either
n=-2k
X-2k, X-2k+2,...,xo 2 x,
X-2k+1, X-2k+3,...,X-1 < x,
(3.3)
or
X-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 either
Y-2k, Y-2k+2,..., Yo > y, = (r – 1)1/Pi
(3.5)
Y-2k+1, Y-2k+3, -..,y-1 < y, = (r – 1)/Pi
or
Y-2k, Y-2k+2, -.., Yo < F2 = (r – 1)'/p
(3.6)
Y-2k+1, Y-2k+3,..,
„Y-1 2 F2 = (r – 1)/P.
%3D
Then {yn}-2k Oscillates about the positive equilibrium point y, = (r – 1)'/2Pi with semicycles of
length one.
n=-2k
Proof. 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," Journal
of 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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F158aed3a-44ed-4147-bb6c-9729d6a7101b%2F10c6df83-60f5-4277-b6f4-99a57fdd992d%2F6cowgrq_processed.png&w=3840&q=75)
Transcribed Image Text:In this paper, the global asymptotic behavior and the periodic character of solutions of
the rational difference equation
axn-1
Xn+1 =
n = 0,1,2,...,
(1.3)
ß+ yITx
i=*n-2i
where the parameters a, ß, y, pi, Pl+1,...pk are nonnegative real numbers, I, k are nonnegative
integers such that I< k, and the initial conditions x-2k,x-2k+1,...,xo are arbitrary nonnegative
real numbers such that
3. A General Oscillation Result
1/2Pi
The change of variables x, = (B/y)"
reduces (1.3) to the difference equation
ryn-1
Pi
n = 0,1,2,...,
(3.1)
Yn+1 =
1+ IIYn-2i
a/ß > 0.
Note that y, = 0 is always an equilibrium point. When r > 1, (3.1) also possesses the
unique positive equilibrium ỹ, = (r – 1)'/P.
where r =
%3D
1/Ek
Theorem B (see [8]). Assume that F E C([0, c0)2k+1
arguments, and nondecreasing in the even arguments. Let x be an equilibrium point of the difference
equation
[0, 00)) is nonincreasing in the odd
Xn+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 either
n=-2k
X-2k, X-2k+2,...,xo 2 x,
X-2k+1, X-2k+3,...,X-1 < x,
(3.3)
or
X-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 either
Y-2k, Y-2k+2,..., Yo > y, = (r – 1)1/Pi
(3.5)
Y-2k+1, Y-2k+3, -..,y-1 < y, = (r – 1)/Pi
or
Y-2k, Y-2k+2, -.., Yo < F2 = (r – 1)'/p
(3.6)
Y-2k+1, Y-2k+3,..,
„Y-1 2 F2 = (r – 1)/P.
%3D
Then {yn}-2k Oscillates about the positive equilibrium point y, = (r – 1)'/2Pi with semicycles of
length one.
n=-2k
Proof. 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," Journal
of 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.
Transcribed Image Text:the positive solutions of the rational difference equation
aSn-q + bSn-r + cSn-s
dSn-q+ eSn-r + f Sn-s ,
(1)
Sn+1 = Sn-p
where 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 that
p > q > r > s > 0.
+b-
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