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 = axn + dxn-1 + exn-2+ gxn-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)

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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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)
а
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 = aXn +
dxn-1 + exn-2+ gxn-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)
а
Transcribed Image Text: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) а 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 = aXn + dxn-1 + exn-2+ gxn-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) а
The objective of this article is to investigate some qualitative behavior of
the solutions of the nonlinear difference equation
бxn-1 + сх,n-2 + fxn-3 + rtn-4
Xn+1 = axn +
n = 0, 1, 2, .. (1)
dxn-1 + en-2 + gxn-3 + sxn-4
where the coefficients a, b, c, d, e, f, g,r, s E (0, 0), while the initial con-
ditions x-4,X –3,X –2, x – 1, xo are arbitrary positive real numbers. Note that
the special cases of Eq.(1) has been studied discussed in [11] when f = g =
0 and Eq. (1) has been studied discussed in [35] in the special case
r = s =
when r = s = 0.
Transcribed Image Text:The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation бxn-1 + сх,n-2 + fxn-3 + rtn-4 Xn+1 = axn + n = 0, 1, 2, .. (1) dxn-1 + en-2 + gxn-3 + sxn-4 where the coefficients a, b, c, d, e, f, g,r, s E (0, 0), while the initial con- ditions x-4,X –3,X –2, x – 1, xo are arbitrary positive real numbers. Note that the special cases of Eq.(1) has been studied discussed in [11] when f = g = 0 and Eq. (1) has been studied discussed in [35] in the special case r = s = when r = s = 0.
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