Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![Theorem 14.If 0< A+B+ C+D<1 and e+ d, then
the equilibrium point x given by (7) of Eq.(1) is global
attractor.
Proof.We consider the following function
by
F(x, y, z, w) = Ax+By+Cz+ Dw+
(50)
(dy- ez)'
ez)?>bez and
where dy # ez, provided that B(dy -
C(dy- ez)? + bey>0. It is easy to verify the condition (i)
of Theorem 3. Let us now verify the condition (ii) of
Theorem 3 as follows:
[F(x, х, х, х) — х (х— X) —
b
K5x-
X
е —
- -
X
[Ks — 1] (е- d)
S x(e- d) [Ks – 1] – b \
2
e-d
1
(51)
[K5 – 1]
where K5 = (A+B+C+D). Since 0 < K5 < 1 and e+ d,
then we deduce from (51) that
[F(x, х, х, х) — х] (х— 3) <0.
(52)
According to Theorem 3, is global attractor. Thus, the
proof is now completed.O
On combining the two Theorems 4 and 14, we have the
result.](https://content.bartleby.com/qna-images/question/3e6fa233-173d-4c75-ac91-9f159b3c4543/96e4f06c-2544-4e6b-afd4-bffa669d8e72/1lyptew_processed.jpeg)
Transcribed Image Text:Theorem 14.If 0< A+B+ C+D<1 and e+ d, then
the equilibrium point x given by (7) of Eq.(1) is global
attractor.
Proof.We consider the following function
by
F(x, y, z, w) = Ax+By+Cz+ Dw+
(50)
(dy- ez)'
ez)?>bez and
where dy # ez, provided that B(dy -
C(dy- ez)? + bey>0. It is easy to verify the condition (i)
of Theorem 3. Let us now verify the condition (ii) of
Theorem 3 as follows:
[F(x, х, х, х) — х (х— X) —
b
K5x-
X
е —
- -
X
[Ks — 1] (е- d)
S x(e- d) [Ks – 1] – b \
2
e-d
1
(51)
[K5 – 1]
where K5 = (A+B+C+D). Since 0 < K5 < 1 and e+ d,
then we deduce from (51) that
[F(x, х, х, х) — х] (х— 3) <0.
(52)
According to Theorem 3, is global attractor. Thus, the
proof is now completed.O
On combining the two Theorems 4 and 14, we have the
result.
![The objective of this article is to investigate some
qualitative behavior of the solutions of the nonlinear
difference equation
bxn-k
Xn+1 = Axn+ Bxŋ-k+Cxn-1+Dxn-o+
[dxn-k– exn-1]
(1)
n = 0,1,2, ...
where the coefficients A, B, C, D, b, d, e E (0,0), while
k, 1 and o are positive integers. The initial conditions
X-g,..., X_1,..., X_k ..., X_1, Xo are arbitrary positive real
numbers such that k <1< o. Note that the special cases
of 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] when
B=C= D=0, and k = 0, b is replaced by – b and in
[33] when B = C = D = 0, 1= 0 and in [32] when
A= C= D=0, 1=0, b is replaced by – b.
6.
•.•9
-
6.
|
(7)
[(A+B+C+ D) – 1] (e – d)
Let
us
now
introduce
a
continuous
function
F: (0, 0)* → (0, ) which is defined by
ai
bu
Theorem 3.[12]. Consider the difference equation (2).
Let xe I be an equilibrium point of Eq. (2). Suppose also
that
(i) F
is a nondecreasing function in each of its
arguments.
(ii) The function F satisfies the negative feedback
property
[F (x, X,х, х) — х (х- %) <0 for all х€1-{},
where I is an open interval of real numbers. Then x is
global attractor for all solutions of Eq. (2).](https://content.bartleby.com/qna-images/question/3e6fa233-173d-4c75-ac91-9f159b3c4543/96e4f06c-2544-4e6b-afd4-bffa669d8e72/apkvba7_processed.png)
Transcribed Image Text:The objective of this article is to investigate some
qualitative behavior of the solutions of the nonlinear
difference equation
bxn-k
Xn+1 = Axn+ Bxŋ-k+Cxn-1+Dxn-o+
[dxn-k– exn-1]
(1)
n = 0,1,2, ...
where the coefficients A, B, C, D, b, d, e E (0,0), while
k, 1 and o are positive integers. The initial conditions
X-g,..., X_1,..., X_k ..., X_1, Xo are arbitrary positive real
numbers such that k <1< o. Note that the special cases
of 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] when
B=C= D=0, and k = 0, b is replaced by – b and in
[33] when B = C = D = 0, 1= 0 and in [32] when
A= C= D=0, 1=0, b is replaced by – b.
6.
•.•9
-
6.
|
(7)
[(A+B+C+ D) – 1] (e – d)
Let
us
now
introduce
a
continuous
function
F: (0, 0)* → (0, ) which is defined by
ai
bu
Theorem 3.[12]. Consider the difference equation (2).
Let xe I be an equilibrium point of Eq. (2). Suppose also
that
(i) F
is a nondecreasing function in each of its
arguments.
(ii) The function F satisfies the negative feedback
property
[F (x, X,х, х) — х (х- %) <0 for all х€1-{},
where I is an open interval of real numbers. Then x is
global attractor for all solutions of Eq. (2).
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