From which we have (a1 + a2 + a3 + a4) D+a5d-(1-A) (B1 + 32 +33 +34) D² = (1 - A) 35 Dd (5.32) and (a₁ + a2 + a3 + a4) d+a5D-(1-A) (B₁ + B2 + B3+ B4) d² = (1 - A) 5 Dd (5.33) From (5.32) and (5.33), we obtain (d-D) {[(a1 + a2 + a3 + a4) - a5] - (1 - A) (B1 + B2+ B3 + B4) (d+ D)} = 0. (5.34) Since A 1 and a5 2 (a1 + a2 + a3 + a4), we deduce from (5.34) that D = d. It follows by Theorem 2, that y of Eq. (1.1) is a global attractor.
From which we have (a1 + a2 + a3 + a4) D+a5d-(1-A) (B1 + 32 +33 +34) D² = (1 - A) 35 Dd (5.32) and (a₁ + a2 + a3 + a4) d+a5D-(1-A) (B₁ + B2 + B3+ B4) d² = (1 - A) 5 Dd (5.33) From (5.32) and (5.33), we obtain (d-D) {[(a1 + a2 + a3 + a4) - a5] - (1 - A) (B1 + B2+ B3 + B4) (d+ D)} = 0. (5.34) Since A 1 and a5 2 (a1 + a2 + a3 + a4), we deduce from (5.34) that D = d. It follows by Theorem 2, that y of Eq. (1.1) is a global attractor.
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
Show me the steps of determine green and the inf is here
![The main focus of this article is to discuss some qualitative behavior of
the solutions of the nonlinear difference equation
a1Ym-1+a2Ym-2 + a3ym-3+ a4Ym-4 + a5Ym-5
Aym+
B1ym-1 + B2ym-2 + B3Ym-3 + B4Ym-4 + BsYm-5
т %3D 0, 1, 2, ...,
Ym+1 =
(1.1)
where the coefficients A, ai, Bi E (0, 00), i = 1, ..., 5, while the initial condi-
tions y-5,y-4,Y–3,Y-2, y-1, yo are arbitrary positive real numbers. Note that
the special case of Eq.(1.1) has been discussed in [4] when az =
B4
when a4 = B4 = a5 = B5 = 0 and Eq.(1.1) has been discussed in [5] in the
special case when az = B5 = 0.
B3 = a4 =
B5 = 0 and Eq.(1.1) has been studied in [8] in the special case
= a5 =
Theorem 2 ([6). Let H : [a, b]k+1 → [a, b] be a continuous function, where
k is a positive integer, and where [a, b] is an interval of real numbers. Con-
sider the difference equation (1.2). Suppose that H satisfies the following
conditions:
1. For each integer i with1 < i < k+ 1; the function H(z1, z2, ..., Zk+1)
is weakly monotonic in zi for fixed z1, z2, ..., Zi-1, Zi+1, ..., Zk+1•
2. If (d, D) is a solution of the system
d = H(d1, d2, ., de+1) and D= H(D1, D2, .., Dk+1),
then d = D, where for each i = 1, 2,
..., k +1, we set
d
di = {
if F is non – decreasing in zi
if F is non – increasing in zi
D
аnd
{
(D if F is non – decreasing in z;
Di =
if
F is non – increasing in zị.
Then there exists exuctly one equilibrium y of Eq.(1.2), and every solution
of Eq. (1.2) converges to y.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7410b99-86dc-4094-add7-8a5727a4f3de%2F6ade2ffa-0920-4a7b-8f3d-845b980a3db7%2Ftyspa3j_processed.png&w=3840&q=75)
Transcribed Image Text:The main focus of this article is to discuss some qualitative behavior of
the solutions of the nonlinear difference equation
a1Ym-1+a2Ym-2 + a3ym-3+ a4Ym-4 + a5Ym-5
Aym+
B1ym-1 + B2ym-2 + B3Ym-3 + B4Ym-4 + BsYm-5
т %3D 0, 1, 2, ...,
Ym+1 =
(1.1)
where the coefficients A, ai, Bi E (0, 00), i = 1, ..., 5, while the initial condi-
tions y-5,y-4,Y–3,Y-2, y-1, yo are arbitrary positive real numbers. Note that
the special case of Eq.(1.1) has been discussed in [4] when az =
B4
when a4 = B4 = a5 = B5 = 0 and Eq.(1.1) has been discussed in [5] in the
special case when az = B5 = 0.
B3 = a4 =
B5 = 0 and Eq.(1.1) has been studied in [8] in the special case
= a5 =
Theorem 2 ([6). Let H : [a, b]k+1 → [a, b] be a continuous function, where
k is a positive integer, and where [a, b] is an interval of real numbers. Con-
sider the difference equation (1.2). Suppose that H satisfies the following
conditions:
1. For each integer i with1 < i < k+ 1; the function H(z1, z2, ..., Zk+1)
is weakly monotonic in zi for fixed z1, z2, ..., Zi-1, Zi+1, ..., Zk+1•
2. If (d, D) is a solution of the system
d = H(d1, d2, ., de+1) and D= H(D1, D2, .., Dk+1),
then d = D, where for each i = 1, 2,
..., k +1, we set
d
di = {
if F is non – decreasing in zi
if F is non – increasing in zi
D
аnd
{
(D if F is non – decreasing in z;
Di =
if
F is non – increasing in zị.
Then there exists exuctly one equilibrium y of Eq.(1.2), and every solution
of Eq. (1.2) converges to y.
![Case 1. Let the function H(uo,
..., u5) is non-decreasing in uo,U1,U2,U3,U4
and non-increasing in uz. Suppose that (d, D) is a solution of the system
D = H(D, D, D, D, D, d)
and
d = H(d, d, d, d, d, D).
Then we get
a1 D+ a2D+a3D+ a4D+ azd
ajd + azd + azd + a4d + a5 D
D = AD+
and
d = Ad+
B1D+ B2D + B3D+ B4D + Bzd
Bid + B2d + B3d + Bąd + B3D
or
(a1 + a2 + a3+a4) D + a5d
(B1 + B2 + B3 + B4) D + Bzd
(a1 + a2 + a3+ a4) d + a5D
(B1 + B + Bз + BA) d + BsD
D (1 – A)
аnd d (1- A)
From which we have
(a1 + a2 + a3 + a4) D+azd-(1 – A) (B1 + B2 + B3 + B4) D² = (1 – A) B5Dd
(5.32)
and
(a1 + a2 + a3 + a4) d+a5D-(1 – A) (ß1 + B2 + B3 + B4) d² = (1 – A) B5 Dd
(5.33)
From (5.32) and (5.33), we obtain
(d – D) {[(a1+ a2 + a3 + a4) – a3] – (1 – A) (B1 + B2 + B3 + B4) (d + D)} :
(5.34)
Since A < 1 and as 2 (a1 + a2 + a3 + a4), we deduce from (5.34) that
D = d. It follows by Theorem 2, that ỹ of Eq.(1.1) is a global attractor.
= 0.
17](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7410b99-86dc-4094-add7-8a5727a4f3de%2F6ade2ffa-0920-4a7b-8f3d-845b980a3db7%2Frpkqc6o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Case 1. Let the function H(uo,
..., u5) is non-decreasing in uo,U1,U2,U3,U4
and non-increasing in uz. Suppose that (d, D) is a solution of the system
D = H(D, D, D, D, D, d)
and
d = H(d, d, d, d, d, D).
Then we get
a1 D+ a2D+a3D+ a4D+ azd
ajd + azd + azd + a4d + a5 D
D = AD+
and
d = Ad+
B1D+ B2D + B3D+ B4D + Bzd
Bid + B2d + B3d + Bąd + B3D
or
(a1 + a2 + a3+a4) D + a5d
(B1 + B2 + B3 + B4) D + Bzd
(a1 + a2 + a3+ a4) d + a5D
(B1 + B + Bз + BA) d + BsD
D (1 – A)
аnd d (1- A)
From which we have
(a1 + a2 + a3 + a4) D+azd-(1 – A) (B1 + B2 + B3 + B4) D² = (1 – A) B5Dd
(5.32)
and
(a1 + a2 + a3 + a4) d+a5D-(1 – A) (ß1 + B2 + B3 + B4) d² = (1 – A) B5 Dd
(5.33)
From (5.32) and (5.33), we obtain
(d – D) {[(a1+ a2 + a3 + a4) – a3] – (1 – A) (B1 + B2 + B3 + B4) (d + D)} :
(5.34)
Since A < 1 and as 2 (a1 + a2 + a3 + a4), we deduce from (5.34) that
D = d. It follows by Theorem 2, that ỹ of Eq.(1.1) is a global attractor.
= 0.
17
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