THEOREM 2.2. The positive equilibrium T1 of the difference equation (1) is locally asymptotically stable if c(1 - a – B) < a. (4) Proof: So, we can write Eq. (2) at the positive equilibrium point a1 a-B) %3D b(1-a- af (1, T1, T1) du = B = P1, af(T1, T1, F1) dv = a = p2 and of(T1, 1, ¤1) c(1-a-8)2 = p3. Then the linearized equation of Eq. (1) about 71 is Yn+1 - P1Yn-k - P2Yn-l - P3Yn-s = 0, According to Theorem 1.6 page 7 in [1], then Eq. (1) is asymptotically stable if and only if |p1|+ \p2| + |p3| < 1. Thus, | c(1-α-β)2 < 1, a and so c(1-a-8)2 <1-а-b, a if a + B < 1, then c(1 – a – B) < a.
THEOREM 2.2. The positive equilibrium T1 of the difference equation (1) is locally asymptotically stable if c(1 - a – B) < a. (4) Proof: So, we can write Eq. (2) at the positive equilibrium point a1 a-B) %3D b(1-a- af (1, T1, T1) du = B = P1, af(T1, T1, F1) dv = a = p2 and of(T1, 1, ¤1) c(1-a-8)2 = p3. Then the linearized equation of Eq. (1) about 71 is Yn+1 - P1Yn-k - P2Yn-l - P3Yn-s = 0, According to Theorem 1.6 page 7 in [1], then Eq. (1) is asymptotically stable if and only if |p1|+ \p2| + |p3| < 1. Thus, | c(1-α-β)2 < 1, a and so c(1-a-8)2 <1-а-b, a if a + B < 1, then c(1 – a – B) < a.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![THEOREM 2.2. The positive equilibrium T1 of the difference equation (1) is locally asymptotically stable if
с (1 — а — В) < а.
(4)
Proof: So, we can write Eq. (2) at the positive equilibrium point 71
%3D
b(1-a-
af (T1, F1, Ti)
du
= B = P1,
af(T1, #1, T1)
dv
= a = p2 and of(T1, T1, T1)
c(1-α-β)2
= p3.
Then the linearized equation of Eq. (1) about ¤1 is
Yn+1 – P1Yn-k – P2Yn-1 – P3Yn-s = 0,
According to Theorem 1.6 page 7 in [1], then Eq. (1) is asymptotically stable if and only if
\P1|+ \p2| + \p3| < 1.
Thus,
18| + |a| + 1-a-B% <1,
and so
c(1-a-B)²
<1-а- b,
a
if a + B < 1, then
с (1 — а — В) < а.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F158aed3a-44ed-4147-bb6c-9729d6a7101b%2Fd8b12922-5d66-4130-b086-9cf0250c440f%2Fgp9vik_processed.jpeg&w=3840&q=75)
Transcribed Image Text:THEOREM 2.2. The positive equilibrium T1 of the difference equation (1) is locally asymptotically stable if
с (1 — а — В) < а.
(4)
Proof: So, we can write Eq. (2) at the positive equilibrium point 71
%3D
b(1-a-
af (T1, F1, Ti)
du
= B = P1,
af(T1, #1, T1)
dv
= a = p2 and of(T1, T1, T1)
c(1-α-β)2
= p3.
Then the linearized equation of Eq. (1) about ¤1 is
Yn+1 – P1Yn-k – P2Yn-1 – P3Yn-s = 0,
According to Theorem 1.6 page 7 in [1], then Eq. (1) is asymptotically stable if and only if
\P1|+ \p2| + \p3| < 1.
Thus,
18| + |a| + 1-a-B% <1,
and so
c(1-a-B)²
<1-а- b,
a
if a + B < 1, then
с (1 — а — В) < а.
Transcribed Image Text:Our goal is to obtain some qualitative behavior of the positive solutions of the difference equation
Bxn-l+ arn-k+
п %3D0, 1, ...,
aIn-t
In+1 =
brn-t+c'
(1)
where the parameters B, a, a, b and c are positive real numbers and the initial conditions x-s, x-s+1, ..., x-1,
xo are positive real numbers where s =
тах{1, k, t}.
Let f : (0, 0)3 (0, o0) be a continuous function defined by
f(u, v, w) = Bu + av +
aw
bw+c
Therefore, it follows that
af(u, v, w)
du
= B,
af(u, v, w)
dv
af(u, v, w)
= a,
(2)
ас
(bw+c)²
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