Theorem 3 Assume that a <1 and |6 (e +g +s) – d (c+ f +r)|+|c (d +g +s) – e (b + f +r)| +|f (d +e +s) – g (b +c+r)|+ |r (d +e +g)– s (b +c+ f)| < (d+e+g) (6+c+ f), (12) then the positive equilibrium point (7) of Eq.(1) is locally asymptotically sta- ble. Proof: It follows by Theorem 1 that Eq.(10) is asymptotically stable if all roots of Eq.(11) lie in the open disk is |A| < 1 that is if |p4|+ |p3| + |p2| + leil+ |po] < 1, d (c +f +r)] (1 – a) [b (e +g +s) |a| + (d +e +g+s) (b +c + f +r) |(1 – a) [c (d +g +s) (d +e +g+s) (b +c +f + r) (1 – a) [f (d+e +s) (d+e +g+s) (b +c+ f +r) (1 – a) [r (d + e + g) (d +e +g+ s) (b+c + f +r) e (b + f +r)]| g (b +c+r)] s (b +c+ f)]| | < 1, and so (1 – a) [b (e +g + s) – d (c + f +r)]| (d +e +g+ s) (6+c + f +r) |(1 – a) [c (d +g +s) – e(b +f +r)]| (d +e +g+s) (6 +c+ f +r) (1 – a) [f (d +e +s) (d +e +g+s) (b +c + f +r) (1 – a) [r (d +e +g) - (d +e +g+s) (b +c + f +r) g (b +c+r)]| (b +c+ f)]| < (1– a), a < 1, or |6 (e +g +s) – d (c+ f +r)|+|c(d +g+s) – e (b+ f +r)| +If (d +e +s) – g (b +c+r)+|r (d +e +g) – s (b +c+ f)| ((d +e +g) (b+c+ f). Ca Thus, the proof is now completed.
Theorem 3 Assume that a <1 and |6 (e +g +s) – d (c+ f +r)|+|c (d +g +s) – e (b + f +r)| +|f (d +e +s) – g (b +c+r)|+ |r (d +e +g)– s (b +c+ f)| < (d+e+g) (6+c+ f), (12) then the positive equilibrium point (7) of Eq.(1) is locally asymptotically sta- ble. Proof: It follows by Theorem 1 that Eq.(10) is asymptotically stable if all roots of Eq.(11) lie in the open disk is |A| < 1 that is if |p4|+ |p3| + |p2| + leil+ |po] < 1, d (c +f +r)] (1 – a) [b (e +g +s) |a| + (d +e +g+s) (b +c + f +r) |(1 – a) [c (d +g +s) (d +e +g+s) (b +c +f + r) (1 – a) [f (d+e +s) (d+e +g+s) (b +c+ f +r) (1 – a) [r (d + e + g) (d +e +g+ s) (b+c + f +r) e (b + f +r)]| g (b +c+r)] s (b +c+ f)]| | < 1, and so (1 – a) [b (e +g + s) – d (c + f +r)]| (d +e +g+ s) (6+c + f +r) |(1 – a) [c (d +g +s) – e(b +f +r)]| (d +e +g+s) (6 +c+ f +r) (1 – a) [f (d +e +s) (d +e +g+s) (b +c + f +r) (1 – a) [r (d +e +g) - (d +e +g+s) (b +c + f +r) g (b +c+r)]| (b +c+ f)]| < (1– a), a < 1, or |6 (e +g +s) – d (c+ f +r)|+|c(d +g+s) – e (b+ f +r)| +If (d +e +s) – g (b +c+r)+|r (d +e +g) – s (b +c+ f)| ((d +e +g) (b+c+ f). Ca Thus, the proof is now completed.
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
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
Multiplying by denominator but where s,r explain this step of determine red
![Theorem 3 Assume that a < 1 and
|b (e +g + s) – d (c+ f +r)|+ |c (d +g +s) – e (b+ f +r)|
+|f (d +e +s) – g (b +c+r)I+ |r (d +e +g)
(d +e+ g) (b+ c+ f),
s (b +с+f)|
(12)
then the positive equilibrium point (7) of Eq. (1) is locally asymptotically sta-
ble.
Proof: It follows by Theorem 1 that Eq.(10) is asymptotically stable if all
roots of Eq.(11) lie in the open disk is |A| < 1 that is if |e4|+ |p3| + |p2|+
\ei|+ \po] < 1,
(1 – a) [b (e +g + s)
|a| +
(d +e +g+ s) (b + с +f+r)
d (c +f +r)]
(1 – a) [c (d +g +s)
(d +e +g+ s) (b +c + f + r)
|(1 – a) [f (d+ e + s)
(d + e +g+ s) (b +c+ ƒ +r)
(1– a) [r (d+ e + 9) – s (b +c+ f).
(d +e +g+s) (b+c + f +r)
< 1,
e (b + f +r)] |
g (b +c+r)]
and so
- - a) [b (e +g + s) – d (c + f +r)]
(d +e +g+ s) (b+ c + f +r)
|(1 – a) [e (d +g + s)
(d +e +g+s) (b +c+ f +r)
|(1 – a) [f (d + e +s)
(d +e +g+s) (b +c + f +r)
(1 – a) [r (d +e +g)
(d +e +g+s) (b +c + f +r)
e (b + f +r)]
g (b +c+r)]
(b +c+ f)]
- S
< (1- а),
а <1,
or
16 (e + g +s) – d (c+ f +r)|+ |c(d +g+s) – e (b+ f +r)|
+\f (d +e +s) – g (b +c+r)|+ |r (d +e + g) – s (b +c+ f)|
(d +e +g) (b+c+ f).)
Thus, the proof is now completed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf606203-a5b8-4153-85ec-20a78972fb86%2F8bd01d8d-25fa-4bcf-88a9-6a114fc1f281%2Fhekrswj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theorem 3 Assume that a < 1 and
|b (e +g + s) – d (c+ f +r)|+ |c (d +g +s) – e (b+ f +r)|
+|f (d +e +s) – g (b +c+r)I+ |r (d +e +g)
(d +e+ g) (b+ c+ f),
s (b +с+f)|
(12)
then the positive equilibrium point (7) of Eq. (1) is locally asymptotically sta-
ble.
Proof: It follows by Theorem 1 that Eq.(10) is asymptotically stable if all
roots of Eq.(11) lie in the open disk is |A| < 1 that is if |e4|+ |p3| + |p2|+
\ei|+ \po] < 1,
(1 – a) [b (e +g + s)
|a| +
(d +e +g+ s) (b + с +f+r)
d (c +f +r)]
(1 – a) [c (d +g +s)
(d +e +g+ s) (b +c + f + r)
|(1 – a) [f (d+ e + s)
(d + e +g+ s) (b +c+ ƒ +r)
(1– a) [r (d+ e + 9) – s (b +c+ f).
(d +e +g+s) (b+c + f +r)
< 1,
e (b + f +r)] |
g (b +c+r)]
and so
- - a) [b (e +g + s) – d (c + f +r)]
(d +e +g+ s) (b+ c + f +r)
|(1 – a) [e (d +g + s)
(d +e +g+s) (b +c+ f +r)
|(1 – a) [f (d + e +s)
(d +e +g+s) (b +c + f +r)
(1 – a) [r (d +e +g)
(d +e +g+s) (b +c + f +r)
e (b + f +r)]
g (b +c+r)]
(b +c+ f)]
- S
< (1- а),
а <1,
or
16 (e + g +s) – d (c+ f +r)|+ |c(d +g+s) – e (b+ f +r)|
+\f (d +e +s) – g (b +c+r)|+ |r (d +e + g) – s (b +c+ f)|
(d +e +g) (b+c+ f).)
Thus, the proof is now completed.
![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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf606203-a5b8-4153-85ec-20a78972fb86%2F8bd01d8d-25fa-4bcf-88a9-6a114fc1f281%2F6nijmi_processed.jpeg&w=3840&q=75)
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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