2 The local stability of the solutions The equilibrium point x of Eq. (1) is the positive solution of the equation Ỹ=(A+B+C+D)x+ x= where d = e. If [(A+B+C+D) − 1] (e – d) > 0, then the only positive equilibrium point Ỹ of Eq. (1) is given by b [(A+B+C+D) − 1] (e – d) * Let us now introduce a continuous F: (0,∞)4 → (0,∞) which is defined by F(uo, u1, U2, U3) = Auo + Bu₁ + Cu₂+ Du3+ provided du₁eu₂. Consequently, we get ƏF(x,x,x,x) = дио JF(x,x,x,x) ди JF(x,x,x,x) 242 bx (d- e)x = C+ = B_ e [(A+B+C+D) −1] (e - d) = A = Po, JF(x,x,x,x) = Ju3 e[(A+B+C+D) −1] (ed) 2 D = P3, function (7) bu₁ (du₁ - eu₂) (8) = P1, P2, (9) where e ‡ d. Thus, the linearized equation of Eq. (1) about x takes the form Cat1 Zn+1-Pozn-P1Zn-k-P2²n-1-P3²n-σ = 0, where po, P1, P2 and p3 are given by (9). (10) (6)

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Chapter2: Second-order Linear Odes
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Explain the determine red and the inf is here

The objective of this article is to investigate some
qualitative behavior of the solutions of the nonlinear
difference equation
bxn– k
X+1 = Axn+ Bxp–k+CXp-1+Dxp-o+
[dxn-k– ex-1
(1)
n= 0, 1,2, .....
where the coefficients A, B, C, D, b, d, e e (0,00), 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 < 0. 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=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.
••..
- b and in [27] when
6.
Transcribed Image Text:The objective of this article is to investigate some qualitative behavior of the solutions of the nonlinear difference equation bxn– k X+1 = Axn+ Bxp–k+CXp-1+Dxp-o+ [dxn-k– ex-1 (1) n= 0, 1,2, ..... where the coefficients A, B, C, D, b, d, e e (0,00), 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 < 0. 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=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. ••.. - b and in [27] when 6.
2 The local stability of the solutions
The equilibrium point x of Eq.(1) is the positive solution
of the equation
x= (A+B+C+D)x+
(6)
(d-e)x'
where d + e. If [(A+B+C+D) – 1] (e – d) > 0, then
the only positive equilibrium point ž of Eq.(1) is given by
|
b
(7)
!!
[(A+B+C+D) – 1] (e – d)
Let
us
now
introduce
a
continuous
function
F: (0,00)4 -
→ (0, 00) which is defined by
bui
F(u0, U1, U2, U3) = Auo + Bun + Cu2+ Duz +
(du – euz)'
(8)
provided du # euz. Consequently, we get
ƏF(x,x,X,X)
= A= Po,
One
e [(A+B+C+D) –1]
(e - d)
ƏF(x,X,x,X)
-.
B-
= P1,
(9)
ƏF(x,x,x,X)
e[(A+B+C+D) –1]
(e - d)
= C+
= P2,
ƏF(x,x,x,X)
d u3
D=P3,
%3D
where et d. Thus, the linearized equation of Eq.(1) about
x takes the form
Zn+1 - Pozn- P1Zn-k - P2 Zn-1- P3Zn-o = 0,
(10)
where po, P1, P2 and p3 are given by (9).
Transcribed Image Text:2 The local stability of the solutions The equilibrium point x of Eq.(1) is the positive solution of the equation x= (A+B+C+D)x+ (6) (d-e)x' where d + e. If [(A+B+C+D) – 1] (e – d) > 0, then the only positive equilibrium point ž of Eq.(1) is given by | b (7) !! [(A+B+C+D) – 1] (e – d) Let us now introduce a continuous function F: (0,00)4 - → (0, 00) which is defined by bui F(u0, U1, U2, U3) = Auo + Bun + Cu2+ Duz + (du – euz)' (8) provided du # euz. Consequently, we get ƏF(x,x,X,X) = A= Po, One e [(A+B+C+D) –1] (e - d) ƏF(x,X,x,X) -. B- = P1, (9) ƏF(x,x,x,X) e[(A+B+C+D) –1] (e - d) = C+ = P2, ƏF(x,x,x,X) d u3 D=P3, %3D where et d. Thus, the linearized equation of Eq.(1) about x takes the form Zn+1 - Pozn- P1Zn-k - P2 Zn-1- P3Zn-o = 0, (10) where po, P1, P2 and p3 are given by (9).
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