Then, the only positive equilibrium point ỹ of Eq.(1.1) is given by Ei-1 di = (2.7) Li=1 provided that A < 1. Now, let us introduce a continuous function H : (0, 00)6 → (0, 0) which is defined by H(uo, ..., uz) = Auo + (2.8) Therefore, it follows that H(uo,..us) А, Ong H(u0..,u5) H(40,..,us) duz a2 (B1u1+E(Biu;)] – B2 [a141+E_3(a;u;)] H(uo,...us) duz H(uo,...us) dug as E (B;u;)+Bgus] – Ba [C1(@;u;)+a5us] H(u0,...,u5) dug as E(B:u;)] – Bs EL(@;u;)]

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Chapter2: Second-order Linear Odes
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Show me the steps of determine yellow and 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
Ут+1 — Аутt
т 3 0, 1, 2, ...,
B1Ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4 + B3Ym-5
(1.1)
where the coefficients A, a;, B; E (0, 0), 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 = B3 = a4 =
= a5 = B5 = 0 and Eq.(1.1) has been studied in [8] in the special case
B4
when a4 =
B4 = a5 = B5 = 0 and Eq.(1.1) has been discussed in [5] in the
special case when az = B5 = 0.
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 Ут+1 — Аутt т 3 0, 1, 2, ..., B1Ym-1+ B2Ym-2 + B3Ym-3 + B4Ym-4 + B3Ym-5 (1.1) where the coefficients A, a;, B; E (0, 0), 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 = B3 = a4 = = a5 = B5 = 0 and Eq.(1.1) has been studied in [8] in the special case B4 when a4 = B4 = a5 = B5 = 0 and Eq.(1.1) has been discussed in [5] in the special case when az = B5 = 0.
2 The local stability of the solutions
In this section, the local stability of the solutions of Eq.(1.1) is investigated.
The equilibrium point ỹ of Eq.(1.1) is the positive solution of the equation
ỹ = Aỹ + Li-1i
(2.6)
Then, the only positive equilibrium point ỹ of Eq.(1.1) is given by
y =
(2.7)
(1 – A) (E )
vi=1
provided that A < 1. Now, let us introduce a continuous function
(0, 00)6 –
H :
→ (0, 00) which is defined by
H(uo,..., u5) = Auo +
(2.8)
Therefore, it follows that
H(uo,..,u5)
duo
A,
H(uo,...,u5)
H(u0,..,u5)
a2
-
(EL (B:u4))"
H(u0,...,u5)
duz
a3 (E (B;u;)+E-4(Biu;)] – Ba (Ei_,(a;u;)+£{-q(a;u;)]
H(uo,...,u5)
as (E-1 (Biu:)+Bzus] – Ba [E-1(a;u;)+a5u5]
H(u0,..,u5)
a5
Transcribed Image Text:2 The local stability of the solutions In this section, the local stability of the solutions of Eq.(1.1) is investigated. The equilibrium point ỹ of Eq.(1.1) is the positive solution of the equation ỹ = Aỹ + Li-1i (2.6) Then, the only positive equilibrium point ỹ of Eq.(1.1) is given by y = (2.7) (1 – A) (E ) vi=1 provided that A < 1. Now, let us introduce a continuous function (0, 00)6 – H : → (0, 00) which is defined by H(uo,..., u5) = Auo + (2.8) Therefore, it follows that H(uo,..,u5) duo A, H(uo,...,u5) H(u0,..,u5) a2 - (EL (B:u4))" H(u0,...,u5) duz a3 (E (B;u;)+E-4(Biu;)] – Ba (Ei_,(a;u;)+£{-q(a;u;)] H(uo,...,u5) as (E-1 (Biu:)+Bzus] – Ba [E-1(a;u;)+a5u5] H(u0,..,u5) a5
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