This work aims to investigate the equilibria, local stability, global attractivity and the exact solutions of the following difference equations Bun-1un-5 Un+1 = Qun-1+ n = 0,1,..., (1) YUn-3 - dun-5 | Bun-1un-5 Un+1 = aun-1 n = 0,1, .., (2) - YUn-3 + dun-5 where the coefficients a, B, y, and & are positive real numbers and the initial con- ditions u; for all i = -5, -4,., 0, are arbitrary non-zero real numbers. We also present the numerical solutions via some 2D graphs. 2. ON THE EQUATION Uln+1 = QUn-1+ Bun-1un-5 Yun-3-dun-5 This section is devoted to study the qualitative behaviors of Eq. (1). The equilibrium point of Eq. (1) is given by

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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
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This work aims to investigate the equilibria, local stability, global attractivity
and the exact solutions of the following difference equations
Bun-1un-5
Un+1 = aUn-1+
n = 0,1, ..,
(1)
YUn-3 - dun-5'
Bun-1un-5
Un+1 = aUn-1-
n = 0,1, ..,
(2)
YUn-3 + dun-5'
where the coefficients a, B, y, and & are positive real numbers and the initial con-
ditions ui for all i = -5, -4, .., 0, are arbitrary non-zero real numbers. We also
present the numerical solutions via some 2D graphs.
2. ON THE EQUATION Un+1 = aUn-1 +
Bun-1un-5
Yun-3-dun-5
(1). The
This section is devoted to study the qualitative behaviors of Eq.
equilibrium point of Eq. (1) is given by
6. EXACT SOLUTION OF EQ. (1) WHEN a =
B=y= 8 = 1
In this section, we investigate the exact solutions of the following rational differ-
ence equation
Un-1un-5
Иn+1 — ит-1 +
n = 0, 1, ...,
(10)
Un-3 - Un-5
where the initial conditions are positive real numbers.
Theorem 5 Let {un}n=-5
be a solution to Eq. (10) and suppose that u-5 =
d, u-1 = e, uo = f. Then, for n = 0, 1,2, .., the
а, и-4 — б, и_з — с, и-2 —
solutions of Eq. (10) are given by the following formulas:
e2n c"
U8n-5 =
(c- e)"(a – c)n'
f2n d"
bn-1(d - f)"(b– d)n'
cn+le2n
n-1
U8n-4 =
U8n-3 =
a" (a – c)" (c – e)"'
dn+1 f2n
b" (b – d)" (d- f)"'
e2n+1cn
U8n-2 =
U8n-1 =
a" (a – c)"(c – e)* '
f2n+1gn
br (b – d)"(d – f)"'
cn+1 e2n+1
U8n
U8n+1
a" (c - e)" (a – c)n+1>
dn+1 f2n+1
b" (d – f)"(b – d)n+1*
U8n+2 =
Proof.
It can be easily observed that the solutions are true for n = 0. We now
assume that n > 0 and that our assumption holds for n – 1. That is,
e2n-2 cn-1
n-2(c – e)n-1(a – c)n-1'
f2n-2 an-1
b2-2 (d – f)n-1(b – d)a-I?
U8n-13 =
U8n-12 =
c"e2n-2
U8n-11
n-1(a - c)a-1(c – e)n–1'
an-1
Transcribed Image Text:This work aims to investigate the equilibria, local stability, global attractivity and the exact solutions of the following difference equations Bun-1un-5 Un+1 = aUn-1+ n = 0,1, .., (1) YUn-3 - dun-5' Bun-1un-5 Un+1 = aUn-1- n = 0,1, .., (2) YUn-3 + dun-5' where the coefficients a, B, y, and & are positive real numbers and the initial con- ditions ui for all i = -5, -4, .., 0, are arbitrary non-zero real numbers. We also present the numerical solutions via some 2D graphs. 2. ON THE EQUATION Un+1 = aUn-1 + Bun-1un-5 Yun-3-dun-5 (1). The This section is devoted to study the qualitative behaviors of Eq. equilibrium point of Eq. (1) is given by 6. EXACT SOLUTION OF EQ. (1) WHEN a = B=y= 8 = 1 In this section, we investigate the exact solutions of the following rational differ- ence equation Un-1un-5 Иn+1 — ит-1 + n = 0, 1, ..., (10) Un-3 - Un-5 where the initial conditions are positive real numbers. Theorem 5 Let {un}n=-5 be a solution to Eq. (10) and suppose that u-5 = d, u-1 = e, uo = f. Then, for n = 0, 1,2, .., the а, и-4 — б, и_з — с, и-2 — solutions of Eq. (10) are given by the following formulas: e2n c" U8n-5 = (c- e)"(a – c)n' f2n d" bn-1(d - f)"(b– d)n' cn+le2n n-1 U8n-4 = U8n-3 = a" (a – c)" (c – e)"' dn+1 f2n b" (b – d)" (d- f)"' e2n+1cn U8n-2 = U8n-1 = a" (a – c)"(c – e)* ' f2n+1gn br (b – d)"(d – f)"' cn+1 e2n+1 U8n U8n+1 a" (c - e)" (a – c)n+1> dn+1 f2n+1 b" (d – f)"(b – d)n+1* U8n+2 = Proof. It can be easily observed that the solutions are true for n = 0. We now assume that n > 0 and that our assumption holds for n – 1. That is, e2n-2 cn-1 n-2(c – e)n-1(a – c)n-1' f2n-2 an-1 b2-2 (d – f)n-1(b – d)a-I? U8n-13 = U8n-12 = c"e2n-2 U8n-11 n-1(a - c)a-1(c – e)n–1' an-1
d" f2n-2
(b – d)"-1(d – f)"-1’
e2n-1 cn-1
И8п-10
bn-1
U8n-9
1(a – c)n-1(c – e)"-I’
f2n-1qn-1
(b – d)n-1(d – f)n-I³
an-
U8n-8 =
bn-1
c"e2n-1
U8n-7 =
n-1(c – e)n-1(a – c)n'
d" f2n-1
br-1 (d – f)n-1(b – d)n
an-
U8n-6 =
Eq. (10) gives us that
U8n-7u8n-11
И8п-5 — и8n-7 +
и8п-9 — и8n-11
cne2n-1
c"e2n-1
c"e2n-2
-1 (с — е)"-1 (а — с)"
an-1(c-e) n-1(a-c)n an-1(a-c)n-1(c-e)n –1
e2n–1 cn-1
c)n-1
an
спе2n -2
an-
(с-е)п-Т
(а-с)п-
an-I
(c-e)n-I
c"e2n-1
e2n-2 n
(c – e)n–1
an-
- c)"
1(e – e)n-1(a – c)" ( - )
а —
an-1
c"e2n-1
e2n-1 n+1
+
– e)n-1(a
e2n cn
— с)"
an-1
(c- e)"(a – c)"
an-1
с —
-'(c – e)" (a – c)™ *
an-1
Moreover, it can be seen from Eq. (10) that
U8n-6U8n-10
И8п—4 — и8n-6 +
И8п-8 — и8n-10
d" f2n-1
d" f2n-1
br-1(d – f)"-1(b – d)"
dn f2n-2
bn-1(d-f)n-1(b-d)n bn-1(b-d)n-1 (d-f)n-I
f2n-1dn-1
bn-1(b-d)n -1(d-f)n-1
dn f2n-2
bn-1(b-d) n
(d-f)n-1
d" f2n-1
br-1 (d — f)"-1(ь — d)"
dr f2n-2
bn-1 (d – f)"-1(b – d)" (à – })
d" f2n-1
bn-1(d – f)"-1(b – d)n
f2m dr
bn-1(d – f)"(b – d)"°
dn+1 f2n-1
bn-1 (d – f)" (b – d)"
Transcribed Image Text:d" f2n-2 (b – d)"-1(d – f)"-1’ e2n-1 cn-1 И8п-10 bn-1 U8n-9 1(a – c)n-1(c – e)"-I’ f2n-1qn-1 (b – d)n-1(d – f)n-I³ an- U8n-8 = bn-1 c"e2n-1 U8n-7 = n-1(c – e)n-1(a – c)n' d" f2n-1 br-1 (d – f)n-1(b – d)n an- U8n-6 = Eq. (10) gives us that U8n-7u8n-11 И8п-5 — и8n-7 + и8п-9 — и8n-11 cne2n-1 c"e2n-1 c"e2n-2 -1 (с — е)"-1 (а — с)" an-1(c-e) n-1(a-c)n an-1(a-c)n-1(c-e)n –1 e2n–1 cn-1 c)n-1 an спе2n -2 an- (с-е)п-Т (а-с)п- an-I (c-e)n-I c"e2n-1 e2n-2 n (c – e)n–1 an- - c)" 1(e – e)n-1(a – c)" ( - ) а — an-1 c"e2n-1 e2n-1 n+1 + – e)n-1(a e2n cn — с)" an-1 (c- e)"(a – c)" an-1 с — -'(c – e)" (a – c)™ * an-1 Moreover, it can be seen from Eq. (10) that U8n-6U8n-10 И8п—4 — и8n-6 + И8п-8 — и8n-10 d" f2n-1 d" f2n-1 br-1(d – f)"-1(b – d)" dn f2n-2 bn-1(d-f)n-1(b-d)n bn-1(b-d)n-1 (d-f)n-I f2n-1dn-1 bn-1(b-d)n -1(d-f)n-1 dn f2n-2 bn-1(b-d) n (d-f)n-1 d" f2n-1 br-1 (d — f)"-1(ь — d)" dr f2n-2 bn-1 (d – f)"-1(b – d)" (à – }) d" f2n-1 bn-1(d – f)"-1(b – d)n f2m dr bn-1(d – f)"(b – d)"° dn+1 f2n-1 bn-1 (d – f)" (b – d)"
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