Page 41 - IJOCTA-15-2

M. M. Parsa, K. Sayevand, H. Jafari, I. Masti / IJOCTA, Vol.15, No.2, pp.225-244 (2025)
Furthermore, the convergence of the series in the (94)
right side of Eq. (107) implies that n −iωh n n iωh n−1 −iωh
µ 1 s e + µ 2 s − µ 1 s e = −µ 1 s e
1
1
n
n
|r | = |R | ≤ η|R | = η|r |. (113)
i
i
n−1
X β β
e
− µ 2 s n−1 + µ 1 s n−1 iωh + b − b
n−l−1 n−l
l=1

n
n
l
Theorem 4. Suppose that E = s e is a so- × s + s l−1 + r .
n iωx j
j
lution of Eq. (94), then (123)
1
γ|r | The above equation can be rewritten as:
n
|s | ≤ , (114)
b β
n−1
where γ is a positive constant. n
(µ 2 − i (2µ 1 sin(ωh))) s
n
Proof. Suppose that E = s e is a solution = (−µ 2 + i (2µ 1 sin(ωh))s n−1 (124)
n iωx j
j
of Eq. (94), therefore n−1 β β
X
n
l
+ b n−l−1 − b n−l s + s l−1 + r .
n
n
n−1
µ 1 E n + µ 2 E − µ 1 E i+1 = −µ 1 E i−1 − µ 2 E n−1 l=1
i−1
i
i
Assuming that R := µ 2 −i (2µ 1 sin(ωh)), we have
n−1
X β β l−1
n−1
l
+ µ 1 E + b − b E + E
i+1 n−l−1 n−l i i
l=1
n
+ R .
i
(115) n n−1 n−1 β β
X
s + s R = b n−l−1 − b n−l
In the above equation, for n = 1, we have
l=1 (125)
1
1
µ 1 E 1 + µ 2 E − µ 1 E 1 = R . (116) l l−1 n
i−1 i i+1 i × s + s + r .
Therefore, from Eqs. (108) and (109) Then
1
1
1 iωh
1 −iωh
µ 1 s e + µ 2 s − µ 1 s e = r , (117)
or
1 n−1
X
n β β

1
1
µ 1 e −iωh + µ 2 − µ 1 e iωh s = r , (118) s + s n−1 ≤ b n−l−1 − b n−l
|R|
l=1 (126)
which is equivalent to
l
n
× |s | + |s l−1 | + |r |.
1
1
(µ 2 − i (2µ 1 sin(ωh))) s = r . (119) It is clear that |R| ≥ 1. Therefore
Assuming that R := µ 2 −i (2µ 1 sin(ωh)), we have
1 1 n n−1 n−1 β β
X
1
|s | = |r |. (120) |s | + |s | ≤ b n−l−1 − b n−l
|R|
l=1 (127)
We know from the previous section that |R| ≥ 1 l l−1 n
× |s | + |s | + |r |.
β
and b = 1, therefore According to the assumption of induction (122)
0
1 1 and Lemma 5, we have
1
1
|s | ≤ |r | = β |r |. (121)
b 0 −γ n−1
X
1
n
|s | ≤ |r | + b β − b β
Now, with the help of mathematical induction, we b β n−2 l=1 n−l−1 n−l
γ|r | γ|r | γ|r |
1 1 1
assume that for k = 2, 3, . . . , n − 1, the following × β + β + γ|r | = β
1
relation is established: b l−1 b l−2 b n−1
γ β n−1 (128)
1
k
|s | ≤ |r |, (122) −b n−1 X β β
b β × β + b n−l−1 − b n−l
k−1 b n−2 l=1
where γ = max{1, η} is a positive constant. We β β !
b b
× n−1 + n−1 + b β ,
n−1
show that it holds for k = n as well. From Eq. b β b β
l−1 l−2
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