Lemma 2. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {zn} E S2 for n > N> no and assume that (2.1) "s+1Ds+1 = . n=N Un s=n Then: Zn (i) {} is decreasing for all n> N; bn (ii) "} is decreasing for all n> N; 1/a Zn (iii) -} is increasing for all n> N. Bn Proof. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {z,} E S2 for all n > N. Since a„A(b,(Azn)ª) is decreasing, we have n-1 b,(Azn)ª > £ a,A(b,(Azs)“) > ApdnA(b,(Azn)“), n>N. s=N as From the last inequality, we obtain a 1 (b,(Azn)ª` A,A(b,(Atn)ª) – ,(Azn)ª Un An A„An+1
Lemma 2. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {zn} E S2 for n > N> no and assume that (2.1) "s+1Ds+1 = . n=N Un s=n Then: Zn (i) {} is decreasing for all n> N; bn (ii) "} is decreasing for all n> N; 1/a Zn (iii) -} is increasing for all n> N. Bn Proof. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {z,} E S2 for all n > N. Since a„A(b,(Azn)ª) is decreasing, we have n-1 b,(Azn)ª > £ a,A(b,(Azs)“) > ApdnA(b,(Azn)“), n>N. s=N as From the last inequality, we obtain a 1 (b,(Azn)ª` A,A(b,(Atn)ª) – ,(Azn)ª Un An A„An+1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Lemma 2. Let {yn} be a positive solution of equation (1.1) with the corresponding
sequence {zn} E S2 for n >N>no and assume that
1
= o.
(2.1)
s+1
n=N
An
s=n
Then:
Zn
-} is decreasing for all n> N;
1/a
(ii)
'Azn
“} is decreasing for all n> N;
Zn
(iii) {} is increasing for all n > N.
Bn
Proof. Let {yn} be a positive solution of equation (1.1) with the corresponding
sequence {z,} E S2 for all n > N. Since a„A(bn(Azn)¤) is decreasing, we have
n-1
a,A(b,(Azs)“)
b,(Azn)ª > E
> A,a,A(b,(Azn)“), n>N.
as
s=N
From the last inequality, we obtain
b(Azn) \
A
A„A(b,(Azn)ª) – b,(Azn)ª1
an
<0
An
A„An+1
Transcribed Image Text:/d Azn is decreasing for all n> N;
In this paper, we are concerned with the asymptotic properties of solutions of the
third order neutral difference equation
A(a,A(b,(Azn)“)) +9nY%+1 = 0,
n> no 20,
(1.1)
where zn = yn+ PnYo(n), a is the ratio of odd positive integers, and the following
conditions are assumed to hold throughout:
(H1) {an}, {bn}, and {qn} are positive real sequences for all n> no;
(H2) {Pn} is a nonnegative real sequence with 0 < Pn <p< 1;
(H3) {o(n)} is a sequence of integers such that o(n) 2n for all n2 no;
(H4) Σ.
= +00 and E=no Va = +00,
%3D
Lemma 2. Let {yn} be a positive solution of equation (1.1) with the corresponding
sequence {zn} E S2 for n > N > no and assume that
1
= 0,
(2.1)
n=N
An
S=n
Then:
(i) {} is decreasing for all n > N;
ug
(ii) {
Zn
(iii) {} is increasing for all n > N.
Bn
Proof. Let {Yn} be a positive solution of equation (1.1) with the corresponding
sequence {zn} € S2 for all n> N. Since a,A(b, (Azn)“) is decreasing, we have
n-1
ba(Azn)" > 4;A(b,(Azs)ª)
> AnanA(bn(Azn)“), n2N.
as
s=N
From the last inequality, we obtain
A„A(b,(Azn)") – bn(Azn)ª1
a(ba(Ača)“) = AnA(b,(Az,)ª) – b.(Az.)ª!
An
AnAn+1
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