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 00 00 Σ 1 Eg.EB+1 = 00, s+1Bs+1 (2.1) n=N Un s=n Then: Zn (i) {} is decreasing for all n > N; C, 1/a b “Azn (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,} € S2 for all n> N. Since a,A(b,(Az,)) 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)") – bn(Az,)“ A, 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 00 00 Σ 1 Eg.EB+1 = 00, s+1Bs+1 (2.1) n=N Un s=n Then: Zn (i) {} is decreasing for all n > N; C, 1/a b “Azn (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,} € S2 for all n> N. Since a,A(b,(Az,)) 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)") – bn(Az,)“ A, A„An+1
Computer Networking: A Top-Down Approach (7th Edition)
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Chapter1: Computer Networks And The Internet
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Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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Transcribed Image Text: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 > 0,
(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 Sp< 1;
(H3) {o(n)} is a sequence of integers such that o(n) >n for all n > no;
(H4) En=no dn
= +0o and En=no Va
'00+ = D/1
Transcribed Image Text:Lemma 2. Let {yn} be a positive solution of equation (1.1) with the corresponding
sequence {zn} E $2 for n > N > no and assume that
00
00
1
(2.1)
n=N an
s=n
Then:
(i) {} is decreasing for all n> N;
1/a
AZn
(ii) {
"} is decreasing for all n > N;
1/a
An
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
ba(Azn)a > a;A(b,(Azs)ª)
> A,a„A(bn(Azn)ª), n>N.
as
s=N
From the last inequality, we obtain
A„A(b„(Azn)“) – b,(Azn)ª1
A
Un
An
A„An+1
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