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n=N un s=n = decreasing for all n>N; Zn "} is decreasing for all n> N;

n=N un s=n = decreasing for all n>N; Zn "} is decreasing for all n> N;

Advanced Engineering Mathematics
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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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
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
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), ɑ 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) > n for all n > no; (H4) Ln=no an = +00 and E=no Va =+∞.
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), ɑ 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) > n for all n > no; (H4) Ln=no an = +00 and E=no Va =+∞.
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