In this paper, we are concerned with the asymptotic properties of solutions of the third order neutral difference equation A(a,A(b,(Azn)“)+4ny+1=0, n> no 2 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 n for all n > no; (H4) Ln=no un = +0o and Ln=no ya = +00, 1/a Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {zn} E So for n > N. Then: (i) (1– p)zn N; (ii) {Zn¢n} is increasing for all n> N. Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres- Theorem 1. If condition (2.1) holds and Ca 00 1 lim sup s+1 s+1 > 1, (2.5) An+1 s=N Ba n+1 s=n+1 n00 then S2 = Ø. 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 = 0. (2.1) n=N Un s=n Then: Zn (i) {} is decreasing for all n> N; bn (ii) { "} is decreasing for all n> N; An Zn (iii) { } is increasing for all n > N. Bn Kiguradze's theorem [1] can be used to describe the structure of the solution space for the nonoscillatory solutions. For example, for the ordinary difference equation (see [13]) A?((Ayn)") +9ny+ = 0, n> no, (1.2) the set K of all positive solutions has the decomposition K - KaIK Theorem 2. Assume that conditions (2.1) and (2.5) hold. If {yn} is a positive solution of equation (1.1), then there are positive constants C, and C2 such that Ciyı (n) < yn < C242(n), (2.9) where n-1 1 Vı (n) and y2(n) = II(1 – R). On s=N Corollary 1. Assume that = 00 (2.10) n=N \s=n and n 00 1 lim sup E(s+1)"(s+2)"q;+(n+2)" (s+1)"qs }> 2". (2.11) n+1 s=N n00 s=n+1 every positive solution {y,} of equation (1.2) is decreasing and satisfies estim- ates of the form (2.9) with p = 0 in Rp. Then Theorem 2. Assume that conditions (2.1) and (2.5) hold. If {yn} is a positive solution of equation (1.1), then there are positive constants Cı and C2 such that Ciyi (n) < yn N. Therefore, (1 — р) Yn 2 On -P) ZNON = C1¥1 (n). On On the other hand, summing equation (1.1) from n to o and applying Lemma 4 (i), we have 00 a„A(b,(Ačn)“) = £ 9. 2 (1– p)“ £9. 's+1 s=n s=n s=n Again summing the last inequality and applying Lemma 4 (ii) gives 1 00 00 -b,(Azn)“ > (1– p)“ E; Yn n a. s=n us t=s Yt+1 or Azn <-Rn, n> N. Zn Summing the last inequality from N to n – 1, we obtain Yn < Zn 2ª. (2.11) n00 n+1 s=N s=n+1 Then every positive solution {y„} of equation (1.2) is decreasing and satisfies estim- ates of the form (2.9) with p = 0 in R,.
In this paper, we are concerned with the asymptotic properties of solutions of the third order neutral difference equation A(a,A(b,(Azn)“)+4ny+1=0, n> no 2 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 n for all n > no; (H4) Ln=no un = +0o and Ln=no ya = +00, 1/a Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding sequence {zn} E So for n > N. Then: (i) (1– p)zn N; (ii) {Zn¢n} is increasing for all n> N. Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres- Theorem 1. If condition (2.1) holds and Ca 00 1 lim sup s+1 s+1 > 1, (2.5) An+1 s=N Ba n+1 s=n+1 n00 then S2 = Ø. 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 = 0. (2.1) n=N Un s=n Then: Zn (i) {} is decreasing for all n> N; bn (ii) { "} is decreasing for all n> N; An Zn (iii) { } is increasing for all n > N. Bn Kiguradze's theorem [1] can be used to describe the structure of the solution space for the nonoscillatory solutions. For example, for the ordinary difference equation (see [13]) A?((Ayn)") +9ny+ = 0, n> no, (1.2) the set K of all positive solutions has the decomposition K - KaIK Theorem 2. Assume that conditions (2.1) and (2.5) hold. If {yn} is a positive solution of equation (1.1), then there are positive constants C, and C2 such that Ciyı (n) < yn < C242(n), (2.9) where n-1 1 Vı (n) and y2(n) = II(1 – R). On s=N Corollary 1. Assume that = 00 (2.10) n=N \s=n and n 00 1 lim sup E(s+1)"(s+2)"q;+(n+2)" (s+1)"qs }> 2". (2.11) n+1 s=N n00 s=n+1 every positive solution {y,} of equation (1.2) is decreasing and satisfies estim- ates of the form (2.9) with p = 0 in Rp. Then Theorem 2. Assume that conditions (2.1) and (2.5) hold. If {yn} is a positive solution of equation (1.1), then there are positive constants Cı and C2 such that Ciyi (n) < yn N. Therefore, (1 — р) Yn 2 On -P) ZNON = C1¥1 (n). On On the other hand, summing equation (1.1) from n to o and applying Lemma 4 (i), we have 00 a„A(b,(Ačn)“) = £ 9. 2 (1– p)“ £9. 's+1 s=n s=n s=n Again summing the last inequality and applying Lemma 4 (ii) gives 1 00 00 -b,(Azn)“ > (1– p)“ E; Yn n a. s=n us t=s Yt+1 or Azn <-Rn, n> N. Zn Summing the last inequality from N to n – 1, we obtain Yn < Zn 2ª. (2.11) n00 n+1 s=N s=n+1 Then every positive solution {y„} of equation (1.2) is decreasing and satisfies estim- ates of the form (2.9) with p = 0 in R,.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![In this paper, we are concerned with the asymptotic properties of solutions of the
third order neutral difference equation
A(a,A(b,(Azn)“)+4ny+1=0, n> no 2 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 <p<1;
(H3) {o(n)} is a sequence of integers such that o(n) > n for all n > no;
(H4) Ln=no un
= +0o and Ln=no ya
= +00,
1/a
Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding
sequence {zn} E So for n > N. Then:
(i) (1– p)zn <Yn < Zn for all n >N;
(ii) {Zn¢n} is increasing for all n> N.
Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres-
Theorem 1. If condition (2.1) holds and
Ca
00
1
lim sup
s+1
s+1
> 1,
(2.5)
An+1 s=N
Ba
n+1 s=n+1
n00
then S2 = Ø.
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
= 0.
(2.1)
n=N Un s=n
Then:
Zn
(i) {} is decreasing for all n> N;
bn
(ii) {
"} is decreasing for all n> N;
An
Zn
(iii) {
} is increasing for all n > N.
Bn
Kiguradze's theorem [1] can be used to describe the structure of the solution space
for the nonoscillatory solutions. For example, for the ordinary difference equation
(see [13])
A?((Ayn)") +9ny+
= 0, n> no,
(1.2)
the set K of all positive solutions has the decomposition
K - KaIK
Theorem 2. Assume that conditions (2.1) and (2.5) hold. If {yn} is a positive
solution of equation (1.1), then there are positive constants C, and C2 such that
Ciyı (n) < yn < C242(n),
(2.9)
where
n-1
1
Vı (n)
and y2(n) = II(1 – R).
On
s=N
Corollary 1. Assume that
= 00
(2.10)
n=N \s=n
and
n
00
1
lim sup
E(s+1)"(s+2)"q;+(n+2)" (s+1)"qs }> 2". (2.11)
n+1
s=N
n00
s=n+1
every positive solution {y,} of equation (1.2) is decreasing and satisfies estim-
ates of the form (2.9) with p = 0 in Rp.
Then](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9d6994c6-9c0b-4312-919e-29acb733a883%2F137af517-e964-47cb-bce6-643e9356db02%2Fj51204_processed.png&w=3840&q=75)
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)“)+4ny+1=0, n> no 2 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 <p<1;
(H3) {o(n)} is a sequence of integers such that o(n) > n for all n > no;
(H4) Ln=no un
= +0o and Ln=no ya
= +00,
1/a
Lemma 4. Let {yn} be a positive solution of equation (1.1) with the corresponding
sequence {zn} E So for n > N. Then:
(i) (1– p)zn <Yn < Zn for all n >N;
(ii) {Zn¢n} is increasing for all n> N.
Proof. Assume that {yn} is a positive solution of equation (1.1) with the corres-
Theorem 1. If condition (2.1) holds and
Ca
00
1
lim sup
s+1
s+1
> 1,
(2.5)
An+1 s=N
Ba
n+1 s=n+1
n00
then S2 = Ø.
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
= 0.
(2.1)
n=N Un s=n
Then:
Zn
(i) {} is decreasing for all n> N;
bn
(ii) {
"} is decreasing for all n> N;
An
Zn
(iii) {
} is increasing for all n > N.
Bn
Kiguradze's theorem [1] can be used to describe the structure of the solution space
for the nonoscillatory solutions. For example, for the ordinary difference equation
(see [13])
A?((Ayn)") +9ny+
= 0, n> no,
(1.2)
the set K of all positive solutions has the decomposition
K - KaIK
Theorem 2. Assume that conditions (2.1) and (2.5) hold. If {yn} is a positive
solution of equation (1.1), then there are positive constants C, and C2 such that
Ciyı (n) < yn < C242(n),
(2.9)
where
n-1
1
Vı (n)
and y2(n) = II(1 – R).
On
s=N
Corollary 1. Assume that
= 00
(2.10)
n=N \s=n
and
n
00
1
lim sup
E(s+1)"(s+2)"q;+(n+2)" (s+1)"qs }> 2". (2.11)
n+1
s=N
n00
s=n+1
every positive solution {y,} of equation (1.2) is decreasing and satisfies estim-
ates of the form (2.9) with p = 0 in Rp.
Then
Transcribed Image Text:Theorem 2. Assume that conditions (2.1) and (2.5) hold. If {yn} is a positive
solution of equation (1.1), then there are positive constants Cı and C2 such that
Ciyi (n) < yn <C2¥2(n),
(2.9)
where
n-1
1
Vı(n):
and Wz(n) —D П(1— R.).
On
s=N
Proof. Assume that {yn} is a positive solution of equation (1.1). Then, by The-
orem 1, {yn} is a Kneser type solution. From Lemma 4, we have that {z,0n} is
increasing for all n > N. Therefore,
(1 — р)
Yn 2
On
-P)
ZNON = C1¥1 (n).
On
On the other hand, summing equation (1.1) from n to o and applying Lemma 4 (i),
we have
00
a„A(b,(Ačn)“) = £ 9. 2 (1– p)“ £9.
's+1
s=n
s=n
s=n
Again summing the last inequality and applying Lemma 4 (ii) gives
1 00
00
-b,(Azn)“ > (1– p)“ E;
Yn n
a.
s=n us t=s Yt+1
or
Azn
<-Rn, n> N.
Zn
Summing the last inequality from N to n – 1, we obtain
Yn < Zn <C2W2(n).
This proves the theorem.
From Theorem 2 we deduce the following result for the half-linear difference equa-
tion (1.2).
Corollary 1. Assume that
00
00
= 00
(2.10)
n=N
and
1
lim sup
E (s+1)"(s+2)ªq,+ (n+2)ª £ (s+1)ªqs } > 2ª. (2.11)
n00
n+1
s=N
s=n+1
Then every positive solution {y„} of equation (1.2) is decreasing and satisfies estim-
ates of the form (2.9) with p = 0 in R,.
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