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doi:10.1155/2010/594783

Research Article

Error Bounds for Asymptotic Solutions of

Second-Order Linear Difference Equations II:

The First Case

L. H. Cao

1, 2

and J. M. Zhang

3

1Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong 2Department of Mathematics, Shenzhen University, Guangdong 518060, China

3Department of Mathematics, Tsinghua University, Beijin 100084, China

Correspondence should be addressed to J. M. Zhang,jzhang@math.tsinghua.edu.cn

Received 13 July 2010; Accepted 27 October 2010

Academic Editor: Rigoberto Medina

Copyrightq2010 L. H. Cao and J. M. Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equationyn2 npanyn1 nqbnyn 0,wherepandqare integers,anandbnhave asymptotic expansions of the forman∼∞s0as/ns,bn

s0bs/ns, for large values ofn,

a0/0,andb0/0.

1. Introduction

Asymptotic expansion of solutions to second-order linear difference equations is an old subject. The earliest work as we know can go back to 1911 when Birkhoff 1first deal with this problem. More than eighty years later, this problem was picked up again by Wong and Li2,3. This time two papers on asymptotic solutions to the following difference equations:

yn2 anyn1 bnyn 0 1.1

yn2 npanyn1 nqbnyn 0 1.2

were published, respectively, where coefficientsanandbnhave asymptotic properties

an∼∞

s0 as

ns, bn

s0 bs

ns, 1.3

(2)

Unlike the method used by Olver4to treat asymptotic solutions of second-order linear differential equations, the method used in Wong and Li’s papers cannot give us way to obtain error bounds of these asymptotic solutions. Only order estimations were given in their papers. The estimations of error bounds for these asymptotic solutions to 1.1

were given in 5 by Zhang et al. But the problem of obtaining error bounds for these asymptotic solutions to 1.2 is still open. The purpose of this and the next paper Error bounds for asymptotic solutions of second-order linear difference equations II: the second case is to estimate error bounds for solutions to 1.2. The idea used in this paper is similar to that of Olver to obtain error bounds to the Liouville-Green WKB asymptotic expansion of solutions to second-order differential equations. It should be pointed out that similar method appeared in some early papers, such as Spigler and Vianello’s papers

6–9.

In Wong and Li’s second paper 3, two different cases were given according to different values of parameters. The first case is devoted to the situation when k > 0, and in the second case ask < 0 wherek 2pq. The whole proof of the result is too long to understand, so we divide the estimations into two parts, part Ithis paperand part IIthe next paper, which correspond to the different two cases of3, respectively.

In the rest of this section, we introduce the main results of3 in the case thatk is positive. In the next section, we give two lemmas on estimations of bounds for solutions to a special summation equation and a first order nonlinear difference equation which will be often used later. Section 3 is devoted to the case when k 1. And in Section 4, we discuss the case when k > 1. The next paper Error bounds for asymptotic solutions of second-order linear difference equations II: the second caseis dedicated to the case when

k <0.

1.1. The Result in [

3

] When

k

1

Whenk 1, from3we know that1.2has two linearly independent solutiony1nand y2n

y1n n−2!qpρn11 ∞

s0 c1

s

ns , 1.4

ρ1− b0 a0

, α1

b0 a2 0

a1 a0

b1 b0 −

pq, c1

0 /0, 1.5

y2n n−2!2nnα2 ∞

s0 c2

s

ns , 1.6

ρ2−a0, α2 a1

0

a1− b0 a0

, c2

0 /0, 1.7

(3)

1.2. The Result in [

3

] When

k >

1

Whenk >1, from3we know that1.2has two linearly independent solutionsy1nand y2n

y1n n−2!q1nnα1 ∞

s0 c1

s

ns , 1.8

ρ1 −ba0 0

, α1 bb1

0 − a1 a0 −

pq, c1

0 /0, 1.9

y2n n−2!2nnα2 ∞

s0 c2

s

ns , 1.10

ρ2−a0, α2 aa1 0

, c2

0 /0. 1.11

In the following sections, we will discuss in detail the error bounds of the proceeding asymptotic solutions of1.2. Before discussing the error bounds, we consider some lemmas.

2. Lemmas

2.1. The Bounds for Solutions to the Summation Equation

We consider firstly a bound of a special solution for the “summary equation”

hn

jn

Kn, jRjjpφjhj1jqψjhj .

2.1

Lemma 2.1. LetKn, j,φj,ψj,Rjbe real or complex functions of integer variablesn, j;pand

qare integers. If there exist nonnegative constantsn1,θ,ς,N,s,t,β,CK,CR,Cφ,Cψ,Cβ,Cαwhich

satisfy

θpsβ−3

2,θqt−2β− 3

2, 2.2

and whenj nn1,

Kn, jCKPn

pjjθ, R

jCRp

jjςN−1,

φ

jCφjs, ψjCψjt,

Pj pj

Cβj−2β,

Pj1

pj

2Cαρ1jβ,

(4)

wherePnandpnare positive functions of integer variablen. Letn0,n2be integers defined by

n2

1

ς

2CK

2CαCφρ1sup

j

11

j

ςNθ−3/2

CψCβ

θ

,

n0max{n1, n2},

2.4

then2.1has a solutionhn, which satisfies

|hn| 2CRCKPnnςNθ−1 ςNθ−2CK

2CαCφρ1supj

11/jςNθ−3/2CψCβ

,

2.5

forNn0.

Proof. Set

h0n 0,

hs1n

jn

Kn, jRjjpφjh

sj1−jqψjhsj ,

s0,1,2, . . . ,

2.6

then

|h1n| ∞

jn

Kn, jRjCKCRPn

jn

jςNθ−1

2CRCK

ςNθPnnςNθ.

2.7

The inequality∞jnjp2/p−1np−1, np−1>0, is used here. Assuming that

|hsnhs−1n| ςN2CRCKθλs−1PnnςNθ−1, 2.8

where

λ ςN2CKθ

2CαCφρ1sup

j

11

j

ςNθ−3/2

CψCβ

(5)

then

|hs1nhsn|

jn

Kn, jjpφjh

sj1hs−1j1jqψjhsjhs−1j

jn

2CRCK2Pn

ςNθpjjθ

jpsC

φλs−1Pj1j1−ςNθ−1

jqtC

ψλs−1PjjςNθ−1

2CRCK

ςNθλsPnnςNθ−1.

2.10

By induction, the inequality holds for any integers. Hence the series

s0

{hs1nhsn}, 2.11

whenλ <1, that is,Nn0max{n1, n2}, is uniformly convergent innwhere

n2

1

ς

2CK

2CαCφρ1sup

j

11

j

ςNθ−3/2

CψCβ

θ

. 2.12

And its sum

hn

s0

{hs1nhsn} 2.13

satisfies

|hn|∞

s0

|hs1nhsn| ςN2CRCKθPnnςNθ−1

s0 λs

2CRCKPnnςNθ−1 ςNθ−2CK

2CαCφρ1supj

11/jςNθ−3/2CψCβ

. 2.14

(6)

2.2. The Bound Estimate of a Solution to

a Nonlinear First-Order Difference Equation

Lemma 2.2. If the functionfnsatisfies

fn 1 A

n2 f1n, 2.15

wheren3|f

1n|B(AandBare constants), whennis large enough, then the following first-order

difference equation

xnxn1 fn,

x∞ 1 2.16

has a solutionxnsuch thatsupn{n2|xn1|}is bounded by a constantC

x, whennis big enough.

Proof. Obviously from the conditions of this lemma, we know that infinite products

k0fn2kand

k0fn2k1are convergent.

xn

k0fn2k

k0fn2k1

. 2.17

is a solution of2.16with the infinite condition. Letgn, k fn2k/fn2k1−1; then whennis large enough,

gn, k4|A|4B n2k3,

n2|xn1|n2

k0fn2k

k0fn2k1

−1

n2∞

k0

1gn, k−1

4|A|4BCx.

2.18

3. Error Bounds in the Case When

k

1

Before giving the estimations of error bounds of solutions to1.2, we rewriteyinas

(7)

with order linear difference equations

εNin2 npanεNin1 nqbnεNin RNin, i1,2, 3.3

3.1. The Error Bound for the Asymptotic Expansion of

y

1

n

Now we firstly estimate the error bound of the asymptotic expansion ofy1nin the case k1. Let

It can be easily verified that

z1n n−2!qpρn11

are two linear independent solutions of the comparative difference equation

(8)

From the definition, we know that the two-term approximation ofxnis

Equation 3.8 is a second-order linear difference equation with two known linear independent solutions. Its coefficients are quite similar to those in3.3. This reminds us to rewrite3.3in the form similar to3.8.

are finite. Equation 3.10 is a inhomogeneous second-order linear difference equation; its solution takes the form of a particular solution added to an arbitrary linear combination of solutions to the associated homogeneous linear difference equation3.8.

From10, any solution of the “summary equation”

(9)

Firstly we consider the denominator inKn, j. We get from3.8

Set the Wronskian of the two solutions of the comparative difference equation as

(10)
(11)
(12)

3.2. The Error Bound for the Asymptotic Expansion of

y

2

(

n

)

Now we estimate the error bound of the asymptotic expansion of the linear independent solutiony2nto the original difference equation ask1. Let

ε2

Nn y1nδNn. 3.33

From3.3, we have

y1n2δNn2 npany1n1δNn1 nqbny1nδNn RN2n. 3.34

Fory1nbeing a solution of1.2, let

ΔNn δNn1−δNn; 3.35

thenΔNnsatisfies the first-order linear difference equation

y1nNn1−nqbny1nΔNn RN2n. 3.36

The solution of3.36is

ΔNn

in

Xn

Xi1

R2

Ni

y1i2

, 3.37

where Xn Xmnj−1mjqbjy1j/y1j 2, Xm is a constant, andmis an integer

which is large enough such that wheninm,

y1i i−2!p−1ρi1iReα11ε 1 1 i

1

2i−2! p−1ρi

1iReα1. 3.38

The two-term approximation ofjqbjy1j/y1j2is

jqbjy

1

j y1

j2

b0j ρ2

1

1α2−α1

j σ

j, 3.39

whereσjis the reminder andσ0supj{j2|σj|}is a constant. From Lemma 3 of5, we obtain

|Xm|b0 ρ2 1

nm

n−1!

m−1!exp−k1n

Reα2−α1

|Xn||Xm|b0 ρ2 1

nm

n−1!

m−1!expk1n

Reα2−α1,

(13)
(14)

LetM0 Ii0−1mi−1!|ρ21|

iiReα2−α1−N; then

n−1

im

i−1!ρ2

ρ1

iiReα2−α1−NM

0nn−1! ρ2

ρ1

nnReα2−α1−N

2n−2!ρ2

ρ1

nnReα2−α1−N1,

3.47

where limn→ ∞n−2!|ρ21|

nnReα2−α1−N2∞.Hence

|δNn|2μn−2!ρρ2 1

nnReα2−α1−N2. 3.48

From3.33, we obtain

ε2

Nny1nδNn

≤2μsup nm

n−2!1−pρ1nn−Reα1y 1n

×n−2!2

nnReα2−N2.

3.49

Thus we complete the estimate of error bounds to asymptotic expansions of solutions of1.2

ask1.

4. Error Bounds in Case When

k >

1

Here we also rewriteyinas

yin LNin εNin, i1,2, 4.1

with

L1

Nn n−2!qpρn11

N−1

s0 c1

s

ns,

L2

Nn n−2!2nnα2

N−1

s0 c2

s

ns ,

4.2

andεNin,i 1,2, are error terms. ThenεNin,i 1,2, satisfy the inhomogeneous second-order linear difference equations

(15)

where

4.1. The Error Bound for the Asymptotic Expansion of

y

1

n

Now let us come to the case when k > 1. This time a difference equation which has two known linear independent solutions is also constructed for the purpose of comparison for

1.2.

is a constant. And the function

(16)

such that

are two linear independent solutions of the difference equation

zn2 np

This difference equation4.13can be regarded as the comparative equation of4.3. Rewriting4.3in the form similar to the comparative difference equation4.13, we get

ε1 4.14 is an inhomogeneous second-order linear difference equation; its solution takes the form of a particular solution added to an arbitrary linear combination of solutions to the associated homogeneous linear difference equation4.13.

From10, any solution of the “summary equation”

(17)

Similar toSection 3.1, we have

we have fromLemma 2.1that

4.2. The Error Bound for the Asymptotic Expansion of

y

2

n

Let

ε2

(18)

From3.3, we have

y1n2δNn2 npany1n1δNn1 nqbny1nδNn RN2n. 4.23

Using the method employed inSection 3.2, it is not difficult to obtain

ε2

Nny1nδNn

2μsup nm

n−2!1−nn−Reα1y12nnReα2−Nk1.

4.24

Now we completed the estimate of the error bounds for asymptotic solutions to second order linear difference equations in the first case. For the second case, we leave it to the second part of this paper: Error Bound for Asymptotic Solutions of Second-order Linear Difference Equation II: the second case.

In the rest of this paper, we would like to give an example to show how to use the results of this paper to obtain error bounds of asymptotic solutons to second-order linear difference equations. Here the difference equation is

yn2−yn1n1

2yn0. 4.25

It is a special case of the equation

n1fnα1xnαxfnαx fnα−1x 0, 4.26

α1, x1, which is satisfied by Tricomi-Carlitz polynomials. By calculation, the constant

CKin3.30is1728/5e2π2. So4.25has a solution

y1n n1

2!n21ε1n, 4.27

forn≥3 with the error termε1nsatisfing

|ε1n|

864 25e

2π2

n−1. 4.28

Acknowledgments

(19)

References

1 G. D. Birkhoff, “General theory of linear difference equations,” Transactions of the American Mathematical Society, vol. 12, no. 2, pp. 243–284, 1911.

2 R. Wong and H. Li, “Asymptotic expansions for second-order linear difference equations,”Journal of Computational and Applied Mathematics, vol. 41, no. 1-2, pp. 65–94, 1992.

3 R. Wong and H. Li, “Asymptotic expansions for second-order linear difference equations. II,”Studies in Applied Mathematics, vol. 87, no. 4, pp. 289–324, 1992.

4 F. W. J. Olver, Asymptotics and Special Functions, Computer Science and Applied Mathematics, Academic Press, New York, NY, USA, 1974.

5 J. M. Zhang, X. C. Li, and C. K. Qu, “Error bounds for asymptotic solutions of second-order linear difference equations,”Journal of Computational and Applied Mathematics, vol. 71, no. 2, pp. 191–212, 1996.

6 R. Spigler and M. Vianello, “Liouville-Green approximations for a class of linear oscillatory difference equations of the second order,”Journal of Computational and Applied Mathematics, vol. 41, no. 1-2, pp. 105–116, 1992.

7 R. Spigler and M. Vianello, “WKBJ-type approximation for finite moments perturbations of the differential equationy

0 and the analogous difference equation,”Journal of Mathematical Analysis and Applications, vol. 169, no. 2, pp. 437–452, 1992.

8 R. Spigler and M. Vianello, “Discrete and continuous Liouville-Green-Olver approximations: a unified treatment via Volterra-Stieltjes integral equations,”SIAM Journal on Mathematical Analysis, vol. 25, no. 2, pp. 720–732, 1994.

9 R. Spigler and M. Vianello, “A survey on the Liouville-GreenWKB approximation for linear difference equations of the second order,” inAdvances in Difference Equations (Veszpr´em, 1995), N. Elaydi, I. Gy ¨ori, and G. Ladas, Eds., pp. 567–577, Gordon and Breach, Amsterdam, The Netherlands, 1997.

References

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