Hindawi Publishing Corporation Advances in Difference Equations Volume 2010, Article ID 594783, 19 pages 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 1 Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, H ong Kong 2 Department of Mathematics, Shenzhen University, Guangdong 518060, China 3 Department 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 Copyright q 2010 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 equation yn  2n p anyn  1n q bnyn0, where p and q are integers, an and bn have asymptotic expansions of the form an ∼  ∞ s0 a s /n s , bn ∼  ∞ s0 b s /n s , for large values of n, a 0 /  0, and b 0 /  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1 first deal with this problem. More than eighty years later, this problem was picked up again by Wong and Li 2, 3. This time two papers on asymptotic solutions to the following difference equations: y  n  2   a  n  y  n  1   b  n  y  n   0 1.1 y  n  2   n p a  n  y  n  1   n q b  n  y  n   0 1.2 were published, respectively, where coefficients an and bn have asymptotic properties a  n  ∼ ∞  s0 a s n s ,b  n  ∼ ∞  s0 b s n s , 1.3 for large values of n, a 0 /  0, b 0 /  0, and p, q ∈ Z. 2 Advances in Difference Equations Unlike t he method used by Olver 4 to 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,twodifferent 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 as k<0 where k  2p − q. The whole proof of the result is too long to understand, so we divide the estimations into two parts, part I this paper and part II the next paper, which correspond to the different two cases of 3, respectively. In the rest of this section, we introduce the main results of 3 in the case that k 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 case is dedicated to the case when k<0. 1.1. The Result in [3] When k  1 When k  1, from 3 we know that 1.2 has two linearly independent solution y 1 n and y 2 n y 1  n    n − 2  !  q−p ρ n 1 n α 1 ∞  s0 c 1 s n s , 1.4 ρ 1  − b 0 a 0 ,α 1  b 0 a 2 0 − a 1 a 0  b 1 b 0 − p  q, c 1 0 /  0, 1.5 y 2  n    n − 2  !  p ρ n 2 n α 2 ∞  s0 c 2 s n s , 1.6 ρ 2  −a 0 ,α 2  1 a 0  a 1 − b 0 a 0  ,c 2 0 /  0, 1.7 for n ≥ 2. Advances in Difference Equations 3 1.2. The Result in [3] When k>1 When k>1, from 3 we know that 1.2 has two linearly independent solutions y 1 n and y 2 n y 1  n    n − 2  !  q−p ρ n 1 n α 1 ∞  s0 c 1 s n s , 1.8 ρ 1  − b 0 a 0 ,α 1  b 1 b 0 − a 1 a 0 − p  q, c 1 0 /  0, 1.9 y 2  n    n − 2  !  p ρ n 2 n α 2 ∞  s0 c 2 s n s , 1.10 ρ 2  −a 0 ,α 2  a 1 a 0 ,c 2 0 /  0. 1.11 In the following sections, we will discuss in detail the error bounds of the proceeding asymptotic solutions of 1.2. 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” h  n   ∞  jn K  n, j  R  j  − j p φ  j  h  j  1  − j q ψ  j  h  j  . 2.1 Lemma 2.1. Let Kn, j, φj, ψj, Rj be real or complex functions of integer variables n, j; p and q are integers. If there exist nonnegative constants n 1 , θ, ς, N, s, t, β, C K , C R , C φ , C ψ , C β , C α which satisfy −θ  p − s − β  − 3 2 , − θ  q − t − 2β  − 3 2 , 2.2 and when j  n  n 1 ,   K  n, j     C K P  n  p  j  j −θ ,   R  j     C R p  j  j −ςN−1 ,   φ  j     C φ j −s ,   ψ  j     C ψ j −t ,      P  j  p  j        C β j −2β ,      P  j  1  p  j        2C α   ρ 1   j −β , 2.3 4 Advances in Difference Equations where Pn and pn are positive functions of integer variable n.Letn 0 , n 2 be integers defined by n 2   1 ς  2C K  2C α C φ   ρ 1   sup j  1  1 j  −ςN−θ−3/2  C ψ C β  − θ   , n 0  max { n 1 ,n 2 } , 2.4 then 2.1 has a solution hn, which satisfies | h  n  |  2C R C K P  n  n −ςN−θ−1 ςN  θ − 2C K  2C α C φ   ρ 1   sup j  1  1/j  −ςN−θ−3/2  C ψ C β  , 2.5 for N  n 0 . Proof. Set h 0  n   0, h s1  n   ∞  jn K  n, j  R  j  − j p φ  j  h s  j  1  − j q ψ  j  h s  j  , s  0, 1, 2, , 2.6 then | h 1  n  |  ∞  jn   K  n, j      R  j     C K C R P  n        ∞  jn j −ςN−θ−1        2C R C K ςN  θ P  n  n −ςN−θ . 2.7 The inequality  ∞ jn j −p 2/p − 1  n −p−1 ,n p − 1 > 0, is used here. Assuming that | h s  n  − h s−1  n  |  2C R C K ςN  θ λ s−1 P  n  n −ςN−θ−1 , 2.8 where λ  2C K ςN  θ  2C α C φ   ρ 1   sup j  1  1 j  −ςN−θ−3/2  C ψ C β  ; 2.9 Advances in Difference Equations 5 then | h s1  n  − h s  n  |  ∞  jn   K  n, j     j p   φ  j      h s  j  1  h s−1  j  1     j q   ψ  j      h s  j  −h s−1  j      ∞  jn 2C R C 2 K P  n   ςN  θ  p  j  j −θ  j p−s C φ λ s−1 P  j  1  j  1  −ςN−θ−1 j q−t C ψ λ s−1 P  j  j −ςN−θ−1   2C R C K ςN  θ λ s P  n  n −ςN−θ−1 . 2.10 By induction, the inequality holds for any integer s. Hence the series ∞  s0 { h s1  n  − h s  n  } , 2.11 when λ<1, that is, N  n 0  max{n 1 ,n 2 }, is uniformly convergent in n where n 2   1 ς  2C K  2C α C φ   ρ 1   sup j  1  1 j  −ςN−θ−3/2  C ψ C β  − θ   . 2.12 And its sum h  n   ∞  s0 { h s1  n  − h s  n  } 2.13 satisfies | h  n  |  ∞  s0 | h s1  n  − h s  n  |  2C R C K ςN  θ P  n  n −ςN−θ−1 ∞  s0 λ s  2C R C K P  n  n −ςN−θ−1 ςN  θ − 2C K  2C α C φ   ρ 1   sup j  1  1/j  −ςN−θ−3/2  C ψ C β  . 2.14 So we get the bound of any solution for the “summary equation” 2.1. Next we consider a nonlinear first-order difference equation. 6 Advances in Difference Equations 2.2. The Bound Estimate of a Solution to a Nonlinear First-Order Difference Equation Lemma 2.2. If the function fn satisfies f  n   1  A n 2  f 1  n  , 2.15 where n 3 |f 1 n|  B (A and B are constants), when n is large enough, then the following first-order difference equation x  n  x  n  1   f  n  , x  ∞   1 2.16 has a solution xn such that sup n {n 2 |xn − 1|} is bounded by a constant C x ,whenn is big enough. Proof. Obviously from the conditions of this lemma, we know that infinite products  ∞ k0 fn  2k and  ∞ k0 fn  2k  1 are convergent. x  n    ∞ k0 f  n  2k   ∞ k0 f  n  2k  1  . 2.17 is a solution of 2.16 with the infinite condition. Let gn, kfn  2k/fn  2k  1 − 1; then when n is large enough, g  n, k   4 | A |  4B  n  2k  3 , n 2 | x  n  − 1 |  n 2      ∞ k0 f  n  2k   ∞ k0 f  n  2k  1  − 1      n 2 ∞  k0  1  g  n, k   − 1   4 | A |  4B  C x . 2.18 3. Error Bounds in the Case When k  1 Before giving the estimations of error bounds of solutions to 1.2, we rewrite y i n as y i  n   L i N  n   ε i N  n  ,i 1, 2, 3.1 Advances in Difference Equations 7 with L 1 N  n    n − 2  !  q−p ρ n 1 n α 1 N−1  s0 c 1 s n s , L 2 N  n    n − 2  !  p ρ n 2 n α 2 N−1  s0 c 2 s n s , 3.2 and ε i N n, i  1, 2, being error terms. Then ε i N n, i  1, 2, satisfy inhomogeneous second- order linear di fference equations ε i N  n  2   n p a  n  ε i N  n  1   n q b  n  ε i N  n   R i N  n  ,i 1, 2, 3.3 where R i N  n   −  L i N  n  2   n p a  n  L i N  n  1   n q b  n  L i N  n   ,i 1, 2. 3.4 We know from 3 that C R 1  sup n  n N      R 1 N  n   n!  q−p ρ n 1 n α 1       ,C R 2  sup n  n N1      R 2 N  n   n!  p ρ n 2 n α 2       . 3.5 3.1. The Error Bound for the Asymptotic Expansion of y 1 n Now we firstly estimate the error bound of the asymptotic expansion of y 1 n in the case k  1. Let x  n   −  1 − 1/n  ρ 2 2  1  2/n  α 2 − n −2 ρ 2 1  1  2/n  α 1  ρ 2  1 − 1/n  1  1/n  α 2 − ρ 1 n −1  1  1/n  α 1   a 0  a 1 /n  , l  n   − n  1 − 1/n  p  ρ 2 2  1  2/n  α 2   a 0  a 1 /n  ρ 2  1  1/n  α 2 x  n  1   x  n  x  n  1  − b 0 − b 1 n . 3.6 It can be easily verified that z 1  n    n − 2  !  q−p ρ n 1 n α 1 ∞  kn x  k  , z 2  n    n − 2  !  p ρ n 2 n α 2 ∞  kn x  k  3.7 are two linear independent solutions of the comparative difference equation z  n  2   n p  a 0  a 1 n  z  n  1   n q  b 0  b 1 n  l  n   z  n   0. 3.8 8 Advances in Difference Equations From the definition, we know that the two-term approximation of xnis x  n   1  a 0  a 1 − b 0 /a 0  − pa 2 0 −  a 1 − pa 0  a 0  b 0 a 2 0 1 n  ω  n  , 3.9 where ωn is the reminder and the coefficient of 1/n is zero. So C x  sup n {n 2 |xn − 1|} is a constant. And ln satisfies C l  sup n {n 2 |ln|} being a constant; here we have made use of the definitions of α i ,ρ i in 1.5, 1.7,and2p − q  1. Equation 3.8 is a second-order linear difference equation with two known linear independent solutions. Its coefficients are quite similar to those in 3.3. This reminds us to rewrite 3.3 in the form similar to 3.8. According to the coefficients in 3.8, we rewrite 3.3 as ε 1 N  n  2   n p  a 0  a 1 n  ε 1 N  n  1   n q  b 0  b 1 n  l  n   ε 1 N  n   R 1 N  n  − n p  a  n  − a 0 − a 1 n  ε 1 N  n  1  − n q  b  n  − b 0 − b 1 n − l  n   ε 1 N  n  , 3.10 where an and bn are such that C a  sup jn  j 2     a  j  − a 0 − a 1 j      ,C b  sup jn  j 2     b  j  − b 0 − b 1 j − l  j       3.11 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 equation3.8. From 10, any solution of the “summary equation” ε 1 N  n   ∞  jn K  n, j   R 1 N  j  − j p  a  j  − a 0 − a 1 j  ε 1 N  j  1  −j q  b  j  − b 0 − b 1 j − l  j   ε 1 N  j   3.12 is a solution of 3.10, where K  n, j   z 1  j  1  z 2  n  − z 1  n  z 2  j  1  z 1  j  2  z 2  j  1  − z 1  j  1  z 2  j  2  . 3.13 Now we estimate the bound of the function Kn, j. Advances in Difference Equations 9 Firstly we consider the denominator in Kn, j.Wegetfrom3.8  z 1  n  2  z 2  n  1  − z 1  n  1  z 2  n  2   n q  b 0  b 1 n  l  n    z 1  n  z 2  n  1  − z 1  n  1  z 2  n   0. 3.14 Set the Wronskian of the two solutions of the comparative difference equation as W  n   z 1  n  1  z 2  n  − z 1  n  z 2  n  1  ; 3.15 we have W  n  1   n q  b 0  b 1 n  l  n   W  n  . 3.16 From 3.16, w e have W  n  1   W  2  n!  q b n−1 0 n  k2  1  b 1 b 0 1 k  l  k  b 0  . 3.17 From Lemma 3 of 5,weobtain exp  −k 1  n  1  Reb 1 /b 0        n  km  1  b 1 b 0 1 k  l  k  b 0        exp  k 1  n  1  Reb 1 /b 0  , 3.18 where k 1      b 1 b 0      1 m  1 6m 2  1 60m 4  ln m   π 2 6 σ 0 , 3.19 σ 0  sup k  k 2     ln  1  b 1 b 0 1 k  l  k  b 0  − b 1 b 0 1 k       m<k<n  ; 3.20 m is an integer which is large enough such that 1 b 1 /b 0 1/klk/b 0 > 0, when k  m. Let C ∗  |  m−1 k2 1 b 1 /b 0 1/klk/b 0 |, for the property of lk,weknowthat C ∗ is a constant. Then we obtain from 3.18 | W  n  1  |  | W  2  |  n!  q    b n−1 0    C ∗ exp  −k 1  n  1  Reb 1 /b 0  . 3.21 10 Advances in Difference Equations Now considering the numerator in Kn, j,weget z 1  j  1  z 2  n  − z 1  n  z 2  j  1    j − 1  !  p−1  n − 2  !  p−1 ∞  kj1 x  k  ∞  kn x  k  ×  ρ j1 1 ρ n 2  j  1  α 1 n α 2  n − 2  ! − ρ n 1 ρ j1 2  j  1  α 2 n α 1  j − 1  !  . 3.22 Here we have made use of q − p  p − 1. From Lemma 2 of 5, we have       ∞  kj1 x  k  ∞  kn x  k         exp  2π 2 3 C x  , 3.23 where C x  sup n {n 2 |xn − 1|} is a constant. For the bound of Kn, j,weset K  n, j    n − 2  !  q−p ρ n 1 n α 1  j!  q−p ρ j 1 j α 1  K  n, j  , 3.24 then     K  n, j      | I |  | II | , 3.25 where | I |         j!  q−p ρ j 1 j α 1  n − 2  !  q−p ρ n 1 n α 1              exp  2π 2 /3  C x  j − 1  !  p−1  n − 2  !  p−1 | W  2  |  j!  q    b j 0    C ∗ exp  −k 1   j  1  Reb 1 /b 0         ×    ρ n 1 ρ j1 2  j  1  α 2 n α 1  j − 1  !    | II |         j!  q−p ρ j 1 j α 1  n − 2  !  q−p ρ n 1 n α 1              exp  2π 2 /3  C x  j − 1  !  p−1  n − 2  !  p−1 | W  2  |  j!  q    b j 0    C ∗ exp  −k 1   j  1  Reb 1 /b 0         ×    ρ j1 1 ρ n 2  j  1  α 1 n α 2  n − 2  !    . 3.26 [...]... 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. .. Difference Equations 3.2 The Error Bound for the Asymptotic Expansion of y2 (n) Now we estimate the error bound of the asymptotic expansion of the linear independent solution y2 n to the original difference equation as k 1 Let 2 εN n y1 n δN n 3.33 From 3.3 , we have y1 n 2 δN n np a n y1 n 2 1 δN n nq b n y1 n δN n 1 2 RN n 3.34 For y1 n being a solution of 1.2 , let ΔN n δN n 1 − δN n ; 3.35 then ΔN... Difference Equations 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. .. complete the estimate of error bounds to asymptotic expansions of solutions of 1.2 as k 1 4 Error Bounds in Case When k > 1 Here we also rewrite yi n as i yi n i LN n εN n , i 1, 2, 4.1 with 1 LN n n−2 ! 2 LN n q−p n α1 ρ1 n n n − 2 ! p ρ2 nα2 i i and εN n , i 1,2, are error terms Then εN n , i order linear difference equations i εN n 2 i np a n εN n 1 N−1 s N−1 s 1 cs , ns 0 4.2 2 cs , ns 0 1,2, satisfy the. .. 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,... Acknowledgments The authors would like to thank Dr Z Wang for his helpful discussions and suggestions The second author thanks Liu Bie Ju Center for Mathematical Science and Department of Mathematics of City University of Hong Kong for their hospitality This work is partially supported by the National Natural Science Foundation of China Grant no 10571121 and Grant no 10471072 and Natural Science Foundation of Guangdong... 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 0 and the analogous difference equation,” Journal of Mathematical Analysis differential equation y and Applications,... 4.2 The Error Bound for the Asymptotic Expansion of y2 n Let 2 εN n y1 n δN n 4.22 18 Advances in Difference Equations From 3.3 , we have y1 n 2 δN n np a n y1 n 2 1 δN n nq b n y1 n δN n 1 2 RN n 4.23 Using the method employed in Section 3.2, it is not difficult to obtain 2 y1 n δN n εN n n−2 ! 2μ sup n m k ρ1 −n − Re α1 y1 n n n ρ2 nRe α2 −N k 1 4.24 Now we completed the estimate of the error bounds. .. Difference Equations 15 where i i RN n − LN n i np a n L N n 2 i nq b n L N n , 1 i 1, 2 4.4 We know from 3 that 1 2 RN n sup nN CR1 n! n , q−p n α1 ρ1 n sup nN CR2 1 n RN n n n! p ρ2 nα2 4.5 4.1 The Error Bound for the Asymptotic Expansion of y1 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... 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 equation 4.13 From 10 , any solution of the “summary equation” 1 εN n ∞ K n, j j n RN j − j p a j − a0 − 1 a1 −l j j 1 εN j 1 , 4.15 where K n, j is a solution of 4.14 z1 j z1 j 1 z2 . 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. 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. Difference Equations Volume 2010, Article ID 594783, 19 pages doi:10.1155/2010/594783 Research Article Error Bounds for Asymptotic Solutions of Second-Order Linear Difference Equations II: The First