This article was downloaded by: [Nipissing University] On: 20 October 2014, At: 02:54 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Difference Equations and Applications Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/gdea20 On Linear Implicit Non-autonomous Systems of Difference Equations a a L.C Loi , N.H Du & P.K Anh a a Faculty of Mathematics, Mechanics and Informatics , Vietnam National University , 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Published online: 17 Sep 2010 To cite this article: L.C Loi , N.H Du & P.K Anh (2002) On Linear Implicit Non-autonomous Systems of Difference Equations, Journal of Difference Equations and Applications, 8:12, 1085-1105, DOI: 10.1080/1023619021000053962 To link to this article: http://dx.doi.org/10.1080/1023619021000053962 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-andconditions Downloaded by [Nipissing University] at 02:54 20 October 2014 Journal of Difference Equations and Applications, 2002 Vol (12), pp 1085–1105 On Linear Implicit Non-autonomous Systems of Difference Equations L.C LOI, N.H DU and P.K ANH* Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam (Received 11 July 2000; Revised 10 April 2001; In final form 27 April 2001) This paper deals with the solvability of initial-value problems (IVPs) and multipoint boundaryvalue problems (MPBVPs) for linear implicit non-autonomous systems of difference equations Keywords: Linear implicit difference equations (LIDEs); Index of LIDEs; Singular-value decompostions (SVDs); IVPs; MPBVPs 1991 Mathematics Subject Classifications: 39A11; 39A10 *Corresponding author Tel.: +740-593-9479 E-mail: anhpk@vnu.edu.vn ISSN 1023-6198 print/ISSN 1563-5120 online q 2002 Taylor & Francis Ltd DOI: 10.1080/1023619021000053962 Downloaded by [Nipissing University] at 02:54 20 October 2014 1086 L.C LOI et al INTRODUCTION Linear implicit difference equations (LIDEs) An xnỵ1 ẳ Bn xn þ qn ð1Þ where An ; Bn [ Rm£m ; qn [ Rm are given and the matrices An are singular for all n [ N, arise in many applications and can be considered as discretizations of linear differential-algebraic equations (DAEs) Atịx0 ỵ Ctịx ẳ qtị 2ị Maărz [2] proposed an unified approach to linear DAEs (Eq (2)) under an assumption on the smoothness of KerA(t ) w.r.t t In this paper, we develop some concepts and techniques of [2] for discrete systems The paper is organized as follows: In section 2, using SVDs of An, we define the notion of index of LIDEs for the case where rankAn ; const: Then we study the solvability of IVPs and MPBVPs for index-1 LIDEs In particular, some discrete analogues of results on MPBVPs for linear DAEs [1] are obtained Section concerns with the solvability of LIDEs in non-constant-rank cases, where KerAn are nested or rankAn are non-decreasing Finally, some illustrative examples are given in section THE CONSTANT-RANK CASE Throughout this section, we assume that rankAn ; r r , m: Consider a SVD of An: (n [ N), where , An ẳ U n Sn V Tnỵ1 ð3Þ where Sn is a diagonal matrix with singular values: snð1Þ $ snð2Þ $ · · · $ ð1Þ ðrÞ sðrÞ n on the main diagonal, i.e Sn ¼ diagðsn ; ; sn ; 0; ; 0Þ and Un, Vn+1 are orthonormal matrices, i.e: U Tn U n ¼ U n U Tn ¼ V Tnỵ1 V nỵ1 ẳ V nỵ1 V Tnỵ1 ẳ I Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1087 Here I and Ik ðk – mÞ denote the m £ m and k £ k- identity matrices, respectively Set QU ! 0 I m2r ; V U I; Qn U V n QV Tn ; Pn U I Qn Obviously Sn Q ẳ and Qnỵ1 is a linear projection onto KerAn Lemma Suppose that the matrix Gn U An ỵ Bn V n QV Tnỵ1 is nonsingular Then there hold the following relations: iị An Pnỵ1 ẳ An 4ị Pnỵ1 ẳ G21 n An 5ị T G21 n Bn Qn ẳ V nỵ1 QV n 6ị iiị iiiị Pnỵ1 G21 n Bn Qn ẳ 0; T Qnỵ1 G21 n Bn Qn ẳ V nỵ1 QV n 7ị (iv) if Gn21 is also non-singular and P~ n U I Qn V n V Tnỵ1 G21 n Bn then ~ n U I P~ n ¼ Qn V n V Tnỵ1 G21 Q B is a projection onto KerA Moreover n n21 n ~ 21 ~ P~ n G21 n21 Bn21 Pn21 ẳ Pn Gn21 Bn21 8ị Proof From the relations An Qnỵ1 ẳ U n Sn V Tnỵ1 V nỵ1 QV Tnỵ1 ẳ U n Sn QV Tnỵ1 ẳ 0; it follows Eq (4) Noting that Gn Pnỵ1 ẳ An ỵ Bn V n V Tnỵ1 Qnỵ1 ịPnỵ1 ẳ An Pnỵ1 and taking into account Eq (4), we get Gn Pnỵ1 ẳ An ; which implies Eq (5) Since Gn An ịV nỵ1 ẳ Bn V n Q; it follows that Bn V n QV Tn ẳ Gn An ịV nỵ1 V Tn ; therefore G21 n Bn Qn ¼ ðI 21 21 T Gn An ịV nỵ1 V n : By virtue of Eq (5), we have Gn Bn Qn ẳ Qnỵ1 V nỵ1 V Tn ẳ V nỵ1 QV Tn : Thus Eq (6) is proved Clearly, relations (7) follow immediately T 21 from Eq (6) Finally, observe that P~ n G21 n21 Bn21 Qn21 V n21 V n Gn21 Bn21 ¼ 21 21 T T P~ n V n QV n21 V n21 V n Gn21 Bn21 ¼ P~ n Qn Gn21 Bn21 : On the other hand, T T P~ n Qn ¼ Qn Qn V n V Tnỵ1 G21 n Bn Qn ẳ Qn Qn V n V nỵ1 V nỵ1 QV n ẳ Qn 2 21 21 21 ~ ~ ~ ~ Qn ¼ 0: Thus, Pn Gn21 Bn21 Pn21 ¼ Pn Gn21 Bn21 Pn Gn21 Bn21 Qn21 V n21 21 ~ 21 ~ ~ 21 £ V Tn G21 this n21 Bn21 ¼ Pn Gn21 Bn21 Pn Qn Gn21 Bn21 ¼ Pn Gn21 Bn21 ; ~ n ¼ An21 Qn V n V Tnỵ1 G21 means Eq (8) Noting that An21 Q B ¼ and n n ~ ¼ Qn V n V T G21 Bn Qn V n V T G21 Bn ¼ using Eq (6), we get Q nỵ1 n nỵ1 n n 21 T ~ Qn V n QV Tnỵ1 G21 A n Bn ẳ V n QV nỵ1 Gn Bn ẳ Qn : Lemma is proved Downloaded by [Nipissing University] at 02:54 20 October 2014 1088 Lemma L.C LOI et al T Let An ẳ U n Sn V Tnỵ1 ẳ Un Sn Vnỵ1 be two SVDs of An Then  n U An ỵ Bn V n QV Tnỵ1 are (i) The matrices Gn U An ỵ Bn V n QV Tnỵ1 and G simultaneously non-singular (ii) If Gn is non-singular, then   T  21 V n QV Tnỵ1 G21 n ẳ Vn QVnỵ1 Gn 9ị ~  21 P~ n G21 n21 ẳ Pn Gn21 10ị and where Pn is defined in Lemma ¯ n the subspace {z : Bn Vn V Tnỵ1 z [ ImAn } and Proof (i) Denote by S suppose that Gn is non-singular For any x [ S n > KerAn there exists T  T z [ Rm such that Bn V n V nỵ1 x ẳ An z; and therefore, Qnỵ1 G21 n Bn Vn Vnỵ1 x ẳ 21 21 Qnỵ1 Gn An z: Using Eq (5), we get Qnỵ1 Gn An z ẳ Qnỵ1 Pnỵ1 z ẳ 0: This means Qnỵ1 G21 n Bn y ẳ 11ị T V n V nỵ1 x: On the other hand, since x [ KerAn, there exists where y U T  nỵ1 z ẳ Vn QV Tnỵ1 z: z [ Rm such that x ẳ Q nỵ1 z; hence y ẳ Vn V nỵ1 Q T T Further, An21 y ẳ Un21 Sn21 Vn V n QV nỵ1 z ẳ 0; therefore y [ KerAn21 m and y ¼ Qn h for some h [ R : By virture of Eqs (7) and (11), we get T Qnỵ1 G21 n Bn Qn h ẳ V nỵ1 QV n h ẳ 0: From the last relation, we find that T QV n h ¼ 0; and hence, V n QV Tn h ¼ 0; or Qn h ¼ 0: It implies that x ẳ T T Vnỵ1 V n y ẳ V nỵ1 V n Qn h ¼ for any x [ S n > KerAn : This means that S n > KerAn ¼ {0}: Now the further proof of part (i) follows the same line  n x ¼ then as the proofs given in Ref [2] Indeed, suppose G T      Bn Vn Vnỵ1 Qnỵ1 x ẳ 2An x [ ImAn ; therefore Qnỵ1 x [ Sn : On the other  nỵ1 x [ KerAn ; hence Q  nỵ1 x [ S n > KerAn ¼ {0}: It follows hand, Q  nỵ1 x ẳ 0: Thus G  n is non-singular that An x ¼ 0; and therefore, x ¼ Q either  n are projections onto KerAn21, (ii) First we note that both Qn and Q T T T     therefore Qn Qn ¼ Qn ; i.e V n QV n Vn QVn ¼ V n QV n : From the last relation, it follows that T Q ¼ V n V n QV Tn V n Q ð12Þ Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1089  n ; substituting Eq (12), we Now we shall prove Eq (9) Instead of Q in G obtain: T T 21    V n QV Tnỵ1 G21 n Gn ẳ V n QV nỵ1 Gn An ỵ Bn Vn QVnỵ1 ị T T 21   ẳ V n QV Tnỵ1 G21 n An ỵ V n QV nỵ1 Gn Bn ðVn Vn Þ T Â ðV n QV Tn ÞV n QV nỵ1 T 21  T ẳ V n QV Tnỵ1 G21 n An ỵ V n QV nỵ1 Gn Bn Qn ịVn QVnỵ1 T Using Eq (5), we get V n QV Tnỵ1 G21 n An ẳ V n V nỵ1 Qnỵ1 Pnỵ1 ẳ 0: Further,  T by virtue of Eq (6), we have V n QV Tnỵ1 G21 n Bn Qn ịVn QVnỵ1 ẳ T  n V n V Tnỵ1 ẳ V n QV Tnỵ1 :  n V n V Tnỵ1 ẳ Q V n QV Tn ịV n QV nỵ1 ẳ Qn Q T 21  T Thus V n QV nỵ1 Gn Gn ẳ V n QV nỵ1 ; which means Eq (9) 21  Finally, to prove Eq (11), we first note that G21 n21 Gn21 ẳ Gn21 An21 ỵ T  Bn21 V n21 QV n Þ: From Eqs (5), (6) and (12), it follows that G21 n21 Gn21 ¼ T 21  T ~   Pn ỵ V n QV n21 Vn21 QVn : Using the last relation, we obtain Pn Gn21 Gn21 ¼ T ~ n is a projection P~ n Pn ỵ P~ n V n QV Tn21 V n21 QV n : Lemma ensures that Q ~ n Qn ¼ Qn and P~ n Qn ¼ 0: Further, P~ n Pn ¼ onto KerAn21, therefore Q T P~ n I Qn ị ẳ P~ n and besides, P~ n V n QV Tn21 V n21 QV n ¼ P~ n Qn V n V Tn21 T  ~ ~ 21 ~  21 V n21 QV n ¼ 0: Thus, P~ n G21 n21 Gn21 ¼ Pn ; hence Pn Gn21 ¼ Pn Gn21 : The proof of Lemma is complete A Lemma guarantees that the following definition does not depend on the chosen SVDs of An Definition The LIDE (Eq (1)) is said to be index-1 if: ðiÞ rankAn ; rð0 , r , mÞ: ðiiÞ The matrices Gn U An ỵ Bn V n QV Tnỵ1 are non-singular for all n [ N T Lemma Let An ẳ U n Sn V Tnỵ1 ẳ Un Sn Vnỵ1 be two SVDs of An Set P^ n U  21 B ; where G  V V T G  n U An ỵ Bn V n QV Tnỵ1 and M nị U I2Q k Qk n 21n nỵ1 n n nị Qk  21  iẳ0 Gn212i Bn212i ; Mk ¼ i¼0 Gn212i Bn212i ð0 # k # n 1ị: Then P~ n ẳ P^ n 13ị ^  nị P~ n M nị k ẳ Pn M k ð14Þ Downloaded by [Nipissing University] at 02:54 20 October 2014 1090 L.C LOI et al Proof Relation (13) follows directly from Eq (9) and the definitions of P~ n and P^ n : Further, using Eq (8), we get ~ P~ n M nị k ẳ Pn k Y P~ n2i G21 n212i Bn212i ð15Þ  21 P^ n2i G n212i Bn212i 16ị iẳ0 ^  nị P^ n M n ¼ Pn k Y i¼0  21 Applying Eq (13) and taking into account Eq (10), we have P^ n2i G n212i ¼ 21 21  n212i ¼ P~ n2i Gn212i : Finally, combining the last relation with P~ n2i G Eqs (13), (15) and (16), we come to Eq (14), as was to be proved Now we are ready to consider IVPs and BVPs for LIDEs A Let LIDE (Eq (1)) be index-1 Then Theorem The IVP: An xnỵ1 ẳ Bn xn ỵ qn n [ Nị 17ị P0 x0 x ị ẳ ð18Þ where x [ R is a given vector, has a unique solution x0 ¼ P~ x QV T1 G21 > q0 > > > T 21 > < x1 ¼ P~ G21  ỵ G21 B0 x q0 Þ V QV G1 q1 P n22 21 21 > x ẳ P~ n M nị  ỵ kẳ0 M nị > n21 x n222k Gk qk ỵ Gn21 qn21 ị > n > > : 2V n QV Tnỵ1 G21 n qn n $ 2ị m£m ð19Þ Moreover, the solution formula (19) is independent of the chosen SVDs of An Proof Multiplying both sides of Eq (17) from the left by Pnỵ1 G21 n and Qnỵ1 G21 ; respectively, and applying Eqs (5) (7), we come to the n following system: { 21 Pnỵ1 xnỵ1 ẳ Pnỵ1 G21 n Bn Pn xn ỵ Pnỵ1 Gn qn 20ị T 21 Qnỵ1 G21 n Bn Pn xn þ V nþ1 QV n xn þ Qnþ1 Gn qn ¼ ð21Þ Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1091 Further, multiplying both sides of Eq (21) from left by V n V Tnỵ1 ; we obtain T 21 V n QV Tnỵ1 G21 n Bn Pn xn ỵ Qn xn ỵ V n QV nỵ1 Gn qn ẳ Denoting Pnxn by un, from the last relation, we find xn ¼ Pn xn ỵ Qn xn ẳ 21 T I V n QV Tnỵ1 G21 n Bn ịun V n QV nỵ1 Gn qn : Thus, xn ẳ P~ n un V n QV Tnỵ1 G21 n qn 22ị Observing that un is a solution of the explicit system of difference equations ( 21 unỵ1 ẳ Pnỵ1 G21 n Bn un ỵ Pnỵ1 Gn qn u0 ẳ u U P0 x we get un ¼ nY 21 Pn2i G21 0 n212i Bn212i u i¼0 n22 n2k22 Y X kẳ0 21 Pn2i G21 n212i Bn212i Pkỵ1 Gn qk iẳ0 ỵ Pn G21 n21 qn21 : Using the last relation and Eq (22) and arguing as in the proof of the Lemma 3, we come to formula (19) Now suppose, we are given two SVDs T of An: An ẳ U n Sn V Tnỵ1 ẳ Un Sn Vnỵ1 : Applying Eqs (9), (10), (13) and (14), we get xn ẳ P~ n M nị  ỵ P~ n n21 x n22 X 21 ~ 21 ~ M nị n2k22 Pkỵ1 Gk qk ỵ Pn Gn21 qn21 kẳ0 V n QV Tnỵ1 G21 n qn  nị ẳ P^ n M  ỵ P^ n n21 x n22 X ^ ^  21  21  nị M n2k22 Pkỵ1 Gk qk ỵ Pn Gn21 qn21 k¼0 T 21  n qn V n QV nỵ1 G This means that the solution formula (19) does not depend on the chosen SVDs Theorem is proved A Downloaded by [Nipissing University] at 02:54 20 October 2014 1092 L.C LOI et al The next problems we shall be concerned with are frequently referred to as MPBVPs: { Ai xiỵ1 ẳ Bi xi ỵ qi n X i ẳ 0; n 23ị C i xi ¼ g ð24Þ i¼0 Suppose that Eq (23) is an index-1 LIDE Proceeding as in the proof of Theorem 1, we get the following solution formula for Eq (23): x1 ẳ P~ M 01ị x ỵ G21 q0 Þ V QV T2 G21 q1 ; ! x0 ¼ P~ x QV T1 G21 q0 ; xi ¼ P~ i M iị 0 ỵ i21 x i22 X 21 21 M iị i2k22 Gk qk ỵ Gi21 qi21 kẳ0 V i QV Tiỵ1 G21 i qi i ẳ 2; n 1ị; x n ẳ Pn M nị 0 n21 x ỵ n22 X ! 21 M nị n2k22 Gk qk ỵ G21 n21 qn21 ỵ Q n j; kẳ0 where x0 ; j [ Rm are arbitrary vectors ðnÞ Put X U P~ ; X i U P~ i M iị i21 i ẳ 1; n 1ị; X n U Pn M n21 and D ẳ Pn n i¼0 C i X i : Obviously, {X i }i¼0 are fundamental solutions of Eq (23), i.e Ai X iỵ1 ẳ Bi X i i ẳ 0; n 1ị: Let Ai ẳ U i Si V Tiỵ1 be a SVD of Ai, where Si ẳ diagsi1ị ; · · ·; sðrÞ i ; 0; · · ·; 0ịi ẳ 0; n 1ị: We recall that Qẳ ! 0 I m2r is a projection onto KerSi : Further, set Qn U V n QV Tn ; V U I; Pn U I Qn and P U I Q: In what follows, we shall deal with the (m £ 2m ) matrix (DjCnQn) with columns of D and CnQn, and the (2m £ 2m ) matrix RU P 0 Qn ! Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1093 Definition A MPBVP for index-1 LIDE is said to be regular if the following regularity condition is valid: KerDjCn Qn ị ẳ KerR 25ị The correctness of Definition is guaranteed by the following lemma Lemma The regularity condition does not depend on the chosen SVDs of Ai i ẳ 0; n 1ị: T Proof Let Ai ẳ Ui Si Viỵ1 be another SVD of Ai i ẳ 0; n 1ị: For the last P  ¼ ni¼0 Ci X i ; X U P^ ; X i U P^ i M  iị SVDs, we define D i21 i ẳ 1; n 1Þ; ðnÞ ðiÞ  n and P^ i ; M  i21 are defined as in Lemma  n21 ; where P n U I Q Xn U P n M This Lemma also ensures that X ¼ P~ ¼ P^ ¼ X ; X i ẳ P~ i M iị i21 ẳ nị nị ^ nM      iị   P^ i M P ẳ X i ẳ 1; n 1ị: Besides, X ẳ P ¼ P ¼ M i n n n i21 n21 Pn n21 ðnÞ    P n P~ n M nị ẳ P M : From the last relation we have, D ¼ C X n n21 n21 i¼0 i i ẳ Pn nị   C X ỵ C X X ị ẳ D ỵ C ð P P ÞM : Now suppose that n n n n n n i¼0 i i n21 T T T   Eq (25) is valid Let ðx0 ; j Þ [ KerðDjC n Qn Þ then  nj ẳ  x0 ỵ C n Q D ð26Þ  n ; from Eq (26), we have Dx0 þ Cn ðP n Observing that Q n ¼ Qn Q nị  Pn ịM n21 x ỵ C n Qn Qn j ¼ 0: Since An21 Pn ¼ An21 ¼ An21 Pn ; it follows An21 ðP n Pn ịM nị  ẳ 0: This means P n Pn ịM nị  ẳ Qn z for some n21 x n21 x m z [ R : Thus Eq (26) is equivalent to the relation Dx0 ỵ Cn Qn z ỵ Q n jị ẳ 0; hence xT0 ; z ỵ Q n jịT ÞT [ KerðDjC n Qn Þ ¼ KerR: The last inclusion ensures that P0 x0 ¼ and Qn ðz ỵ Q n jị ẳ Using relations ~ iị  (7), we find Xi P0 ¼ X i P0 ¼ P~ i M ðiÞ i21 P0 ¼ Pi M i21 ẳ X i ẳ Xi and by the nị nị    same relations Xn P0 ẳ Pn M n21 P0 ¼ Pn Pn M n21 P0 ¼ P n Pn M nị n21 ẳ P nị nị  x0 ¼ ni¼0 C i X i x ¼  n21 ¼ X n : This implies D P n M n21 ¼ P n M Pn  n j ¼ Since   ¼ 0: From Eq (26), it follows that C n Q i¼0 C i Xi P0 x     Qn Qn ẳ Qn and Qn z ỵ Qn jị ẳ 0; we get Qn j ¼ 2Qn z; and therefore Cn Qn z ¼ 0: On the other hand, as X i x0 ¼ X i P0 x ¼ i ẳ 0; nị; we find that Dx0 ỵ C n Qn z ¼ 0: This means ðxT0 ; z T ịT [ KerDjCn Qn ị ẳ KerR;  where hence Q n j ¼ X Qn z ¼ 0; therefore ðxT0 ; j T ÞT [ KerR; R U P 0  Qn ! Downloaded by [Nipissing University] at 02:54 20 October 2014 1094 L.C LOI et al  n Þ , KerR is proved To show the converse  nQ Thus, the inclusion Ker ðDjC  there hold the inclusion, we observe that for arbitrary ðxT0 ; j T ÞT [ KerR;  n j ¼ 0: This implies that Xi x ¼ X i x ¼ relations P0 x ¼ P0 Px0 ¼ and Q  n j ¼ 0; it follows D  x0 ¼ 0: Further, since Cn Q  x0 ỵ X i P0 x ẳ 0; therefore D T T T    Cn Qn j ẳ or x0 ; j ị [ KerðDjCn Qn Þ: The proof of Lemma is complete A Theorem A necessary and sufficient condition for the unique solvability of MPBVP Eqs (23) and (24) is its regularity Proof It suffices to prove that the regularity of MPBVP Eqs (23) and (24) is equivalent to the fact that the corresponding homogenous MPBVP: Ai xiỵ1 ẳ Bi xi i ẳ 0; n 1ị n X C i xi ẳ 27ị 28ị iẳ0 has only trivial solutions Suppose first that problem (27), (28) has only trivial solution and let ðxT0 ; j T ÞT [ KerðDjC n Qn ị: Putting x*i U X i x0 i ẳ n 0; n 1Þ and x*n U X n x ỵ Qn j; we find that {x*i }iẳ0 is a solution of Eqs (27) and (28) From the assumption, it follows x*i ẳ i ẳ 0; nị: In particular X x0 ¼ and X n x ỵ Qn j ẳ 0: Since X x0 ¼ ðI Q0 V V T1 G21 x0 ¼ 0; or Q0 V V T1 G21  ¼ x we get P0 x ¼ 0; B0 Þ B0 x consequently X n x ¼ X n P0 x ¼ 0: Thus, Qn j ¼ 0; hence ðxT0 ; j T ÞT [ KerR: This means KerðDjCn Qn Þ , KerR: To prove the converse inclusion, let ðxT0 ; j T ịT [ KerR: Then P0 x0 ẳ 0; therefore X i x ¼ X i P0 x ¼ 0ði ¼ 0; nÞ; hence Dx0 ¼ 0: On the other hand, j [ KerQn ; consequently ðxT0 ; j T ÞT [ KerðDjC n Qn Þ: Now suppose that MPBVP Eqs (23) and (24) is regular and let {xi }ni¼0 be a solution of Eqs (27) and (28) Then there exist x0 ; j [ Rm such that xi ¼ X i x0 ; ði ¼ 0; n 1ị and xn ẳ X n x ỵ Qn j: From condition (28), it follows Dx0 ỵ Cn Qn j ẳ or xT0 ; j T ịT [ KerðDjCn Qn Þ: The regularity condition ensures that ðxT0 ; j T ịT [ KerR; or P0 x0 ẳ and Qn j ¼ 0: Since X i x0 ¼ X i P0 x ¼ ði ¼ 0; nị it follows xi ẳ i ẳ 0; nÞ: Thus problem (27), (28) has only trivial solution Theorem is proved A Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1095 THE NON-CONSTANT-RANK CASE We begin this section by considering a LIDE (Eq (1)) with nesting kernels KerAnỵ1 , KerAn 29ị Let Qn be linear projections onto KerAn and Pn U I Qn : Lemma Condition (29) is equivalent to the relations: Pn Pnỵ1 ẳ Pn 30ị Proof Suppose that Eq (30) is fulfilled For any, x [ KerAnỵ1 ; we have Qnỵ1 x ẳ x or Pnỵ1 x ẳ 0; hence Pn Pnỵ1 x ẳ 0: From Eq (30), it follows Pn x ¼ or x [ KerAn : This means Eq (29) Conversely, let Eq (29) be valid For any x [ Rm ; Qnỵ1 x [ KerAnỵ1 , KerAn : Then Qn Qnỵ1 x ẳ Qnỵ1 x: The last equality is equivalent to Pn Pnỵ1 x ¼ Pn x: The proof of Lemma is complete A Theorem Suppose that condition (29) is fulfilled Moreover, assume that the matrices Gn U An ỵ Bn Qn are non-singular for all n [ N: Then LIDE (Eq (1)) is solvable for any right-hand side qn and there holds a solution formula: x0 ¼ ðI Q0 G21 x0 Q0 G21 B0 Þ q0 nY 21 xn ẳ I Qn G21 n Bn ị{ Pn212i G21 0 n212i Bn212i x iẳ0 ỵ n22 X n222k Y k¼0 i¼0 21 Pn212i G21 n212i Bn212i ðPk Gk qk ỵ Qk jk ị 21 ỵ Pn21 G21 n21 qn21 ỵ Qn21 jn21 } Qn Gn qn n $ 1ị; where ji [ Rm i ẳ 0; n 1Þ are arbitrary vectors ð31Þ Downloaded by [Nipissing University] at 02:54 20 October 2014 1096 L.C LOI et al Proof By arguing as in the proof of Theorem 1, we see that LIDE (Eq (1)) is equivalent to the system: 21 Pn xnỵ1 Pn G21 n Bn Pn x n ẳ Pn G n q n 32ị 21 xn ẳ I Qn G21 n Bn ịPn xn Qn Gn qn ð33Þ { Denoting Pnxn by un and taking into account relation (30) in Lemma 5, from Eq (32), we have 21 Pn unỵ1 Pn G21 n B n un ¼ Pn G n qn The last equation has solutions of the form 21 unỵ1 ẳ Pn G21 n Bn u n ỵ Pn G n q n ỵ Q n j n 34ị where jn [ Rm are arbitrary vectors From Eq (34), we easily deduce that un ¼ nY 21 Pn212i G21 n212i Bn212i u0 iẳ0 ỵ n22 X n222k Y kẳ0 iẳ0 21 Pn212i G21 n212i Bn212i Pk Gk qk ỵ Qk jk ị ỵ Pn21 G21 n21 qn21 ỵ Qn21 jn21 : Now formula (31) is insured from Eqs (34) and (33) Theorem is proved A Remarks Under the assumptions of Theorem 3, the initial-value problem (17), (18)) always has solutions xn depending on arbitrary vectors ji ði ¼ 0; n 1Þ and projections Qn In particular, when KerAn are constant, then rankAn ; r; Pn ; P; Qn ; Q and LIDE (Eq (1)) is index-1 21 According to relations (7), Pi G21 i Bi Qi21 ji21 ¼ PGi Bi Qji21 ¼ and solution formula (31) is the same as Eq (19) We end this section by considering a direct consequence of Theorem Let An ¼ U n Sn V Tnỵ1 be a SVD of An, where Sn ẳ diagsn1ị ; ; snrn ị ; 0; ; 0Þ and r n U rankAn : Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1097 Corollary Consider a LIDE with non-decreasing ranks: rankAnỵ1 $ rankAn : Suppose that for all n [ N; the matrices An ỵ Bn V n Qn V Tnỵ1 ; where Qn ẳ ! 0 I m2rn are non-singular Than the LIDE (Eq (1)) is solvable for any righthand side qn Proof From Eq (1), it follows U n Sn V Tnỵ1 xnỵ1 ẳ Bn V n V Tn xn ỵ qn : Denoting V Tn xn by yn ; we find Sn ynỵ1 ẳ U Tn Bn V n yn ỵ U Tn qn 35ị Obviously, the inequality r nỵ1 $ r n implies the inclusion KerSnỵ1 , KerSn : Further, since Sn ỵ U Tn Bn V n Qn ẳ U Tn An ỵ Bn V n Qn V Tnỵ1 ịV nỵ1 ; from the assumption of Corollary, it follows that the matrices Gn ẳ Sn ỵ U Tn Bn V n Qn are all non-singular Now Theorem ensures that LIDE (Eq (35)) has solutions yn, therefore xn ¼ V n yn will be solutions of LIDE (Eq (1)) The proof of Corollary is complete A EXAMPLE We shall illustrate the main results of the preceding sections by using some simple examples Because of limitation of space, we shall not give lengthy details Example An ¼ Consider Eq (1) with the following data: 1 n n ! ; Bn ¼ n n n n21 ! ; qn U q ẳ 21 ! 36ị Downloaded by [Nipissing University] at 02:54 20 October 2014 1098 L.C LOI et al In this case, rankAn ¼ 1; An ¼ U n Sn V Tnỵ1 ; 1 U n ẳ p @ ỵ n2 n V nỵ1 1 @ ¼ pffiffiffi 21 n 21 p Sn ẳ ỵ 2n @ A; A; A: Further, Q¼ Q0 ¼ Q; 0 0 ! ; V U I; 21 1 Qn ¼ V n Qn V Tn ¼ @ 21 1@ Pn ¼ 1 1 A; A: Simple calculation shows that B G0 ẳ A0 ỵ B0 V QV T1 ẳ @ 1 p1 Gn ẳ An ỵ Bn V n QV Tnỵ1 ẳ @ p C A; 21= 1 n ỵ 1=2 n 1=2 A: Since Gnðn $ 0Þ are non-singular, Eq (1) with data (36) is an index-1 LIDE Theorem ensures the unique solvability of the corresponding IVP Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1099 Using formula (19), we find that the IVP (Eqs (17) and (18)) has a unique solution n22 X 2n 2 2n ỵ 2; 22n ỵ 2nịT x0 ẳ x01ị ; 21ịT ; xn ẳ n 1ị! k! kẳ0 ỵ 2n ỵ 3; 22n 2 1ÞT Example data: Consider a three-point BVP (Eqs (23) and (24)) with the 2i B B B Ai ¼ B B @ i i 21 i 2i B B B Bi ¼ B B @ i C C 1C C; C A 1 B B B C0 ¼ B B @ 0 C C 21 C C; C A B C B C B C qi ¼ B i C ; B C @ A i2 1 B B B C2 ¼ B B @ 37ị i ẳ 0; 21 0 22 21 C C C C; C A C C 1C C; C A 1 B B B C1 ¼ B B @ 1 B C B C B C g ¼ B0C B C @ A 21 21 C C 0C C; C A ð38Þ Downloaded by [Nipissing University] at 02:54 20 October 2014 1100 L.C LOI et al Let Ai ¼ U i Si V Tiỵ1 be a SVD of Ai, where B B U0 ¼ B B0 @ 1 B B V1 ¼ B B @ 0 C C C C; A 21 0 C pffiffiffi C 0C C; A 0 B B S0 ¼ B B0 @ 0 pffiffiffi pffiffiffi C C 1= 1= C; C pffiffiffi pffiffiffi A 21= 1= pffiffiffi pffiffiffi pffiffiffi pffiffiffi ỵ 3ị=3 ỵ 3ị 3 1ị=6 B p p p B 21 ỵ 3ị=3 ỵ 3ị 1= U1 ¼ B B @ pffiffiffi pffiffiffi pffiffiffi 1=ð3 þ 3Þ 3ð þ 1Þ=6 0 C C pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi C 3 C; C A 0 p p 3ỵ B B B S1 ¼ B B @ 0 1= pffiffiffi 21= pffiffiffi C C 21= C; C p A 1= p p 3ỵ B B pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi B V ¼ B 21 ỵ 3ị=2 ỵ B @ p p p ỵ 3ị=2 ỵ p p p ỵ 3ị= ỵ p p 1= ỵ 21= p p 6ỵ2 Further, 0 B Q¼B @0 C 0C A; 0 V U I; Qi ¼ V i QV i C pffiffiffi C 1= C C C pffiffiffi A 1= Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1101 Calculating G0, G1, we find: B G0 ẳ A0 ỵ B0 V QV T1 ¼ B @ 0 B G1 ẳ A1 ỵ B1 V QV T2 ẳ B @ 1 p p C ỵ 1= 21 ỵ C; p p A 1= 1= 21 2 C 0C A Obviously, G0, G1, both are non-singular Thus LIDE with data (37) is index-1 Now we study the regularity of problems (23) and (24) with data (37) and (38) Since 1 0 B C B C; X ¼ P~ ¼ I QV T1 G21 B0 ¼ @ A 0 0 21 B 21 B 21 X ẳ P~ M 01ị ẳ ðI V QV T2 G21 B1 ÞG0 B0 ¼ @ 21 21 B 21 B 21 X ẳ P~ M 12ị ẳ I V QV T2 ịG21 B1 G0 B0 ẳ @ C 0C A; 1 C 0C A 21 21 B D ẳ C0 X ỵ C1 X ỵ C2 X ¼ @ 25 0C A; C2 Q2 ¼ C V QV T2 B 21=2 ¼@ 1 22 21=2 C A 1 we have: 0 Downloaded by [Nipissing University] at 02:54 20 October 2014 1102 L.C LOI et al Thus, 21 0 B DjC Q2 ị ẳ B @4 25 0 21=2 22 0 0 0 0 0 0 0 0 0 0 1=2 0 1=2 1 C 21=2 C A and P R¼ B B0 B ! B B0 B ¼B Q2 B0 B B0 B @ 0 C C C C C C C C C 1=2 C C A 1=2 Observing that KerDjC2 Q2 ị ẳ KerR; we come to the conclusion that the three-point BVP is regular, hence, it has a unique solution Simple calculation gives x0 ẳ 13; 13; 0ịT ; x1 ẳ 1; 22; 22ịT ; x2 ẳ 21; 28; 25ịT : Example Let us consider LIDE (Eq (1)) with the data: 1 0 C B C B Bnỵ1 n 0C C B C; An ¼ B C B B n nn ỵ 1ị nn 1ị C C B A @ nỵ1 n nn 1ị 0 n B B Bnỵ1 B Bn ¼ B B Bn B @ 2n 2n n21 2n nỵ2 nỵ1 21 qn ẳ 1; 1; 1; 1ÞT ; ðn $ 0Þ n C C 2n C C C C n ỵ 1C C A ð39Þ Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1103 Since KerA0 ¼ Span{ð0; 0; 0; 1ÞT ; ð0; 0; 1; 0ÞT ; 0; 1; 0; 0ịT }; KerA1 ẳ Span{0; 0; 0; 1ịT ; 0; 0; 1; 0ịT }; KerAn ẳ Span{0; 0; 0; 1ÞT }ðn $ 2Þ: It follows that KerA0 KerA1 KerA2 ẳ KerA n n $ 3ị: Obviously, 0 B B B0 B Q0 ¼ B B B0 B @ 0 B B B0 B Qn ¼ B B B0 B @ 0 0 0 0 C C 0C C C; C 0C C A 0 0 0 0 0 B B B0 B Q1 ¼ B B B0 B @ 0 0 C C 0C C C; C 0C C A 0 0 and C C 0C C C C 0C C A ; Q are projections onto KerA0, KerA1 and KerAn ðn $ 2Þ; respectively Further, B B B B G0 ẳ A0 ỵ B0 Q0 ¼ B B B 21 B @ 1 B B B2 B G1 ẳ A1 ỵ B1 Q1 ¼ B B B0 B @ 2 21 21 C C 0C C C; C 1C C A 1 C C 21 C C C C C C A 21 Downloaded by [Nipissing University] at 02:54 20 October 2014 1104 L.C LOI et al and 0 n C 2n C C C nn ỵ 1ị nn 1ị n ỵ C A n nn 1ị B Bnỵ1 B G n ẳ An ỵ Bn Q n ẳ B Bn @ nỵ1 n ðn $ 2Þ Since Gn ðn $ 0Þ are non-singular and KerAn are nested, Theorem ensures the solvability of LIDE (Eq (1)) with data (39) Example Consider LIDE (Eq (1)) with the following data: n 0 C C nC C; A n B B An ¼ B B0 @ 1 B B Bn ¼ B Bn @ nỵ1 1 C C nỵ1C C; A n 1 n B C B C B C qn ẳ B C @ A 21 40ị The matrices An have SVDs: An ¼ U n Sn V Tnỵ1 ; where p B 2 U n ¼ pffiffiffi B @ 0 V nỵ1 ẳ p n þ1 C C A; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2n þ B Sn ¼ B @ 21 B Bn @ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2 ỵ 2n C C A 0 C n 0C A; 0 Downloaded by [Nipissing University] at 02:54 20 October 2014 LINEAR IMPLICIT DIFFERENCE EQUATION 1105 Observing that rankA0 ¼ , rankAn ¼ ðn $ 1Þ we get a LIDE with nondecreasing ranks Putting 1 0 0 0 B C B C C B C Q0 ¼ B @ A ; Q n U Q ¼ @ 0 A ; Pn ¼ I Q n 0 0 n $ 0ị; V ẳ I: We find 1 21 B ~ ¼ A0 ỵ B0 V Q0 V T1 ẳ B G @ 1 C 21 C A 21 and 2nan B ~ n ¼ An ỵ Bn V n Qn V Tnỵ1 ẳ B nn 2 2nịan G @ ỵ n2n 3ịan n ỵ 2an 2n n ịan n ð2n 3Þan C nC A ~ ¼ 21; detG ~n ¼ where an U n 2 2n ỵ 2ị21=2 n ỵ 1ị21=2 : Since detG n2n 1ịn ỵ 1ịan ðn $ 1Þ; the Corollary of Theorem implies the solvability of LIDE (Eq (1)) with data (40) Acknowledgements This work has been partially supported by the Vietnam Fundamental research programme under the contract No 1.3.6 References [1] P K Anh, Multipoint boundary-value problems for transferable differential-algebraic equations I Linear case, Vietnam J Math J Clin Invest., 25(4) (1997), 347358 [2] R Maărz, On linear differentialalgebraic equations and linearizations, Appl Numer Math., 18 (1995), 267–292  ... [Nipissing University] at 02:54 20 October 2014 Journal of Difference Equations and Applications, 2002 Vol (12), pp 1085–1105 On Linear Implicit Non-autonomous Systems of Difference Equations L.C LOI,... of initial-value problems (IVPs) and multipoint boundaryvalue problems (MPBVPs) for linear implicit non-autonomous systems of difference equations Keywords: Linear implicit difference equations. .. differential-algebraic equations I Linear case, Vietnam J Math J Clin Invest., 25(4) (1997), 347358 [2] R Maărz, On linear differential–algebraic equations and linearizations, Appl Numer Math., 18 (1995),