ON THE SOLVABILITY OF INITIAL-VALUE PROBLEMS FOR NONLINEAR IMPLICIT DIFFERENCE EQUATIONS PHAM KY ANH AND HA THI NGOC YEN Received 18 February 2004 Our aim is twofold. First, we propose a natural definition of index for linear nonau- tonomous implicit difference equations, which is similar to that of linear differential- algebraic equations. Then we extend this index notion to a class of nonlinear implicit difference equations and prove some existence theorems for their initial-value problems. 1. Introduction Implicit difference equations (IDEs) arise in various applications, such as the Leontief dynamic model of a multisector economy, the Leslie population growth model, and so forth. On the other hand, IDEs may be regarded as discrete analogues of differential- algebraic equations (DAEs) which have already attracted much attention of researchers. Recently [1, 3], a notion of index 1 linear implicit difference equations (LIDEs) has been introduced and the solvability of initial-value problems (IVPs), as well as multipoint boundary-value problems (MBVPs) for index 1 LIDEs, has been studied. In this paper, we propose a natural definition of index for LIDEs so t hat it can be extended to a class of nonlinear IDEs. The paper is organized as follows. Section 2 is concerned with index 1 LIDEs and their reduction to ordinary difference equations. In Section 3,westudythe index concept and the solvability of IVPs for nonlinear IDEs. The result of this paper can be considered as a discrete version of the corresponding result of [4]. 2. Index 1 linear implicit differenc e equations Let Q be an arbitrary projection onto a given subspace N of dimension m −r (1  r  m −1) in R m .Further,let{v i } r 1 and {v j } m r+1 be any bases of KerQ and N, respectively. Denote by V = (v 1 , ,v m ) a column matrix and denote ˜ Q = diag(O r ,I m−r ), where O r and I m−r stand for r ×r zero matrix and (m −r) ×(m −r) identit y matrix, respectively. Then V is nonsingular, Q = V ˜ QV −1 , and this decomposition depends on the choice of the bases {v i } m 1 , that is, on V. Now, suppose N α and N β are two subspaces of the same dimension m −r (1  r  m −1) in R m .ThenanyprojectionsQ α and Q β onto N α and N β can be decomposed Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:3 (2004) 195–200 2000 Mathematics Subject Classification: 34A09, 39A10 URL: http://dx.doi.org/10.1155/S1687183904402015 196 IVPs for nonlinear implicit difference equations as Q α = V α ˜ QV −1 α and Q β = V β ˜ QV −1 β , respectively. Define an operator connecting two subspaces N α and N β (connecting operator, for short) Q αβ := V α ˜ QV −1 β .Clearly, Q αβ = Q α Q αβ = Q αβ Q β = Q α V α V −1 β = V α V −1 β Q β , Q αβ Q βα = Q α , Q βα Q αβ = Q β . (2.1) We consider a system of LIDEs A n x n+1 + B n x n = q n (n  0), (2.2) where A n ,B n ∈ R m×m , q n ∈ R m are given and rank A n ≡ r (1  r  m −1) for all n  0. Let Q n be any projection onto KerA n ,P n = I −Q n and consider decompositions Q n = V n ˜ QV −1 n (n  0). For definiteness, we put A −1 := A 0 , Q −1 := Q 0 , P −1 := P 0 ,andV −1 := V 0 . Thus, the connecting operators Q n−1,n := V n−1 ˜ QV −1 n are determined for all n  0. Recall that a linear DAE A(t)x  + B(t)x = q(t), t ∈ J := [t 0 ,T], where A,B ∈ C(J, R m×m ), q ∈ R m ,issaidtobeofindex1ortransferable(see[4]) if there exists a smooth projection Q ∈ C 1 (J,R m×m )ontoKerA(t) such that the mat rix G(t) = A(t)+B(t)Q(t)is nonsingular for all t ∈ J. It is proved that the index 1 property (transferability) of lin- ear DAEs does not depend on the choice of smooth projections and is equivalent to the condition S(t) ∩Ker A(t) ={0},whereS(t):={ξ ∈ R m : B(t)ξ ∈ ImA(t)}. A similar result can be established for LIDEs, namely, the following lemma. Lemma 2.1. The matrix G n := A n + B n Q n−1,n is nonsingular if and only if S n ∩Ker A n−1 ={0}, (2.3) where, as in the DAE case, S n :={ξ ∈ R m : B n ξ ∈ ImA n }. The proof of Lemma 2.1 repeatsthatof[3, Lemma 1] with some obvious changes, and uses the fact that condition (2.3) holds if and only if V n V −1 n−1 S n ∩Ker A n ={0}. Since condition (2.3) does not depend on the representation of connecting operators, we get the following corollary. Corollary 2.2. The nonsingularity of G n does not depend on the choice of connecting op- erator, that is, if Q n−1,n := V n−1 ˜ QV −1 n and ¯ Q n−1,n := ¯ V n−1 ˜ Q ¯ V −1 n , then both matrices G n := A n + B n Q n−1,n and ¯ G n := A n + B n ¯ Q n−1,n are singular or nonsingular simultaneously. Corollary 2.2 confirmsthatitsuffices to restrict our consideration to orthogonal pro- jections onto KerA n ,aswasdonein[3]. However, in the mentioned paper, a singular- value decomposition (SVD) of A n is employed for constructing an orthogonal projection Q n onto KerA n and it seems not to be convenient for a further extension of the index notion to nonlinear cases. Corollary 2.2 also allows us to introduce the following notion of index 1 LIDEs, which is quite similar to that of index 1 (transferable) linear DAEs. Definit ion 2.3. The LIDEs (2.2) are said to be of index 1 if, for al l n  0, (i) rank A n = r; (ii) G n := A n + B n Q n−1,n is nonsingular. P. K. Anh and H. T. N. Yen 197 The main difference between linear index 1 DAEs and linear index 1 IDEs is the fact that the pencil {A(t),B(t)} inthecontinuouscaseisalwaysofindex1forallt ∈ J, while for n  1, {A n ,B n } is not necessarily of index 1. Now, we describe shortly the decomposition technique for index 1 LIDEs. Performing P n G −1 n and Q n G −1 n on both sides of ( 2.2 ), respectively, we get P n x n+1 + P n G −1 n B n x n = P n G −1 n q n , (2.4) Q n G −1 n B n x n = Q −1 n G n q n . (2.5) Further, denoting u n =P n−1 x n , v n =Q n−1 x n (n  0) and observing that P n G −1 n B n Q n−1 x n = P n G −1 n B n Q n−1,n Q n,n−1 x n = P n Q n,n−1 x n = P n Q n Q n,n−1 x n = 0, we find P n G −1 n B n x n = P n G −1 n B n u n .Thus,(2.4) becomes an ordinary difference equation u n+1 + P n G −1 n B n u n = P n G −1 n q n . (2.6) Since Q n G −1 n B n Q n−1 x n =Q n G −1 n B n Q n−1,n Q n,n−1 x n =Q n,n−1 x n =V n V −1 n−1 Q n−1 x n =V n V −1 n−1 v n , (2.5)isreducedto v n = V n−1 V −1 n  Q n G −1 n q n −Q n G −1 n B n u n  . (2.7) Finally, x n = u n + v n =  I −Q n−1,n G −1 n B n  u n + Q n−1,n G −1 n q n . (2.8) Thus, if (2.2) is of index 1, then, for given u 0 = P −1 x 0 = P 0 x 0 ,wecancomputeu n+1 , v n ,andx n (n  0) by (2.6), (2.7), and (2.8), respectively. As in the DAEs case, we only need to initialize the P 0 -component of x 0 . Further, putting n = 0in(2.8) and noting that V −1 = V 0 , u 0 = P −1 x 0 = P 0 x 0 , we find that a consistent initial value x 0 must satisfy a “hidden” constraint, namely, Q 0 (I + G −1 0 B 0 P 0 )x 0 = Q 0 G −1 0 q 0 . 3. Nonlinear implicit difference equations We begin this section by recalling the following version of the Hadamard theorem on homeomorphism. Theorem 3.1 [2, page 222]. Suppose F ∈ C 1 (X,Y) is a local homeomorphism between two Banach spaces X,Y and ζ(R):= inf xR ([F  (x)] −1 ) −1 .Thenif  ∞ 0 ζ(R)dR = +∞, F is a (global) homeomorphism of X into Y. In part icular, if [F  (x)] −1   αx + β for all x ∈ X,whereα  0, β>0, then F is a homeomorphism of X into Y. Further, suppose F = T + H,whereT ∈ C 1 (X,Y), [T  (x)] −1   γ,forallx ∈ X,andH(x) −H(y)  Lx − y,forallx, y ∈ X,thenif Lγ < 1, F is a homeomorphism of X into Y. ConsiderasystemofnonlinearIDEs f n  x n+1 ,x n  = 0(n  0), (3.1) where f n : R m → R m are given vector functions. 198 IVPs for nonlinear implicit difference equations Definit ion 3.2. Equation (3.1)issaidtobeofindex1if (i) the function f n is continuously di fferentiable, moreover, Ker(∂f n /∂y)(y,x) =N n , dimN n = m −r,foralln  0, y,x ∈R m ,where1 r  m −1; (ii) the matrix G n = (∂f n /∂y)(y,x)+(∂f n /∂x)(y, x)Q n−1,n (n  0) is nonsingular. Here, w e put N −1 = N 0 , V −1 = V 0 , Q −1 = Q 0 , and denote by Q n−1,n an operator con- necting two subspaces N n−1 ,N n . In the remainder of this paper, for the sake of simplicity, the norm of R m is assumed to be Euclidean. Theorem 3.3. Let (3.1) b e of index 1. Moreover, suppose that   G −1 n (y, x)    α n y+ β n x+ γ n ∀y, x ∈R m , ∀n  0, (3.2) where α n ,β n  0, γ n > 0 are constants. Then the problem of finding x n from (3.1)andthe initial condition P 0 x 0 = p 0 (3.3) has a unique s olution. Proof. Since f n  x n+1 ,x n  − f n  P n x n+1 ,x n  =  1 0 ∂f n ∂y  P n x n+1 + tQ n x n+1 ,x n  Q n x n+1 dt =0, (3.4) equation (3.1)becomes f n  P n x n+1 ,P n−1 x n + Q n−1 x n  = 0(n  0). (3.5) Suppose u n = P n−1 x n (n  0) is found (for n = 0, u 0 = P −1 x 0 = P 0 x 0 = p 0 is given). We have to find u = P n x n+1 ∈ ImP n ⊂ R r and v = Q n−1 x n ∈ ImQ n−1 ⊂ R m−r .Defineanop- erator F : R m → R m by F : z := (u T ,v T ) T → f n (u,u n + v). Let w = (∆u T ,∆v T ) T ,where ∆u ∈ImP n , ∆v ∈ ImQ n−1 ,thenF  (z)w = (∂f n /∂y)(u,u n + v)∆u +(∂f n /∂x)(u,u n + v)∆v. Consider the linearized equation F  (z)w = q, (3.6) where q ∈ R m is an arbitrary fixed vector. First, observe that G n P n = (∂f n /∂y)P n +(∂f n / ∂x)Q n−1,n Q n P n = (∂f n /∂y)P n = ∂f n /∂y,henceG −1 n (∂f n /∂y) = P n and G n Q n = (∂f n / ∂x)Q n−1,n Q n = (∂f n /∂x)Q n−1,n , therefore G −1 n (∂f n /∂x)Q n−1,n = Q n ,whereG n , ∂f n /∂y, ∂f n /∂x are valued at (u,u n + v). Further, since P n ∆u=∆u, ∆v =Q n−1 ∆v =Q n−1,n Q n,n−1 ∆v, then by the action of G −1 n on both sides of (3.6) and using the last observations, we get ∆u + Q n,n−1 ∆v = G −1 n  u,u n + v  q. (3.7) P. K. Anh and H. T. N. Yen 199 Now, applying P n and Q n to both sides of (3.7), respectively, we find ∆u = P n G −1 n q and Q n,n−1 ∆v = Q n G −1 n q. T he last equality leads to ∆v = V n−1 V −1 n Q n G −1 n q.Thus,(3.6)hasa unique solution w = (∆u T ,∆v T ) T .Moreover,∆u  P n G −1 n q and ∆v  V n−1 V −1 n Q n G −1 n q, that is, F  (z) has a bounded inverse. A simple calculation shows that [F  (z)] −1   ω n z+ δ n ,whereω n = √ 2ρ n max{α n ,β n }, δ n = ρ n (γ n + β n u n ), and ρ n = (P n  2 + V n−1 V −1 n Q n  2 ) 1/2 . By the Hadamard theorem on homeomorphism, (3.1) has a unique solution P n x n+1 and Q n−1 x n for all n  0. This completes the proof of Theorem 3.3.  In the next theorem, without loss of generality, we will use orthogonal projections onto N n , that is, Q n = V n ˜ QV T n and V n V T n = V T n V n = I. In this case, Q n−1,n = V n−1 ˜ QV T n and Q n =P n =V n =1. Theorem 3.4. Suppose f n (y, x) =g n (y, x)+h n (y, x),where (i) g n (y, x) is continuously differentiable, moreover Ker ∂g n ∂y (y, x) =N n ,dimN n = m−r, ∀n  0, ∀x, y ∈R m ; (3.8) (ii) G n (y, x) =(∂g n /∂y)(y,x)+(∂g n /∂x)(y, x)Q n−1,n (n  0) has uniformly bounded in- verses, that is, G −1 n (y, x) γ n for all n  0, y,x ∈R m ; (iii) h n (y, x) =h n (P n y,x) for all n  0, y,x ∈ R m ; (iv) h n (y, x) −h n ( ¯ y, ¯ x)  L n (y − ¯ y 2 + x − ¯ x 2 ) 1/2 for all n  0, y,x, ¯ y, ¯ x ∈ R m . Then, if γ n L n < 1/ √ 2 for all n  0,theIVP(3.1), (3.3)hasauniquesolution. Proof. Using the notations of Theorem 3.3, we define two operators T(z) =g n (u,u n + v) and H(z) = h n (u,u n + v), where, as before, z := (u T ,v T ) T , u := P n x n+1 , v := Q n−1 x n ,and u n := P n−1 x n .FromtheproofofTheorem 3.3, it follows that [T  (z)] −1   √ 2γ n .Onthe other hand, H(z) is Lipschitz continuous with a Lipschitz constant L n and √ 2γ n L n < 1. Thus, the mapping F(z) = T(z)+H(z) is a homeomorphism of X onto Y , therefore the IVP (3.1), (3.3) has a unique solution.  Corollary 3.5. Suppose f n (y, x) = A n y + B n x + h n (y, x),whereA n ,B n ∈ R m×m ,andh n : R m ×R m → R m satisfy the following conditions: (i) rank A n ≡ r and the matrix G n = A n + B n Q n−1,n is nonsingular for all n  0,where Q n−1,n is a connecting operator of Ker A n−1 and Ker A n , A −1 := A 0 ; (ii) h n (y, x) is continuously differentiable, moreover Ker A n ⊂ Ker ∂f n ∂y (y, x) ∀n  0, ∀y,x ∈R m ,   h n (y, x) −h n ( ¯ y, ¯ x)    L n  y − ¯ y 2 + x − ¯ x 2  1/2 , ∀n  0, ∀y, x, ¯ y, ¯ x ∈ R m . (3.9) Then, if L n G −1 n  < 1/ √ 2,theIVP(3.1), (3.3)isuniquelysolvable. It can be shown that the explicit Euler method applied to nonlinear transferable DAEs [4] leads to nonlinear index 1 IDEs. This and other problems related to connections be- tween DAEs and IDEs will be discussed in our forthcoming paper. 200 IVPs for nonlinear implicit difference equations Acknowledgment The first author thanks Prof. Shao Xiumin and Prof. Liang Guoping for their hospitality during his visit to the Institute of Mathematics, Chinese Academy of Sciences. References [1] P.K.AnhandL.C.Loi,On multipoint boundary-value problems for linear implicit non- autonomous systems of difference equations, Vietnam J. Math. 29 (2001), no. 3, 281–286. [2] M. S. Berger, Nonlinearity and Functional Analysis, Lectures on Nonlinear Problems in Mathe- matical Analysis. Pure and Applied Mathematics, Academic Press, New York, 1977. [3] L.C.Loi,N.H.Du,andP.K.Anh,On linear implicit non-autonomous systems of difference equations,J.Difference Equ. Appl. 8 (2002), no. 12, 1085–1105. [4] R. M ¨ arz, On linear differential-algebraic equations and linearizations, Appl. Numer. Math. 18 (1995), no. 1–3, 267–292. Pham Ky Anh: Department of Mathematics, Mechanics, and Informatics, College of Science, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam E-mail address: anhpk@vnu.edu.vn Ha Thi Ngoc Yen: Depar tment of Applied Mathematics, Hanoi University of Technology, 1 Dai Co Viet, 10000 Hanoi, Vietnam E-mail address: hangocyen02@yahoo.com . ON THE SOLVABILITY OF INITIAL-VALUE PROBLEMS FOR NONLINEAR IMPLICIT DIFFERENCE EQUATIONS PHAM KY ANH AND HA THI NGOC YEN Received 18 February 2004 Our aim is twofold. First, we. and the solvability of IVPs for nonlinear IDEs. The result of this paper can be considered as a discrete version of the corresponding result of [4]. 2. Index 1 linear implicit differenc e equations Let. definition of index for linear nonau- tonomous implicit difference equations, which is similar to that of linear differential- algebraic equations. Then we extend this index notion to a class of nonlinear