FOR NONLINEAR IMPLICIT DIFFERENCE EQUATIONS

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 that it can be extended to a class of nonlinear IDEs. The paper is organized as follows.Section 2is concerned with index 1 LIDEs and their reduction to ordinary difference equations. InSection 3, we study the 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 difference equations

LetQbe an arbitrary projection onto a given subspaceNof dimensionm−r(1r m−1) inR^m. Further, let{v_i}^r1and{v_j}^m_r+1be any bases of KerQandN, respectively.

Denote byV =(v1,. . .,v_m) a column matrix and denote ˜Q=diag(O_r,I_m−r), whereO_r andIm−rstand forr×rzero matrix and (m−r)×(m−r) identity matrix, respectively.

ThenV is nonsingular,Q=VQV˜ ⁻¹, and this decomposition depends on the choice of the bases{v_i}^m1, that is, onV.

Now, supposeNα andNβ are two subspaces of the same dimensionm−r(1r m−1) inR^m. Then any projections Qα and Qβ ontoNα andNβ 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

as Q_α=V_αQV˜ _α⁻¹ and Q_β=V_βQV˜ _β⁻¹, respectively. Define an operator connecting two subspacesN_αandN_β(connecting operator, for short)Q_αβ:=V_αQV˜ _β⁻¹. Clearly,

Qαβ=QαQαβ=QαβQβ=QαVαV_β⁻¹=VαV_β⁻¹Qβ,

Q_αβQ_βα=Q_α, Q_βαQ_αβ=Q_β. (2.1) We consider a system of LIDEs

A_nx_n+1+B_nx_n=q_n (n0), (2.2) whereA_n,B_n∈R^m×m,q_n∈R^mare given and rankA_n≡r(1rm−1) for alln0.

Let Qn be any projection onto KerAn,Pn=I−Qn and consider decompositionsQn= VnQV˜ _n⁻¹ (n0). For definiteness, we putA₋1:=A0,Q₋1:=Q0,P₋1:=P0, andV₋1:= V0. Thus, the connecting operatorsQ_n−1,n:=V_n−1QV˜ _n⁻¹are determined for alln0.

Recall that a linear DAE A(t)x+B(t)x=q(t), t∈J:=[t0,T], where A,B∈C(J, R^m×m),q∈R^m, is said to be of index 1 or transferable (see [4]) if there exists a smooth projectionQ∈C¹(J,R^m×m) onto KerA(t) such that the matrixG(t)=A(t) +B(t)Q(t) is nonsingular for allt∈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 conditionS(t)∩KerA(t)= {0}, whereS(t) := {ξ∈R^m:B(t)ξ∈ImA(t)}.

A similar result can be established for LIDEs, namely, the following lemma.

Lemma2.1. The matrixG_n:=A_n+B_nQ_n−1,nis nonsingular if and only if

Sn∩KerAn−1= {0}, (2.3)

where, as in the DAE case,Sn:= {ξ∈R^m:Bnξ∈ImAn}.

The proof ofLemma 2.1repeats that of [3, Lemma 1] with some obvious changes, and uses the fact that condition (2.3) holds if and only ifV_nV_n⁻₋¹₁S_n∩KerA_n= {0}.

Since condition (2.3) does not depend on the representation of connecting operators, we get the following corollary.

Corollary2.2. The nonsingularity ofGndoes not depend on the choice of connecting op- erator, that is, ifQ_n−1,n:=V_n−1QV˜ _n⁻¹andQ¯_n−1,n:=V¯_n−1Q˜V¯_n⁻¹, then both matricesG_n:= An+BnQn−1,nandG¯n:=An+BnQ¯n−1,nare singular or nonsingular simultaneously.

Corollary 2.2confirms that it suffices to restrict our consideration to orthogonal pro- jections onto KerAn, as was done in [3]. However, in the mentioned paper, a singular- value decomposition (SVD) ofA_nis employed for constructing an orthogonal projection Q_nonto KerA_nand it seems not to be convenient for a further extension of the index notion to nonlinear cases.Corollary 2.2also allows us to introduce the following notion of index 1 LIDEs, which is quite similar to that of index 1 (transferable) linear DAEs.

Definition 2.3. The LIDEs (2.2) are said to be of index 1 if, for alln0, (i) rankAn=r;

(ii)G_n:=A_n+B_nQ_n−1,n is nonsingular.

The main difference between linear index 1 DAEs and linear index 1 IDEs is the fact that the pencil{A(t),B(t)}in the continuous case is always of index 1 for allt∈J, while forn1,{An,Bn}is not necessarily of index 1.

Now, we describe shortly the decomposition technique for index 1 LIDEs. Performing P_nG⁻_n¹andQ_nG⁻_n¹on both sides of (2.2), respectively, we get

P_nx_n+1+P_nG⁻_n¹B_nx_n=P_nG⁻_n¹q_n, (2.4) Q_nG⁻_n¹B_nx_n=Q⁻_n¹G_nq_n. (2.5) Further, denotingu_n=P_n−1x_n,v_n=Q_n−1x_n(n0) and observing thatP_nG⁻_n¹B_nQ_n−1x_n= P_nG⁻_n¹B_nQ_n−1,nQ_n,n−1x_n = P_nQ_n,n−1x_n = P_nQ_nQ_n,n−1x_n = 0, we find P_nG⁻_n¹B_nx_n = PnG⁻_n¹Bnun. Thus, (2.4) becomes an ordinary difference equation

un+1+PnG⁻_n¹Bnun=PnG⁻_n¹qn. (2.6) SinceQnG⁻_n¹BnQn−1xn=QnG⁻_n¹BnQn−1,nQn,n−1xn=Qn,n−1xn=VnV_n⁻₋¹1Qn−1xn=VnV_n⁻₋¹1vn, (2.5) is reduced to

vn=Vn−1V_n⁻¹QnG⁻_n¹qn−QnG⁻_n¹Bnun

. (2.7)

Finally,

xn=un+vn=

I−Qn−1,nG⁻_n¹Bn

un+Qn−1,nG⁻_n¹qn. (2.8) Thus, if (2.2) is of index 1, then, for givenu0=P₋1x0=P0x0, we can computeun+1, vn, andxn(n0) by (2.6), (2.7), and (2.8), respectively. As in the DAEs case, we only need to initialize the P0-component of x0. Further, puttingn=0 in (2.8) and noting thatV₋1=V0,u0=P₋1x0=P0x0, we find that a consistent initial valuex0must satisfy a

“hidden” constraint, namely,Q0(I+G⁻0¹B0P0)x0=Q0G⁻0¹q0. 3. Nonlinear implicit difference equations

We begin this section by recalling the following version of the Hadamard theorem on homeomorphism.

Theorem3.1 [2, page 222]. SupposeF∈C¹(X,Y)is a local homeomorphism between two Banach spacesX,Yandζ(R) :=infxR([F(x)]⁻¹)⁻¹. Then if₀^∞ζ(R)dR=+∞,Fis a (global) homeomorphism ofXintoY.

In particular, if [F(x)]⁻¹αx+β for all x∈X, whereα0, β >0, then F is a homeomorphism ofX intoY. Further, supposeF=T+H, where T∈C¹(X,Y), [T(x)]⁻¹γ, for allx∈X, andH(x)−H(y)Lx−y, for allx,y∈X, then if Lγ <1,Fis a homeomorphism ofXintoY.

Consider a system of nonlinear IDEs fn

xn+1,xn

=0 (n0), (3.1)

where f_n:R^m→R^mare given vector functions.

Definition 3.2. Equation (3.1) is said to be of index 1 if

(i) the function fnis continuously differentiable, moreover, Ker(∂ fn/∂y)(y,x)=Nn, dimNn=m−r, for alln0,y,x∈R^m, where 1rm−1;

(ii) the matrixG_n=(∂ f_n/∂y)(y,x) + (∂ f_n/∂x)(y,x)Q_n−1,n(n0) is nonsingular.

Here, we putN₋1=N0,V₋1=V0,Q₋1=Q0, and denote byQn−1,nan operator con- necting two subspacesNn−1,Nn.

In the remainder of this paper, for the sake of simplicity, the norm ofR^mis assumed to be Euclidean.

Theorem3.3. Let (3.1) be of index 1. Moreover, suppose that

G⁻_n¹(y,x)α_ny+β_nx+γ_{n ∀}y,x_∈R^m,∀n0, (3.2)

whereαn,βn0,γn>0are constants. Then the problem of findingxnfrom (3.1) and the initial condition

P0x0=p0 (3.3)

has a unique solution.

Proof. Since fn

xn+1,xn

−fn

Pnxn+1,xn

= ₁

0

∂ fn

∂y

Pnxn+1+tQnxn+1,xn

Qnxn+1dt=0, (3.4) equation (3.1) becomes

fn

Pnxn+1,Pn−1xn+Qn−1xn

=0 (n0). (3.5)

Supposeun=Pn−1xn (n0) is found (forn=0,u0=P₋1x0=P0x0=p0 is given). We have to findu=P_nx_n+1∈ImP_n⊂R^r andv=Q_n−1x_n∈ImQ_n−1⊂R^m−r. Define an op- erator F:R^m →R^m by F:z:=(u^T,v^T)^T → fn(u,un+v). Letw=(∆u^T,∆v^T)^T, where

∆u∈ImPn,∆v∈ImQn−1, thenF(z)w=(∂ fn/∂y)(u,un+v)∆u+ (∂ fn/∂x)(u,un+v)∆v.

Consider the linearized equation

F(z)w=q, (3.6)

whereq∈R^m is an arbitrary fixed vector. First, observe thatGnPn=(∂ fn/∂y)Pn+ (∂ fn/

∂x)Qn−1,nQnPn=(∂ fn/∂y)Pn=∂ fn/∂y, hence G⁻_n¹(∂ fn/∂y)=Pn and GnQn=(∂ fn/

∂x)Q_n−1,nQ_n=(∂ f_n/∂x)Q_n−1,n, therefore G⁻_n¹(∂ f_n/∂x)Q_n−1,n=Q_n, where G_n, ∂ f_n/∂y,

∂ fn/∂xare valued at (u,un+v). Further, sincePn∆u=∆u,∆v=Qn−1∆v=Qn−1,nQn,n−1∆v, then by the action ofG⁻_n¹on both sides of (3.6) and using the last observations, we get

∆u+Qn,n−1∆v=G⁻_n¹u,un+vq. (3.7)

Now, applyingP_nandQ_nto both sides of (3.7), respectively, we find∆u=P_nG⁻_n¹qand Q_n,n−1∆v=Q_nG⁻_n¹q. The last equality leads to∆v=V_n−1V_n⁻¹Q_nG⁻_n¹q. Thus, (3.6) has a unique solution w=(∆u^T,∆v^T)^T. Moreover, ∆uPnG⁻_n¹q and ∆v Vn−1V_n⁻¹QnG⁻_n¹q, that is,F(z) has a bounded inverse. A simple calculation shows that[F(z)]⁻¹ω_nz+δ_n, whereω_n=√

2ρ_nmax{α_n,β_n},δ_n=ρ_n(γ_n+β_nu_n), and ρn=(Pn²+Vn−1V_n⁻¹Qn²)^1/2. By the Hadamard theorem on homeomorphism, (3.1) has a unique solution Pnxn+1 and Qn−1xn for all n0. This completes the proof of

Theorem 3.3.

In the next theorem, without loss of generality, we will use orthogonal projections ontoNn, that is,Qn=VnQV˜ _n^T andVnV_n^T=V_n^TVn=I. In this case,Qn−1,n=Vn−1QV˜ _n^T andQn = Pn = Vn =1.

Theorem3.4. Suppose fn(y,x)=gn(y,x) +hn(y,x), where (i)g_n(y,x)is continuously differentiable, moreover

Ker∂g_n

∂y(y,x)=N_n, dimN_n=m−r, ∀n0,∀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(n0)has uniformly bounded in-

verses, that is,G⁻_n¹(y,x)γnfor alln0,y,x∈R^m; (iii)hn(y,x)=hn(Pny,x)for alln0,y,x∈R^m;

(iv)h_n(y,x)−h_n( ¯y, ¯x)L_n(y−y¯²+x−x¯²)^1/2for alln0,y,x, ¯y, ¯x∈R^m. Then, ifγnLn<1/^√2for alln0, the IVP (3.1), (3.3) has a unique solution.

Proof. Using the notations ofTheorem 3.3, we define two operatorsT(z)=gn(u,un+v) andH(z)=hn(u,un+v), where, as before,z:=(u^T,v^T)^T,u:=Pnxn+1,v:=Qn−1xn, and u_n:=P_n−1x_n. From the proof ofTheorem 3.3, it follows that[T(z)]^−1√2γ_n. On the other hand,H(z) is Lipschitz continuous with a Lipschitz constantLnand^√2γnLn<1.

Thus, the mappingF(z)=T(z) +H(z) is a homeomorphism ofXontoY, therefore the

IVP (3.1), (3.3) has a unique solution.

Corollary3.5. Suppose f_n(y,x)=A_ny+B_nx+h_n(y,x), whereA_n,B_n∈R^m×m, andh_n: R^m×R^m→R^msatisfy the following conditions:

(i) rankAn≡rand the matrixGn=An+BnQn−1,nis nonsingular for alln0, where Q_n−1,nis a connecting operator ofKerA_n−1andKerA_n,A₋1:=A0;

(ii)hn(y,x)is continuously differentiable, moreover KerA_n⊂Ker∂ fn

∂y(y,x) ∀n0,∀y,x∈R^m, hn(y,x)−hn( ¯y, ¯x)Ln

y−y¯²+x−x¯²_1/2

, ∀n0,∀y,x, ¯y, ¯x∈R^m. (3.9)

Then, ifLnG⁻_n¹<1/^√2, the IVP (3.1), (3.3) is uniquely solvable.

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.

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

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[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, and P. 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:[email protected]

Ha Thi Ngoc Yen: Department of Applied Mathematics, Hanoi University of Technology, 1 Dai Co Viet, 10000 Hanoi, Vietnam

E-mail address:[email protected]