ON THE SOLVABILITY OF INITIAL-VALUE PROBLEMS FOR NONLINEAR IMPLICIT DIFFERENCE EQUATIONS PHAM KY ANH pdf

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 exten

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 di fferential-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

LetQ be an arbitrary projection onto a given subspace N of dimension m − r (1
r

m −1) inRm Further, let{ v i } r

1and{ v j } m

r+1 be any bases of KerQ and N, respectively.

Denote byV =(v1, , v m) a column matrix and denote ˜Q =diag(O r,I m−r), whereO r

andI m−rstand forr × r zero matrix and (m − r) ×(m − r) identity matrix, respectively.

ThenV is nonsingular, Q = V ˜ QV −1, and this decomposition depends on the choice of the bases{ v i } m

1, that is, onV

Now, supposeN α andN β are two subspaces of the same dimensionm − r (1
r

m −1) inRm Then any projections Q α and Q β ontoN α andN β can be decomposed

Copyright©2004 Hindawi Publishing Corporation

Advances in Di fference Equations 2004:3 (2004) 195–200

2000 Mathematics Subject Classification: 34A09, 39A10

URL: http://dx.doi.org/10.1155/S1687183904402015

as Q α = V α QV˜ −1

α and Q β = V β QV˜ −1

β , respectively Define an operator connecting two subspacesN αandN 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 β −1= V α V β −1Q 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 (n0), (2.2) whereA n, B n ∈Rm×m,q n ∈Rmare given and rankA n ≡ r (1
r
m −1) for alln0 Let Q n be any projection onto KerA n, P n = I − Q n and consider decompositionsQ n =

V n QV˜ −1

n (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˜ −1

n are 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,

Rm×m),q ∈Rm, is said to be of index 1 or transferable (see [4]) if there exists a smooth projectionQ ∈ C1(J,Rm×m) onto KerA(t) such that the matrix G(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 ∈Rm: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

where, as in the DAE case, S n:= { ξ ∈Rm:B n ξ ∈ImA n }

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 n V −1

n−1S n ∩KerA 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−1QV˜ −1

n and ¯ Q n−1,n:= V¯n−1Q ¯˜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 2.2confirms that it suffices to restrict our consideration to orthogonal pro-jections onto KerA n, 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) rankA n = r;

(ii)G n:= A n+B n Q 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,{ 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 andQ 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

Q n G −1

n B n x n = Q −1

Further, denotingu n = P n−1x n, v n = Q n−1x n(n0) and observing thatP n G −1

n B n Q n−1x n =

P n G −1

n B n Q n−1,n Q n,n−1x n = P n Q n,n−1x n = P n Q n Q n,n−1x 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

SinceQ n G −1

n B n Q n−1x n = Q n G −1

n B n Q n−1,n Q n,n−1x n = Q n,n−1x n = V n V n− −11Q n−1x n = V n V n− −11v n,

(2.5) is reduced to

v n = V n−1V n −1

Q n G − n1q n − Q n G − n1B n u n



Finally,

x n = u n+v n =
I − Q n−1,n G − n1B n



u n+Q n−1,n G − n1q n (2.8) Thus, if (2.2) is of index 1, then, for givenu0= P −1x0= P0x0, we can computeu n+1,

v n, and x n(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 −1B0P0)x0= Q0G −1q0

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 ∈ C1(X, Y ) is a local homeomorphism between two Banach spaces X, Y and ζ(R) : =infx
R([F (x)] −1)−1 Then if∞

0 ζ(R)dR =+∞ , F is a (global) homeomorphism of X into Y

In particular, if [F (x)] −1
α  x +β for all x ∈ X, where α0, β > 0, then F

is a homeomorphism ofX into Y Further, suppose F = T + H, where T ∈ C1(X, Y ),

[T (x)] −1
γ, for all x ∈ X, and  H(x) − H(y) 
L  x − y , for allx, y ∈ X, then if

Lγ < 1, F is a homeomorphism of X into Y

Consider a system of nonlinear IDEs

f n

x n+1, x n

where f n:Rm →Rmare given vector functions

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

(i) the function f nis continuously differentiable, moreover, Ker(∂ fn /∂y)(y, x) = N n,

dimN n = m − r, for all n0,y, x ∈Rm, where 1
r
m −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 byQ n−1,nan operator con-necting two subspacesN n−1,N n.

In the remainder of this paper, for the sake of simplicity, the norm ofRmis assumed

to be Euclidean

Theorem 3.3 Let ( 3.1) be of index 1 Moreover, suppose that

G −1

n (y, x)
α n  y +β n  x +γ n ∀ y, x ∈Rm,∀ n0, (3.2)

where α n, β n0, γ n > 0 are constants Then the problem of finding x n from (3.1) and the initial condition

has a unique solution.

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−1x n+Q n−1x n



Supposeu n = P n−1x n (n0) is found (forn =0,u0= P −1x0= P0x0= p0 is given) We have to findu = P n x n+1 ∈ImP n ⊂Rr andv = Q n−1x n ∈ImQ n−1⊂Rm−r Define an op-erator F :Rm →Rm 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 + (∂ fn /∂x)(u, u n+v) ∆v.

Consider the linearized equation

whereq ∈Rm is an arbitrary fixed vector First, observe thatG 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, hence G −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, where G 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 ofG −1

n on both sides of (3.6) and using the last observations, we get

∆u + Qn,n−1∆v = G − n1

u, u n+v

Now, applyingP nandQ nto 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 The last equality leads to ∆v = V n−1V −1

n Q n G −1

n q Thus, (3.6) has a unique solution w =(∆uT,∆vT)T Moreover,  ∆u 
 P n  G −1

n  q  and  ∆v 

 V n−1V −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 = √2 nmax{ α n,β n },δ n = ρ n(γ n+β n  u n ), and

ρ n =( P n 2+ V n−1V −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−1x n for all n0 This completes the proof of

In the next theorem, without loss of generality, we will use orthogonal projections ontoN n, that is, Q n = V n QV˜ T andV n V T = V T V n = I In this case, Q n−1,n = V n−1QV˜ T

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 di fferentiable, moreover

Ker∂g n

∂y(y, x) = N n, dimN n = m − r, ∀ n0,∀ x, y ∈Rm; (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 −1

n (y, x) 
γ n for all n0, y, x ∈Rm ;

(iii)h n( y, x) = h n( P n y, x) for all n0, y, x ∈Rm ;

(iv) h n( y, x) − h n( ¯ y, ¯x) 
L n(  y − ¯y 2+ x − ¯x 2)1/2 for all n0, y, x, ¯y, ¯x ∈Rm Then, if γ n L n < 1/ √

2 for all n0, the IVP ( 3.1), (3.3) has a unique solution.

Proof Using the notations ofTheorem 3.3, we define two operatorsT(z) = g n( u, u n+v)

andH(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−1x n, and

u n:= P n−1x n From the proof ofTheorem 3.3, it follows that[T (z)] −1
√2 n On the other hand,H(z) is Lipschitz continuous with a Lipschitz constant L nand√

2 n L n < 1.

Thus, the mappingF(z) = T(z) + H(z) is a homeomorphism of X onto Y , therefore the

Corollary 3.5 Suppose f n( y, x) = A n y + B n x + h n( y, x), where A n, B n ∈Rm×m , and h n:

Rm ×Rm →Rm satisfy the following conditions:

(i) rankA n ≡ r and the matrix G n = A n+B n Q n−1,n is nonsingular for all n0, where

Q n−1,n is a connecting operator of KerA n−1and KerA n , A −1:= A0;

(ii)h n( y, x) is continuously differentiable, moreover

KerA n ⊂Ker∂ f n

∂y(y, x) ∀ n0,∀ y, x ∈Rm,

h n( y, x) − h n( ¯ y, ¯x)
L n

 y − ¯y 2+ x − ¯x 2  1/2

, ∀ n0,∀ y, x, ¯y, ¯x ∈Rm

(3.9)

Then, if L n  G −1

n  < 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

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 Anh and L 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, and P K Anh, On linear implicit non-autonomous systems of di fference equations, J Difference Equ Appl 8 (2002), no 12, 1085–1105.

[4] R M¨arz, On linear di fferential-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: Department of Applied Mathematics, Hanoi University of Technology, 1 Dai Co Viet, 10000 Hanoi, Vietnam

E-mail address:hangocyen02@yahoo.com