Hindawi Publishing Corporation Advances in Difference Equations Volume 2008, Article ID 316207,20 pages doi:10.1155/2008/316207

Research Article

Semilinear Evolution Equations of Second Order via Maximal Regularity

Claudio Cuevas1 and Carlos Lizama2

1 Departamento de Matemática, Universidade Federal de Pernambuco, Avenue Prof. Luiz Freire, S/N, Recife, 50540-740 PE, Brazil

2 Departamento de Matematica, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago, Chile

Correspondence should be addressed to Carlos Lizama, clizama@usach.cl

Received 26 October 2007; Revised 23 January 2008; Accepted 4 February 2008

Recommended by Alberto Cabada

This paper deals with the existence and stability of solutions for semilinear second-order evolution equations on Banach spaces by using recent characterizations of discrete maximal regularity.

Copyright © 2008 C. Cuevas and C. Lizama. 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.

1. Introduction

Let A be a bounded linear operator defined on a complex Banach space X. In this article, we are concerned with the study of existence of bounded solutions and stability for the semilinear problem

by means of the knowledge of maximal regularity properties for the vector-valued discrete time evolution equation

with initial conditions x0 = 0 and x1 = 0.

The theory of dynamical systems described by the difference equations has attracted a good deal of interest in the last decade due to the various applications of their qualitative properties; see [1-5].

In this paper, we prove a very general theorem on the existence of bounded solutions for the semilinear problem (1.1) on lp(Z+; X) spaces. The general framework for the proof of this statement uses a new approach based on discrete maximal regularity.

a2xn - Axn — f (n, xn, Axn), n e Z+,

A2xn Axn — fn, n e Z+,

In the continuous case, it is well known that the study of maximal regularity is very useful for treating semilinear and quasilinear problems (see, e.g., Amann [6], Denk et al. [7], Clement et al. [8], the survey by Arendt [9], and the bibliography therein). Maximal regularity has also been studied in the finite difference setting. Blunck considered in [10, 11] maximal regularity for linear difference equations of first order; see also Portal [12, 13]. In [14], maximal regularity on discrete Holder spaces for finite difference operators subject to Dirichlet boundary conditions in one and two dimensions is proved. Furthermore, the authors investigated maximal regularity in discrete Holder spaces for the Crank-Nicolson scheme. In [15], maximal regularity for linear parabolic difference equations is treated, whereas in [16] a characterization in terms of ^-boundedness properties of the resolvent operator for linear second-order difference equations was given; see also the recent paper by Kalton and Portal [17], where they discussed maximal regularity of power-bounded operators and relate the discrete to the continuous time problem for analytic semigroups. However, for nonlinear discrete time evolution equations like (1.1), this new approach appears not to be considered in the literature.

The paper is organized as follows. Section 2 provides an explanation for the basic notations and definitions to be used in the article. In Section 3, we prove the existence of bounded solutions whose second discrete derivative is in lp (1 < p < +<x>) for the semilinear problem

(1.1) by using maximal regularity and a contraction principle. We also get some a priori estimates for the solutions xn and their discrete derivatives axn and a2xn. Such estimates will follow from the discrete Gronwall inequality [1] (see also [18,19]). In Section 4, we give a criterion for stability of (1.1). Finally, in Section 5 we deal with local perturbations of the system

(1.2).

2. Discrete maximal regularity

Let X be a Banach space. Let Z+ denote the set of nonnegative integer numbers and let a be the forward difference operator of the first order, that is, for each x : Z+ ^ X and n e Z+, axn =

Xn+1 xn.

We consider the second-order difference equation

A2xn - (I - T)xn = fn, Vn e Z+,

x0 = x, ax0 = xi - x0 = y,

where T e B(X), A2xn = a(axn), and f : Z+ ^ X.

Denote C(0) = I, the identity operator on X, and define

[n/2] / n \

C(n) = Z Lj (I - T )k, for n = 1,2..........(2.2)

and C(n) = C(-n), for n = -1, -2,.... We define also S(0) = 0,

[(n-1)/2] / n \

S(n)= g (2k + J (I - T)k, (2 3)

for n = 1,2,..., and S(n) = -S(-n), for n = -1, -2,....

Considering the above notations, it was proved in [16] that the (unique) solution of (2.1) is given by

xm+i = C(m)x + S(m)y + (S*f)m. (2.4)

Moreover,

axm+i = (I - T)S(m)x + C(m)y + (C*f )m. (2.5)

The following definition is the natural extension of the concept of maximal regularity from the continuous case (cf., [16]).

Definition 2.1. Let 1 < p < +<x>. One says that an operator T e B(X) has discrete maximal regularity if KTf := 2U=1(I - T)S(k)fn-k defines a bounded operator KT e B(lp(Z+,X)).

As a consequence of the definition, if T e B(X) has discrete maximal regularity, then T has discrete lp-maximal regularity, that is, for each (fn) e lp(Z+; X) we have (a2xn) e lp(Z+; X), where (xn) is the solution of the equation

a2xn - (I - T)xn = fn, Vn e Z+, xo = 0, x1 = 0. (2.6)

Moreover,

a2xn = £(I - T )S(k) fn-1-k + fn.

We introduce the means

||(x1,...,xn)|K := X

j e{-1,11'

for xi,...,xn e X.

Definition 2.2. Let X and Y be Banach spaces. A subset T of B(X, Y) is called ^-bounded if there exists a constant c > 0 such that

||(T1x1,...,Tnxn)|K < c||(x1,...,xn)||R (2.9)

for all T1,...,Tn e T, x1,...,xn e X, n e N. The least c such that (2.9) is satisfied is called the R-bound of T and is denoted by R(T).

An equivalent definition using the Rademacher functions can be found in [7]. We note that R-boundedness clearly implies boundedness. If X = Y, the notion of R-boundedness is strictly stronger than boundedness unless the underlying space is isomorphic to a Hilbert space [20, Proposition 1.17]. Some useful criteria for R-boundedness are provided in [7,20, 21].

Remark 2.3. (a) Let S, T c B(X,Y) be R-bounded sets, then S + T := {S + T : S e S,T el) is R-bounded.

(b) Let T c B(X,Y) and S c B(Y,Z) be R-bounded sets, then S-T := {S-T : S e S, T eT) c B(X,Z) is R-bounded and

R(S ■ T) < R(S) ■ R(T). (2.10)

(c) Also, each subset M c B(X) of the form M = {XI : 1 e q) is R-bounded whenever q c C is bounded. This follows from Kahane's contraction principle (see [20, 22] or [7]).

A Banach space X is said to be UMD if the Hilbert transform is bounded on Lp(R, X) for some (and then all) p e (1, to). Here, the Hilbert transform H of a function f e S(R,X), the Schwartz space of rapidly decreasing X-valued functions, is defined by

Hf := n PV( t) (Z11)

These spaces are also called HT spaces. It is a well-known theorem that the set of Banach spaces of class HT coincides with the class of UMD spaces. This has been shown by Bourgain [23] and Burkholder [24].

Recall that T e B(X) is called analytic if the set

{n(T -1)Tn : n e N} (2.12)

is bounded. For recent and related results on analytic operators we refer the reader to [25]. The characterization of discrete maximal regularity for second-order difference equations by R-boundedness properties of the resolvent operator T reads as follows (see [16]).

Theorem 2.4. Let X be a UMD space and let T e B(X) be analytic. Then, the following assertions are equivalent.

(i) T has discrete maximal regularity of order 2.

(ii) {(1 - 1)2R((X - 1)2,I - T) : |X| = 1,1 /1} is R-bounded.

Observe that from the point of view of applications, the above-given characterization provides a workable criterion; see Section 4 below. We remark that the concept of R-boundedness plays a fundamental role in recent works by Clement-Da Prato [26], Clement et al. [22], Weis [27,28], Arendt-Bu [20, 29], and Keyantuo-Lizama [30-32].

3. Semilinear second-order evolution equations

In this section, our aim is to investigate the existence of bounded solutions, whose second discrete derivative is in ip for semilinear evolution equations via discrete maximal regularity. Next, we consider the following second-order evolution equation:

a2xn - Axn = f(n,xn, Axn), n e Z+, x0 = 0, xi = 0, (3.1)

which is equivalent to

xn+2 - 2xn+T + Txn = f(n,xn, Axn), Vn e Z+, x0 = 0, xT = 0, (3.2)

where T := I - A.

To establish the next result, we need to introduce the following assumption. Assumption 3.1. Suppose that the following conditions hold.

(i) The function f : Z+ x X x X ^ X satisfy the Lipschitz condition on X x X, that is

for all z,w e X x X and n e Z+, we get ||f (n,z) - f (n,w)\\X < an\\z - w\\XxX, where a := (an) e h(Z+).

(ii) f (,0,0) e k(Z+,X).

We remark that the condition a e h (Z) in (i) is satisfied quite often in applications. For example, it appears when we study asymptotic behavior of discrete Volterra systems which describe processes whose current state is determined by their entire history. These processes are encountered in models of materials with memory, in various problems of heredity or epidemics, in theory of viscoelasticity, and in solving optimal control problems (see, e.g., [33,34]). We began with the following property which will be useful in the proof of our main

result.

Lemma 3.2. Let (an) be a sequence of positive real numbers. For all n,l e Z+, one has

m=0 \j=0 / + a \j=0

n-1 / m-1 \ 1 / n-1

^a^^a^ < ¿ + 1(2aj) . (3.3)

Proof. Putting Am := Xm=01aj, we obtain

j=0 ^ j'

(l + 1) A m+1 - Am Alm =

A m+1 Am (,Am + Am Am + ''' + AmAm

1 + Alm

< (Am+1 - Am) (Am+1 + Alm+1Am + ••• + Am+1 Altn1 + A^) (3.4)

= Al+1 Al+1 = Am+1 - Am .

Hence,

n 1 n 1

(l + 1)£ (Am+1 - Am)Alm < £ (A^+1 - A+1) = Al+. (3.5)

m=0 m=0

Denote by W0 the Banach space of all sequences V = (Vn) belonging to l^(Z+,X) such that Vo = V1 = 0 and a2V e lp(Z+,X) equipped with the norm |||V||| = \VW^ + \a2V\\p. We will say that T e B(X) is S-bounded if S e l^(Z+;X). With the above notations, we have the following main result.

Theorem 3.3. Assume that Assumption 3.1 holds. In addition, suppose that T is S-bounded and that it has discrete maximal regularity. Then, there is a unique bounded solution x = (xn) of (3.1) such that (a2xn) e lp (Z+, X). Moreover, one has the following a priori estimates for the solution:

sup[||xn||X + ||axn||X] < 3M||f(■,0,0)||1e3MWl, neZ+ (3.6)

||a2x||p < C^f(■,0,0)||1e6MWaWl, 1 <p< +<x>,

where M := supneZ+ \S(n)\ and C > 0.

Proof. Let V be a sequence in W0 . Then, using Assumption 3.1 we obtain that the function g := f (-,V, AV) is in lp(Z+,X). In fact, we have

Wgfp = £ ||f(n,V„, aV„) ||x

< X(llf (n,Vn, aV„) -f(n,0,0)||x + ||f(n,0,0)||x)p

< 2p£ ||f (n, Vn, aVn) - f (n, 0,0) ||X + 2^ ||f (n, 0,0) ||

n=0 n=0

< 2^«n ^V^ AVn) HXxx + ||f (n, o, 0) ||

2||f (n, 0,0)|X =2 ||f (n, 0,0)|X_1|f (n, 0,0)|

n=0 n=0

< ||f(■,0,0)|OT^||f(n,0,0)||x

= ||f (■, 0,0)||0-1||f (■, 0,0)||i.

Analogously, we have

an < lailOoNla|

On the other hand,

II (V- AVn) llxxx = IVnllx + IVn+1 - Vn|lx < 21 Vn|lx + I\Vn+4X < 31 Vll

Hence,

(3.10)

< 6p|VfoXan + 2p||f(■,0,0)||0-1||f(■,0,0)|i

< 6pWVWOWaWO 1WaWi + 2p||f(■,0,0)if(v0,0)||

proving that g e lp (Z+,x).

Since T has discrete maximal regularity, the Cauchy problem

zn+2 — 2zn+1 + Tzn = gn,

Z0 = zi = 0

(3.11)

has a unique solution (zn) such that (a2zn) e lp (Z+,X), which is given by

zn = [XV ]„ = <

if n = 0,1,

£S(k)f (n — 1 — k, Vn—i—k, AVn—i—if n > 2.

(3.13)

We now show that the operator X : Wqp ^ wQ,p has a unique fixed point. To verify that X is well defined, we have only to show that XV e lU(Z+,X). In fact, we use Assumption 3.1 as above and M := swpneZ+ ||S(n)|| to obtain

£S(k)f (n — 1 — k, Vn—1—k, AVn—1—k) k=1

< M^\\f(n — 1 — k, Vn—1—k, AVn—1—k) — f (n — 1 — k,0,0) ||X + M^ ||f (n — 1 — k,0,0) ||X

< M^>n—1— k |KVn—1—k, aVn—1—k) |XXX + M^ ||f (j, 0,0) ||

n—2 n—2

< 3M||V||„£a, + M^||f(j,0,0)||x

j=0 j=0

< M[3||V|U|a||1 + |f(■,0,0)^].

It proves that the space Wq is invariant under X.

(3.14)

Let V and V be in Wq . In view of Assumption 3.1 (i) and M < u, we have initially as in

(3.14)

II [XV]n — [XV]

£S(k)(f (n — 1 — k,Vn—1—k, aVn—1—k) — fin — 1 — k, Vn—1—k, aVn—1—k))

< ^an—1—k|| ((V — V)n—1—k, a(V — V)n—1—k

(3.15)

n—1—k XXX

k=1 n—2

Hence, we obtain

= M^a, || ((V — V),, a(V — Vj ||xxx < 3M||a||11| V — V||„.

||XV — XV|u< 3M|a|1 HI V — V|||. On the other hand, using the fact that S(1) = I, we observe first that

(3.16)

a[XV]n = f (n — 1, Vn—1, aVn—1) + £ (S(k + 1) — S(k)) f (n — 1 — k, Vn—1—k, aVn—1—k), n > 1.

Since S(2) = 2I,we get a2[KV]n = f 0n,Vn, AVn) - f (n - 1,V„_1, AVn-1) + (S(2) -1)f (n - 1,Vn-1, AVn-1)

+ £ (S(k + 2) - 2S(k + 1)+ S(k)) f(n - 1 - k, Vn-1-k, AVn-1-k) k=1

= f(n, Vn, AVn) + £ (S(k + 2) - 2S(k + 1) + TS(k)) f(n - 1 - k, Vn-1-k, AVn-1-k) k=1

+ £(I - T)S(k)f(n - 1 - k, Vn-1-k, aVn-1-k).

T , TT k, AVn-1-k)

(3.18)

Taking into account that zn+1 = (S*g)n is solution of (3.12), we get the following identity:

2 (S(k + 2) - 2S(k + 1)+ TS(k)) f(n - 1 - k, Vn-1-k, AVn-1-k) = 0. (3.19)

Using (3.19), we obtain for n > 1

a2[KV]n = f(n,Vn, aVn) +£(I - T)S(k)f(n - 1 - k,Vn-1-k, aVn-1-O, (3.20)

whence, for n > 1,

a2[KV]n - a2[KV]n

= f (n, Vn, AVn) - f (n, Vn, AVn)

+ £(I - T)S(k)(J(n - 1 - k,Vn-1-k, aVn-1-k) - f(n - 1 - k,Vn-1-k, aVn-1-k)). k=1

(3.21)

Furthermore, using the fact that a2[KV]0 = f (0,0,0), the above identity, and then Minkowskii's inequality, we get

||a2KV - a2KV||p

= ( ||f (0,0,0) - f (0,0,0)||X + £||a2[KV]n - a2[KV]n!px '

£ Hf(n, Vn, AVn) - f (n, Vn, AVn) ||X

2(I - T)S(k)(f(n-1-k,Vn-1-k, AVn-1-0 -f(n-1-k,Vn-1-k, AVVn-1-^)

Since KT is bounded on lp (Z+,X), using Assumption 3.1, we obtain

|a2KV - a2KV|i < (1 +||Kr|

< (i + 1 |Kt|

£11/(n, Vn, AVn) - f (n, Vn, AVn) IIX

ZalU((V - V)n, A(V - V)„) IIXxX

(3.23)

(3.24)

< 3(1 + ||Kt II) Ml||V - V^n^-

Hence, we obtain from (3.16) and (3.23)

||IKV - XVIII = ||kv -kV|L + II a2kv - a2xV||p

< 3M||a||i|||V - V||| + 3(1 + ||Kt||)||a||i|||V -= 3(M + 1 + ||Kr||)||«||i|||V - V|||

= amv - %

where a := 3M||a||1 and b := 1 + (1 + ||KT||)M-1.

Next, we consider the iterates of the operator K. Let V and V be in WgP. Taking into account that S(1) = I, S(0) = 0, and Vo = V1 = Vo = V^l = 0, we observe first that for n > 2

a[KV]n - a[KVIn

= £ (J(k + 1) - S(k)) f (n - 1 - k, Vn-1-k, aVn-1-0 - f(n - 1 - k, Vn-1-k, AVn-1-k)) k=0

= 2 (S(n - k) -S(n - k - 1)) f(k, Vk, AVk) - f(k, Vk, aVk)), k=1

whence

(3.25)

a[KV]n - a[KV]n||x < 2M^II f (k, Vk, AVk) - f (k, Vkt aVk) |

< 2M^>k IK(V - V)^ A(V - V)k) IIxxx-

(3.26)

On the other hand, from (3.15) we get

|| [KV]n - [KV]J|x < M]?ak|K(V - V)k, A(V - V)k) ||xxx- (3.27)

Using estimates (3.26) and (3.27), we obtain for n > 2

IK [KV - KV ]n, a[KV - KV ] n) IIxxx < 3^«k IK (V - v )k, a(V - V )k) IIxxx- (3.28)

Next, using [XV]0 = [XV]1 = 0 and estimates (3.28) and (3.10), we obtain

||[K2V]„ - [K2V]JX < M^\\f(j, [XV ]j, a[XV];) - fj, [XV], a[xv];) ||x

j=0 n-2

< M^a, \ ([XV - XVI],, a[XV - XVI],) \

U IIXxX

n-1 /1-1

< 3M22 aA^aK (V - V )i, a(V - V )£) ||

=1 \ i=i

(3.29)

< 2(3M)2 £aA ||V - V||„.

Since [X2V]0 = [X2V]1 = 0, we get

||X2V-X2VH^ < 2(3Myay^2

(3.30)

Furthermore, using the identity

a2[X2V]n - ^[^n

= f (n, [XV]n, a[XV]n)-f (n, [XV]n, a[XV]n)

+ £(I -T )S(k) fn-1-k, [XV ]n-1-k,A[XV ]n-1-^-f(n-1 - k, [XV ]n-1-k, a[XV L+0), k=1

(3.31)

the fact that a2[X2V]0 = f (0,0,0) for all V e W20p, and Lemma 3.2, we obtain

||a2X2V - a2X2V|L

(||a2[X2V]0 - a2[X2V]0||X + j||a2[X2V]n - a2[X2V]n||X)

< (1 + ||Kt||)

< (1 + ||Kt||)

< 3M(1 + ||Kt||)

< 32M(1 + ||Kt||)

X ||f (n [XV] n, A [XV] n) - f (n [XV n, A [XV n) ||

|>pn ||([XV -XV]n, a[XV - XVV]n) ||

(3.32)

re / n-1

Zapn(]?ak||([V - V]k, a[V - Vk ||

n=1 \k=1

' re /n-1 \ P

ZtilZak) ||V - V||

n=0 \k=0 / 2

< 32M(1 + ||Kt||) a}) ||V - V||e

C. Cuevas and C. Lizama 11 whence

||a2K2V - a2K2v||p < 2(3My«yo^1 + IKr ID Mr1 in V - V|||. (3.33) From estimates (3.30) and (3.33), we get

|||K2V-K2V|||< ba2|||V - V|||, (3.34)

with a and b defined as above. Taking into account (3.26), (3.28), (3.29), and (3.10), we can infer that

3 / '--1 ^2

IK [K2V - K2V] a [K2V - K2V].) |XxX < 2(3M)2 2aA | V - VH®. (3.35)

Next, using estimate (3.35) and Lemma 3.2, we get

3V n - [K Vnllx < MI> IK K2V - K V A [K2V - K V,) IIxxx

1 n-1 / j-1 \

< 1 (3M)^ aA^aA

2 j=0 V=1 /

1 n-i tr1 \2

MIX ) IV - VU® (3.36)

Hence,

i® - 6 Using (3.35), we get

< 6 (3M)3(£ a,) IV - vy„.

~ 1 i ~ ||K3V-K3^ < -(3MyayO^|IV - V\\\. (3.37)

a2K3V - a2K3V|I < (1 + I|Kr|

H( [K2 V - K2 V] „, a [K2 V - K2 V] J IXxx

whence

< 3(3M)2(1 +IIKrII) ^ ja^j IIV - VIIe

|a2K3V - a2K3\/Ip < g(3Myay0^1 + IKrI)M^V - V|||. (3.39)

(3.38)

From estimates (3.37) and (3.39), we get

|||K3V -K3V||| < -a31|| V - V|||. (3.40)

An induction argument shows us that

|||KnV -Kn

< b^V -

n! 111

(3.41)

Since ban/n! < 1 for n sufficiently large, by the fixed point iteration method K has a unique

fixed point V e W0 . Let V be the unique fixed point of K, then by Assumption 3.1 we have

£S(k)f (n - 1 - k, Vn-1-k, AVn-1-k)

< \\f(k,Vk, aVk) - f (k, 0,0)\X + 11/(k, 0,0)\\,

k=0 k=0

< MI>\KVk, AVk) \\XxX + M\\f (v0,0) \\1,

(3.42)

hence,

\ \ Vn \ \ X < M\\f (', 0, 0)\1 + ME«k\\(AVk) \\xxX'

(3.43)

On the other hand, we have

2 (S(k + 1) - S(k)) f(n - 1 - k, Vn-1-k, aVn-1-k) k=1

n-1 n-1

< 2M£«k \ KVk, AVk) \\xxX + 2ME \ \f (k, 0, °) \ \ X,

(3.44)

\\ aVn\X < 2M\\f (■,0,0)Hx + 2M^«k\\ (Vk, aVk)\\

From (3.43) and (3.45), we get

\\ (Vn, AVn) \\xxX < 3M\\f (■, 0,0) \ \ j + 3M^>k \ (Vk, aVk

(3.45)

(3.46)

T hen, by application of the discrete Gronwall inequality [1, Corollary 4.12, page 183], we get

\\ ( Vn, AVn) \\ XxX < 3M\\f (', 0, 0) W^C1 + 3M«j)

< 3M\\f (■,0,0)\^e3

= 3M\\f (■, 0,0)\1e

„3M«i

(3.47)

< 3M\\f (•,0,0)|^

sup [|| (Vn, aV„)||XxX] < 3M||/(■, 0, OUi^M1. (3.48)

Finally, by (3.20) we obtain

a2Vn = /(n,Vn, aVn) + £(I — T)S(k)/(n — 1 — k,Vn—i—k, AVn—i—k). (3.49)

Hence, using the fact that a2V0 = /(0,0,0) and proceeding analogously as in (3.23), we get

|a2Vniix;

OTO S 1

/ (0,0,0)||X + £ |a2Vnfx)

< ||/(0,0,0)||x + ( E|a2Vnfx'

/ to \1/p / to \1/p

< ||/(0,0,0)||x + ( £||/(n,Vn, aVn) fx) + ||Kr || ( £||/(n,Vn, AVn)

\n=1 / \n=1

/ to \1/p / to \1/p

< 2(£||/(n,Vn, aVn)||x ) + UKr||m/(n,Vn, aVn)||x

\n=0 / \n=0

< (2 +||Kr||)£||/(n,Vn, AVn)||x,

(3.50)

(3.51)

where, by Assumption 3.1 and (3.48),

YU/inVn, AVn) ||x < Xak || (Vk, a Vk) ||xxx + ||/(', 0, °) || 1

n=0 k=0

< 3MH«H1||/(■,0,0)||1e3M|a"1 + ||/(■,0,0)||1 <||/(■, 0,0)||/MM1.

This ends the proof of the theorem. □

In view of Theorem 2.4, we obtain the following result valid on UMD spaces.

Corollary 3.4. Let x be a UMD space. Assume that Assumption 3.1 holds and suppose T e B(x) is an analytic S-bounded operator such that the set {(A — 1)2R((X — 1)2,I — T) : |X| = 1, X/1} is R-bounded. Then, there is a unique bounded solution x = (xn) of (3.1) such that (a2xn) e lp(Z+,x). Moreover, the a priori estimates (3.6) hold.

Example 3.5. Consider the semilinear problem

a2xn — (I — T)xn = qn/(xn), n e Z+, x0 = x1 = 0, (3.52)

where f is defined and satisfies a Lipschitz condition with constant L on a Hilbert space H. In addition, suppose (qn) e l1(Z+). Then, Assumption 3.1 is satisfied. In our case, applying the preceding result, we obtain that if T e B(H) is an analytic S-bounded operator such that the set {(A - 1)2R((1 - 1)2,1 - T) : |A| = 1,1 /1} is bounded, then there exists a unique bounded solution x = (xn) of (3.52) such that (a2xn) e lp(Z+,H). Moreover,

max{sup[||x„||H + ||axn||H], ||a2x|| 1 < C\\f (0)\H||q||1e6LMNl1. (3.53)

UeZ+ J

In particular, taking T = I the identity operator, we obtain the following scalar result which complements those in the work of Drozdowicz and Popenda [2].

Corollary 3.6. Suppose f is defined and satisfies a Lipschitz condition with constant L on a Hilbert space H. Let (qn) e h (Z+, H), then the equation

a2x„ = qnf (xn) (3.54)

has a unique bounded solution x = (xn) such that (a2xn) e lp(Z+,H) and (3.53) holds.

We remark that the above result holds in the finite dimensional case where it is new and covers a wide range of difference equations.

4. A criterion for stability

The following result provides a new criterion to verify the stability of discrete semilinear systems. Note that the characterization of maximal regularity is the key to give conditions based only on the data of a given system.

Theorem 4.1. Let X be a UMD space. Assume that Assumption 3.1 holds and suppose T e B(X) is analytic and 1 e p(T). Then, the system (3.1) is stable, that is the solution (xn) of (3.1) is such that xn — 0 as n —> to.

Proof. It is assumed that T is analytic (which implies that the spectrum is contained in the unit disc and the point 1, see [10]) and that 1 is not in the spectrum, then in view of [27, Proposition 3.6] , the set

{(X - 1)2R((X - 1)2, I - T) : |X| = 1, 1 = 1} (4.1)

is R-bounded, because (X - 1)2R((X - 1)2, I - T) is an analytic function in a neighborhood of the circle. The S-boundedness assumption of the operator T follows from maximal regularity and the fact that I - T is invertible. In fact, we get the following estimate:

sup\\S(n)\\ < \\(I - TniH^T

By Corollary 3.4, there exists a unique bounded solution xn of (3.1) such that (a2xn) e Zp(Z+,X). Then, a2xn ^ 0 as n ^ to. Next, observe that Assumption 3.1 and estimate (3.10) imply

\\f(n,xn, axn) ||X < \\f(n,xn, Axn) - f (n,0,0)||X + ||f (n, 0,0)\\X

< iXfi^iXn, axn) Wxxx + ||f (n, 0,0)||

< ansupnez+ \Kxn, Axn) \\xxX + || f (n, 0,0)||

< 3«nWxW„ + \\f(n,0,0)Wx.

Since (f (■,0,0)) e Zi(Z+,X) and (an) e Zi(Z+),we obtainthatf (n,xn, Axn) ^ 0as n ^to. Then, the result follows from the fact that 1 e p(T) and (3.1). □

From the point of view of applications, we specialize to Hilbert spaces. The following corollary provides easy-to-check conditions for stability.

Corollary 4.2. Let H be a Hilbert space. Let T e B(H) such that ||T|| < 1. Suppose that Assumption 3.1 holds in H. Then, the system (3.1) is stable.

Proof. First, we note that each Hilbert space is UMD, and then the concept of R-boundedness and boundedness coincide; see [7]. Since ||T|| < 1, we get that T is analytic and 1 e p(T). Furthermore, for j1j = 1, 1 = 1, the inequality

||(1 - 1)R (1 - 1)2,I - T)|| =

(X -1)2 j

< j1 - 1j2 (4.4)

<j1 - 2j - ||T||< 1 - ||T|| ( )

1(1 - 2) n=0\ 1(1 - 2),

shows that the set (4.1) is bounded . □

Of course, the same result holds in the finite dimensional case.

5. Local perturbations

In the process of obtaining our next result, we will require the following assumption. Assumption 5.1. The following conditions hold.

(i)* The function f (n,z) is locally Lipschitz with respect to z e X x X; that is for each positive number R, for all n e Z+, and z,w e X x X, ||z||XxX < R, ||w||XxX < R

||f (n,z) - f (n,w)||x < Z(n,R)||z - wUxxx, (5.1)

where i : Z+ x [0, to) ^ [0, to) is a nondecreasing function with respect to the second variable.

(ii)* There is a positive number a such that xTO=oi(n, a) < +to.

(iii)* f (■, 0,0) e i1(Z+,X).

We need to introduce some basic notations. We denote by Wm the Banach space of all sequences V = (Vn) belonging to i00{Z+,X), such that Vn = 0if0 < n < m,and a2V e iv(Z+,X) equipped with the norm ||| • |||. For A > 0, denote by WmP[A] the ball |||V||| < A in main result in this section is the following local version of Theorem 3.3.

Theorem 5.2. Suppose that the following conditions are satisfied.

(a)* Assumption 5.1 holds.

(b)* T is an S-bounded operator and it has discrete maximal regularity.

Then, there are a positive constant m e N and a unique bounded solution x = (xn) of (3.1) for n > m such that xn = 0 if 0 < n < m and the sequence (a2xn) belongs to £p(Z+, X). Moreover, one has

||x|U + l|a2x||p < a, (5.2)

where a is the constant of condition (ii)*.

Proof. Let [ e (0,1/3). Using (iii)* and (ii)*, there are n and n2 in N such that

(M + 2 + ||Kt ||)£||/(j, 0, 0)||x < [a, (5.3)

T := [ + M + 2 + ||Kt||) £ £(j, a) < -, (5.4)

where M := supneZ+ ||S(n)|.

Let V be a sequence in WmP[a/3], with m = max(ni,n2). A short argument similar to (3.7) and involving Assumption 5.1 shows that the sequence

0, if 0 < n < m,

gn := ^ (5.5)

/(n, Vn, AVn), if n>m,

belongs to ip. By the discrete maximal regularity, the Cauchy problem (3.12) with gn defined as in (5.5) has a unique solution (zn) such that (a2zn) e lp (Z+,X), which is given by

Zn = [KV)]n = ^

0, if 0 < n < m,

n-l-m (5.6)

2 S(k)/(n - 1 - k, Vn-1-k, AVn-1-k), if n > m + 1.

— 2 p We will prove that KV belongs to Wm [a/3]. In fact, since

|| (Vj, AVj) ||xxx < 3||V||re < 31|V||| < a, (5.7)

C. Cuevas and C. Lizama we have by Assumption 5.1

II[KV Ux = ME IIfC/V aV=) llx

< MEIIfC/VA= - f (=a°)Wx + MEWf (=0°)lb

j=m j=m

n-2 n-2

< MZlj a) IKV='a= II xxx + ME Wf ='0'0) Wx

< M^l(= a) a + M^ Wf (= 0' 0)Wx

==m = =m

Proceeding in a way similar to (3.20), we get for n > m

A2[XV]n = f(n,Vn, AVn) + 2 (I - T)S(k)f(n - 1 - k'Vn-1-k, AVn-1-0- (5.9)

Hence,

l|a2KV|l =

\\f(m,Vm, AVr

S IIa2[kA:v]nW

< I\f(m,Vm' AVm

2 Wf{n,Vn, aVn) + X (I - T )S(k) f(n - 1 - k'Vn-1-k' aVn-1-0 Wx

< Wf(m,Vm, aVm) IIx + (1 + IIKtW)

< (2 + IIKT IDZWfinVn' AVn) Wx.

XIIf (n'Vn, aVn) II

Therefore, using (5.8) we get

IIA2K^VIIp < (2 +WKtW)

Then, inequalities (5.8) and (5.11) together with (5.3) and (5.4) imply

^l(j,a)a + £Hf=, 0,0) I,

(5.10)

(5.11)

|||KV|||< (M + 2 + WKt W^ £(=' a) a + (M + 2 + WKt W^Wf=' 0' 0) Wx

j=m j=m

<(1 - Aa + ^a = 3 a'

proving that KV belongs to Wm [a/3]. In an essentially similar way to the proof of

Theorem 3.3, for all V and W in Wm [a/3], we prove that

||KV -KWlire < 3M^£(j, a) IV - W|||, (5.13)

IIA2KV - A2KW||p < 3(1 + \\Kt||)£¿(j,a)|||V - W|||, (5.14)

whence

|||KV -KW||| < 3(M + 1 + \\Kt||)2£(j, a) ||| V - W||| = 3(T-p)||| V - W|||. (5.15)

Since 3(T - p) < 1, K is a 3(T - ^-contraction. This completes the proof of the theorem. □ This enables us to prove, as an application, the following corollary.

Corollary 5.3. Let Bi : X x X ^ X, i = 1,2, be two bounded bilinear operators, y e ^(Z+,X), and a,p e £i(Z+,R). In addition, suppose that T is a S-bounded operator and has discrete maximal regularity. Then, there is a unique bounded solution x such that (a2x) e lp(Z+,X) for the equation

xn+2 - 2xn+i + Txn = yn + a„Bi(xn,xn) + p„B2(Axn, Axn). (5.16)

Proof. Take l(n,R) := 2R(\an\ + |pn|)(|B1| + ||B2|). Then, Ere=o^(n, 1) < +re. Note also that f (n, 0,0) = yn belongs to ^1(Z+,X). Hence, Assumption 5.1 is satisfied. □

Remark 5.4. We observe that under the hypotheses of the above local theorem and corollary, the same type of conclusions on stability of solutions proved in Section 4 remains true.

Acknowledgments

The authors would like to thank the referees for the careful reading of the manuscript and their many useful comments and suggestions. The first author is partially supported by CNPq/Brazil under Grant no. 300068/2005-0. The second author is partially financed by Proyecto Anillo ACT-13 and CNPq/Brazil under Grant no. 300702/2007-08.

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