Semilinear Evolution Equations of Second Order via Maximal Regularity

Volume 2008, Article ID 316207,20pages doi:10.1155/2008/316207

Research Article

Semilinear Evolution Equations of Second Order via Maximal Regularity

Claudio Cuevas¹and Carlos Lizama²

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

2Departamento de Matem´atica, 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.

Copyrightq2008 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

LetAbe a bounded linear operator defined on a complex Banach spaceX. In this article, we are concerned with the study of existence of bounded solutions and stability for the semilinear problem

Δ²x_n−Ax_nfn, xn,Δxn, n∈Z, 1.1 by means of the knowledge of maximal regularity properties for the vector-valued discrete time evolution equation

Δ²xn−Axnfn, n∈Z, 1.2 with initial conditionsx₀0 andx₁0.

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; see1–5 .

In this paper, we prove a very general theorem on the existence of bounded solutions for the semilinear problem1.1onlpZ;Xspaces. The general framework for the proof of this statement uses a new approach based on discrete maximal regularity.

In the continuous case, it is well known that the study of maximal regularity is very useful for treating semilinear and quasilinear problemssee, e.g., Amann6 , Denk et al.7 , Cl´ement et al.8 , the survey by Arendt9 , 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 Portal12,13 . In14 , max- imal regularity on discrete H ¨older spaces for finite difference operators subject to Dirichlet boundary conditions in one and two dimensions is proved. Furthermore, the authors inves- tigated maximal regularity in discrete H ¨older spaces for the Crank-Nicolson scheme. In15 , maximal regularity for linear parabolic difference equations is treated, whereas in16 a char- acterization in terms ofR-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 like1.1, this new approach appears not to be considered in the litera- ture.

The paper is organized as follows.Section 2provides an explanation for the basic nota- tions and definitions to be used in the article. InSection 3, we prove the existence of bounded solutions whose second discrete derivative is inl_p1 < p < ∞for the semilinear problem 1.1by using maximal regularity and a contraction principle. We also get some a priori es- timates for the solutionsxnand their discrete derivativesΔxn andΔ²xn. Such estimates will follow from the discrete Gronwall inequality1 see also18,19 . InSection 4, we give a cri- terion for stability of1.1. Finally, inSection 5we deal with local perturbations of the system 1.2.

2. Discrete maximal regularity

LetXbe a Banach space. LetZdenote the set of nonnegative integer numbers and letΔbe the forward difference operator of the first order, that is, for eachx:Z →Xandn∈Z,Δxn xn1−xn.We consider the second-order difference equation

Δ²xn−I−Txnfn, ∀n∈Z,

x₀x, Δx0x₁−x₀y, 2.1

whereT ∈ BX, Δ²x_n ΔΔxn,andf:Z→X.

DenoteC0 I, the identity operator onX, and define

Cn ^n/2

k0

n 2k

I−T^k, forn1,2, . . . , 2.2

andCn C−n, forn−1,−2, . . . .We define alsoS0 0,

Sn

n−1/2

k0

n 2k1

I−T^k, 2.3 forn1,2, . . . ,andSn −S−n,forn−1,−2, . . . .

Considering the above notations, it was proved in16 that theuniquesolution of2.1 is given by

x_m1CmxSmy S∗f_m. 2.4

Moreover,

Δxm1 I−TSmxCmy C∗f_m. 2.5 The following definition is the natural extension of the concept of maximal regularity from the continuous casecf.,16 .

Definition 2.1. Let 1 < p < ∞. One says that an operator T ∈ BXhas discrete maximal regularity ifKTf:_n

k1I−TSkfn−kdefines a bounded operatorKT∈ BlpZ, X.

As a consequence of the definition, ifT ∈ BXhas discrete maximal regularity, thenT has discretelp-maximal regularity, that is, for eachfn∈lpZ;Xwe haveΔ²xn∈lpZ;X, wherexnis the solution of the equation

Δ²x_n−I−Txnf_n, ∀n∈Z, x₀0, x₁0. 2.6

Moreover,

Δ²xnⁿ⁻¹

k1

I−TSkfn−1−kfn. 2.7

We introduce the means

x1, . . . , xn_R: 1 2ⁿ

j∈{−1,1}ⁿ

n j1

jxj

2.8

forx1, . . . , xn∈X.

Definition 2.2. LetXandYbe Banach spaces. A subsetTofBX, Yis calledR-bounded if there exists a constantc≥0 such that

T1x₁, . . . , Tnxn_R≤cx1, . . . , xn_R 2.9 for allT1, . . . , Tn ∈ T, x1, . . . , xn∈X, n∈N.The leastcsuch that2.9is satisfied is called the R-bound ofTand is denoted byRT.

An equivalent definition using the Rademacher functions can be found in7 . We note thatR-boundedness clearly implies boundedness. IfX Y, the notion ofR-boundedness is strictly stronger than boundedness unless the underlying space is isomorphic to a Hilbert space 20, Proposition 1.17 . Some useful criteria forR-boundedness are provided in7,20,21 .

Remark 2.3. aLetS,T ⊂ BX, YbeR-bounded sets, thenST:{ST :S ∈ S, T ∈ T}is R-bounded.

bLetT ⊂ BX, YandS ⊂ BY, ZbeR-bounded sets, thenS · T:{S ·T : S ∈ S, T ∈ T} ⊂ BX, ZisR-bounded and

RS · T≤RS·RT. 2.10

cAlso, each subsetM⊂ BXof the formM{λI :λ ∈Ω}isR-bounded whenever Ω⊂Cis bounded. This follows from Kahane’s contraction principlesee20,22 or7 .

A Banach spaceXis said to be UMD if the Hilbert transform is bounded onL^pR, Xfor someand then allp ∈1,∞.Here, the Hilbert transformH of a functionf ∈ SR, X, the Schwartz space of rapidly decreasingX-valued functions, is defined by

Hf: 1 πPV

1 t

∗f. 2.11

These spaces are also calledHT spaces. It is a well-known theorem that the set of Banach spaces of classHTcoincides with the class of UMD spaces. This has been shown by Bourgain 23 and Burkholder24 .

Recall thatT ∈ BXis called analytic if the set nT−ITⁿ:n∈N

2.12 is bounded. For recent and related results on analytic operators we refer the reader to25 . The characterization of discrete maximal regularity for second-order difference equations by R-boundedness properties of the resolvent operatorT reads as followssee16 .

Theorem 2.4. LetXbe a UMD space and letT ∈ BXbe analytic. Then, the following assertions are equivalent.

iT has discrete maximal regularity of order 2.

ii λ−1²Rλ−1², I−T:|λ|1, λ /1

isR-bounded.

Observe that from the point of view of applications, the above-given characteriza- tion provides a workable criterion; see Section 4 below. We remark that the concept of R- boundedness plays a fundamental role in recent works by Cl´ement-Da Prato26 , Cl´ement et al.22 , Weis27,28 , Arendt-Bu20,29 , and Keyantuo-Lizama30–32 .

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_pfor semilinear evolution equations via discrete maximal regularity.

Next, we consider the following second-order evolution equation:

Δ²xn−Axnf

n, xn,Δxn

, n∈Z, x00, x10, 3.1 which is equivalent to

xn2−2xn1Txnf

n, xn,Δxn

, ∀n∈Z, x00, x10, 3.2 whereT :I−A.

To establish the next result, we need to introduce the following assumption.

Assumption 3.1. Suppose that the following conditions hold.

iThe function f : Z×X×X → X satisfy the Lipschitz condition on X×X, that is for allz, w ∈X×Xandn∈Z, we getfn, z−fn, w_X ≤αnz−wX×X,where α: αn∈l1Z.

iif·,0,0∈l₁Z, X.

We remark that the conditionα∈l1Ziniis 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 epi- demics, in theory of viscoelasticity, and in solving optimal control problemssee, e.g.,33,34 . We began with the following property which will be useful in the proof of our main result.

Lemma 3.2. Letαnbe a sequence of positive real numbers. For alln, l∈Z, one has n−1

m0

αm

_m−1

j0

αj

l

≤ 1 l1

_n−1

j0

αj

l1

. 3.3

Proof. PuttingAm:_m−1

j0 αj, we obtain l1

Am1−Am A^l_m

Am1−Am

A^l_mA^l−1_m Am· · ·AmA^l−1_m A^l_m

≤

Am1−Am

A^l_m1A^l−1_m1Am· · ·Am1A^l−1_m A^l_m A^l1_m1−A^l1_m .

3.4

Hence,

l1ⁿ⁻¹

m0

A_m1−A_m

A^l_m≤ⁿ⁻¹

m0

A^l1_m1−A^l1_m

A^l1_n . 3.5

Denote byW^2,p₀ the Banach space of all sequencesV Vnbelonging tol∞Z, Xsuch thatV0 V1 0 andΔ²V ∈lpZ, Xequipped with the norm|||V||| V∞Δ²Vp.We will say thatT ∈ BXisS-bounded ifS ∈l_∞Z;X.With the above notations, we have the following main result.

Theorem 3.3. Assume thatAssumption 3.1holds. In addition, suppose thatT isS-bounded and that it has discrete maximal regularity. Then, there is a unique bounded solutionx xnof 3.1such that Δ²xn∈lpZ, X. Moreover, one has the following a priori estimates for the solution:

sup

n∈Z

xn_XΔxn_X

≤3Mf·,0,0₁e^3Mα1, Δ²x_p≤Cf·,0,0

1e^6Mα1, 1< p <∞,

3.6 whereM:sup_n∈ZSnandC >0.

Proof. Let V be a sequence inW^2,p₀ . Then, usingAssumption 3.1 we obtain that the function g:f·, V·,ΔV·is inlpZ, X. In fact, we have

g^pp^∞

n0

f

n, V_n,ΔVn^p_X

≤^∞

n0

f

n, Vn,ΔVn

−fn,0,0_Xfn,0,0_Xp

≤2^p ∞ n0

f

n, Vn,ΔVn

−fn,0,0^p_X2^p ∞ n0

fn,0,0^p_X

≤2^p ∞ n0

α^p_nV_n,ΔVn^p_X×X2^p ∞ n0

fn,0,0^p_X,

3.7

where

∞ n0

fn,0,0^p_X^∞

n0

fn,0,0^p−1_X fn,0,0_X

≤f·,0,0^p−1_∞ ^∞

n0

fn,0,0_X f·,0,0^p−1_∞ f·,0,0

1.

3.8

Analogously, we have

∞ n0

α^p_n≤α^p−1_∞ α

1. 3.9

On the other hand,

V_n,ΔVn_X×XV_nXV_n1−V_nX≤2V_nXV_n1X≤3V_∞. 3.10 Hence,

g^pp≤6^pV^p_∞^∞

n0

α^p_n2^pf·,0,0^p−1_∞ f·,0,0

1

≤6^pV^p∞α^p−1∞ α12^pf·,0,0^p−1_∞ f·,0,0₁,

3.11

proving thatg∈lpZ, X.

SinceThas discrete maximal regularity, the Cauchy problem z_n2−2z_n1Tz_ng_n,

z0z10 3.12

has a unique solutionznsuch thatΔ²zn∈lpZ, X, which is given by

zn KV _n

⎧⎪

⎪⎨

⎪⎪

⎩

0, ifn0,1,

n−1 k1

Skf

n−1−k, V_n−1−k,ΔVn−1−k

, ifn≥2. 3.13

We now show that the operatorK:W^2,p₀ → W^2,p₀ has a unique fixed point. To verify thatKis well defined, we have only to show thatKV ∈ l∞Z, X.In fact, we useAssumption 3.1as above andM:sup_n∈ZSnto obtain

n−1

k1

Skf

n−1−k, V_n−1−k,ΔVn−1−k _X

≤Mⁿ⁻¹

k1

f

n−1−k, V_n−1−k,ΔVn−1−k

−fn−1−k,0,0_XMⁿ⁻¹

k1

fn−1−k,0,0_X

≤Mⁿ⁻¹

k1

α_n−1−kV_n−1−k,ΔVn−1−k_X×XMⁿ⁻²

j0

fj,0,0_X

≤3MV∞ n−2

j0

α_jMⁿ⁻²

j0

fj,0,0_X

≤M

3V∞α1f·,0,0₁ .

3.14 It proves that the spaceW^2,p₀ is invariant underK.

LetV andVbe inW^2,p₀ . In view ofAssumption 3.1iandM <∞, we have initially as in 3.14KV _n−KV _nX

n−1 k1

Sk f

n−1−k, Vn−1−k,ΔVn−1−k

−f

n−1−k,Vn−1−k,ΔVn−1−k _X

≤Mⁿ⁻¹

k1

αn−1−kV −V_n−1−k,ΔV −V_{n−1−kX×X}

Mⁿ⁻²

j0

α_jV −V_j,ΔV −V_jX×X ≤3Mα1V −V∞.

3.15

Hence, we obtain

KV − KV∞≤3Mα1V −V. 3.16 On the other hand, using the fact thatS1 I, we observe first that

ΔKV _nf

n−1, Vn−1,ΔVn−1 ⁿ⁻¹

k1

Sk1− Sk f

n−1−k, Vn−1−k,ΔVn−1−k

, n≥1.

3.17

SinceS2 2I, we get Δ²KV _nf

n, V_n,ΔVn

−f

n−1, V_n−1,ΔVn−1

S2−I f

n−1, V_n−1,ΔVn−1

ⁿ⁻¹

k1

Sk2−2Sk1 Skf

n−1−k, Vn−1−k,ΔVn−1−k

f

n, Vn,ΔVn n−1

k1

Sk2−2Sk1 TSk f

n−1−k, Vn−1−k,ΔVn−1−k

ⁿ⁻¹

k1

I−TSkf

n−1−k, Vn−1−k,ΔVn−1−k .

3.18 Taking into account thatzn1 S∗gnis solution of3.12, we get the following identity:

n−1 k1

Sk2−2Sk1 TSk f

n−1−k, Vn−1−k,ΔVn−1−k

0. 3.19

Using3.19, we obtain forn≥1 Δ²KV _nf

n, V_n,ΔVn ⁿ⁻¹

k1

I−TSkf

n−1−k, V_n−1−k,ΔVn−1−k

, 3.20

whence, forn≥1, Δ²KV _n−Δ²KV _n

f

n, V_n,ΔVn

−f

n,V_n,ΔV_n ⁿ⁻¹

k1

I−TSk f

n−1−k, Vn−1−k,ΔVn−1−k

−f

n−1−k,Vn−1−k,ΔVn−1−k .

3.21 Furthermore, using the fact that Δ²KV ₀ f0,0,0, the above identity, and then Minkowskii’s inequality, we get

Δ²KV −Δ²KV_p

f0,0,0−f0,0,0^p_X^∞

n1

Δ²KV _n−Δ²KV _n^p_X1/p

≤ _∞

n1

f

n, Vn,ΔVn−fn,Vn,ΔVn^p_X1/p

_∞

n1

n−1

k1

I−TSk f

n−1−k, Vn−1−k,ΔVn−1−k

−f

n−1−k,Vn−1−k,ΔVn−1−k

p X

1/p

. 3.22

SinceKTis bounded onlpZ, X, usingAssumption 3.1, we obtain Δ²KV −Δ²KV_p≤

1K_T∞

n1

fn, Vn,ΔVn−fn,V_n,ΔV_n^p_X1/p

≤

1KT_∞

n1

α^p_nV −V_n,ΔV −V_n^p_X×X1/p

≤3

1K_Tα1V −V_∞.

3.23

Hence, we obtain from3.16and3.23

KV − KVKV − KV_∞Δ²KV −Δ²KV_p

≤3Mα1V −V3

1KTα1V −V 3

M1K_Tα1V −V abV −V,

3.24

wherea:3Mα1andb:1 1KTM⁻¹.

Next, we consider the iterates of the operatorK. LetV andV be inW^2,p₀ . Taking into account thatS1 I,S0 0,andV0V1V0V10, we observe first that forn≥2

ΔKV _n−ΔKV _n ⁿ⁻¹

k0

Sk1− Sk

fn−1−k, Vn−1−k,ΔVn−1−k

−f

n−1−k,Vn−1−k,ΔVn−1−k

ⁿ⁻¹

k1

Sn−k− Sn−k−1 f

k, Vk,ΔVk−f

k,Vk,ΔVk ,

3.25 whence

ΔKV _n−ΔKV _nX≤2Mⁿ⁻¹

k1

f

k, V_k,ΔVk

−f

k,V_k,ΔV_kX

≤2Mⁿ⁻¹

k1

αkV −Vk,ΔV −Vk_X×X.

3.26

On the other hand, from3.15we get KV _n−KV _nX≤Mⁿ⁻²

k1

αkV −V_k,ΔV −V_kX×X. 3.27

Using estimates3.26and3.27, we obtain forn≥2 KV − KV _n,ΔKV − KV _nX×X≤3Mⁿ⁻¹

k1

αkV −V_k,ΔV −V_kX×X. 3.28

Next, usingKV ₀ KV ₁0 and estimates3.28and3.10, we obtain K²V _n−K²V _nX≤Mⁿ⁻²

j0

f

j,KV _j,ΔKV _j

−f

j,KV _j,ΔKV _jX

≤Mⁿ⁻²

j1

αjKV − KV _j,ΔKV − KV _jX×X

≤3M²ⁿ⁻¹

j1

α_{j j−1}

i1

α_iV −V_i,ΔV −V_iX×X

≤1 23M²

_n−1

τ1

ατ

₂

V −V_∞.

3.29

SinceK²V ₀ K²V ₁0, we get

K²V − K²V_∞≤ 1 2

3Mα1

₂V −V. 3.30 Furthermore, using the identity

Δ² K²V

n−Δ² K²V

n

f

n,KV _n,ΔKV _n

−f

n,KV _n,ΔKV _n ⁿ⁻¹

k1

I−TSk f

n−1−k,KV _n−1−k,ΔKV _n−1−k

−f

n−1−k,KV _n−1−k,ΔKV _n−1−k , 3.31 the fact thatΔ²K²V ₀f0,0,0for allV ∈ W^2,p₀ , andLemma 3.2, we obtain

Δ²K²V −Δ²K²V_p Δ²

K²V

0−Δ² K²V

0^p_X^∞

n1

Δ² K²V

n−Δ² K²V

n^p_X1/p

≤

1K_T∞

n1

f n,

KV

n,Δ KV

n

−f n,

KV

n,Δ KV

n^p_X1/p

≤

1KT_∞

n1

α^p_nKV − KV

n,Δ

KV − KV

n^p_X×X1/p

≤3M

1KT_∞

n1

α^p_{n n−1}

k1

αkV −V _k,ΔV −V _kX×Xp1/p

≤3²M

1KT_∞

n0

α^pn

_n−1

k0

αk

p

V −V^p_∞1/p

≤3²M

1KT1 2

_∞

j0

αj

2

V −V_∞,

3.32

whence

Δ²K²V −Δ²K²V_p≤1 2

3Mα1

2

1KTM⁻¹V −V. 3.33 From estimates3.30and3.33, we get

K²V − K²V≤ b

2a²V −V, 3.34

withaandb defined as above. Taking into account3.26,3.28,3.29, and3.10, we can infer that

K²V − K²V

j,Δ

K²V − K²V

j_X×X≤ 3 23M²

_j−1

τ1

ατ

₂

V −V_∞. 3.35

Next, using estimate3.35andLemma 3.2, we get K³V

n− K³V

n_X ≤Mⁿ⁻²

j1

αjK²V − K²V

j,Δ

K²V − K²V

j_X×X

≤1

23M³ⁿ⁻¹

j0

α_{j j−1}

τ1

α_{τ 2}

V −V_∞

≤1 63M³

_n−1

j1

αj

3

V −V_∞.

3.36

Hence,

K³V − K³V_∞≤ 1 6

3Mα13V −V. 3.37

Using3.35, we get

Δ²K³V −Δ²K³V_p≤

1KT_∞

n1

α^pnK²V − K²V

n,Δ

K²V − K²V

n^p_X×X1/p

≤33M²

1K_T1 6

_∞

j0

α_j 3

V −V_∞,

3.38 whence

Δ²K³V −Δ²K³V_p≤1 6

3Mα1

3

1K_TM⁻¹V −V. 3.39 From estimates3.37and3.39, we get

K³V − K³V≤ b

3!a³V −V. 3.40

An induction argument shows us that

KⁿV − KⁿV≤ b

n!aⁿV −V. 3.41

Sincebaⁿ/n! < 1 fornsufficiently large, by the fixed point iteration methodKhas a unique fixed pointV ∈ W^2,p₀ . LetV be the unique fixed point ofK, then byAssumption 3.1we have

V_nX

n−1 k1

Skf

n−1−k, V_n−1−k,ΔVn−1−k _X

≤Mⁿ⁻²

k0

f

k, Vk,ΔVk

−fk,0,0_XMⁿ⁻²

k0

fk,0,0_X

≤Mⁿ⁻²

k0

αkVk,ΔVk_X×XMf·,0,0

1,

3.42

hence,

V_nX ≤Mf·,0,0

1Mⁿ⁻¹

k0

α_kV_k,ΔVk_X×X. 3.43 On the other hand, we have

ΔVn_X

n−1

k1

Sk1− Sk f

n−1−k, V_n−1−k,ΔVn−1−k _X

≤2Mⁿ⁻¹

k0

αkVk,ΔVk_X×X2Mⁿ⁻¹

k0

fk,0,0_X,

3.44

hence

ΔVn_X ≤2Mf·,0,0₁2Mⁿ⁻¹

k0

αkVk,ΔVk_X×X. 3.45

From3.43and3.45, we get

V_n,ΔVn_X×X ≤3Mf·,0,0

13Mⁿ⁻¹

k0

α_kV_k,ΔVk_X×X. 3.46

T hen, by application of the discrete Gronwall inequality1, Corollary 4.12, page 183 , we get V_n,ΔVn_X×X≤3Mf·,0,0

1

n−1 j0

13Mα_j

≤3Mf·,0,0

1

n−1 j0

e^3Mαj

3Mf·,0,0

1e^3M

_n−1

j0αj

≤3Mf·,0,0

1e^3Mα1.

3.47

Then,

sup

n∈Z

Vn,ΔVn_X×X

≤3Mf·,0,0₁e^3Mα1. 3.48

Finally, by3.20we obtain

Δ²Vnf

n, Vn,ΔVn ⁿ⁻¹

k1

I−TSkf

n−1−k, Vn−1−k,ΔVn−1−k

. 3.49

Hence, using the fact thatΔ²V₀f0,0,0and proceeding analogously as in3.23, we get Δ²V_pf0,0,0^p_X^∞

n1

Δ²V_n^p_X1/p

≤f0,0,0_{X ∞}

n1

Δ²V_n^p_X1/p

≤f0,0,0_{X ∞}

n1

fn, Vn,ΔVn^p_X1/p

KT_∞

n1

fn, Vn,ΔVn^p_X1/p

≤2 _∞

n0

f

n, Vn,ΔVn^p_X1/p

KT_∞

n0

f

n, Vn,ΔVn^p_X1/p

≤

2K_T^∞

n0

f

n, V_n,ΔVn_X,

3.50 where, byAssumption 3.1and3.48,

∞ n0

f

n, Vn,ΔVn_X≤^∞

k0

αkVk,ΔVk_X×Xf·,0,0

1

≤3Mα1f·,0,0

1e^3Mα1f·,0,0

1

≤f·,0,0₁e^6Mα1.

3.51

This ends the proof of the theorem.

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

Corollary 3.4. LetX be a UMD space. Assume thatAssumption 3.1holds and supposeT ∈ BX is an analyticS-bounded operator such that the set{λ−1²Rλ−1², I−T : |λ| 1, λ /1}is R-bounded. Then, there is a unique bounded solutionx xnof 3.1such thatΔ²x_n∈l_pZ, X.

Moreover, the a priori estimates3.6hold.

Example 3.5. Consider the semilinear problem

Δ²x_n−I−Txnq_nfxn, n∈Z, x₀x₁0, 3.52

wheref is defined and satisfies a Lipschitz condition with constantLon a Hilbert spaceH.

In addition, supposeqn∈l1Z.Then,Assumption 3.1is satisfied. In our case, applying the preceding result, we obtain that ifT ∈ BHis an analyticS-bounded operator such that the set{λ−1²Rλ−1², I−T:|λ|1, λ /1}is bounded, then there exists a unique bounded solutionx xnof3.52such thatΔ²xn∈lpZ, H.Moreover,

max

sup

n∈Z

x_nHΔxn_H

,Δ²x_p

≤Cf0_Hq1e^6LMq1. 3.53

In particular, takingT Ithe identity operator, we obtain the following scalar result which complements those in the work of Drozdowicz and Popenda2 .

Corollary 3.6. Supposef is defined and satisfies a Lipschitz condition with constantLon a Hilbert spaceH. Letqn∈l₁Z, H, then the equation

Δ²x_nq_nfxn 3.54

has a unique bounded solutionx xnsuch thatΔ²xn∈lpZ, Hand3.53holds.

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 sys- tems. 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. LetXbe a UMD space. Assume thatAssumption 3.1holds and supposeT ∈ BXis analytic and 1∈ρT.Then, the system3.1is stable, that is the solutionxnof 3.1is such that xn→0 asn→ ∞.

Proof. It is assumed thatTis analyticwhich implies that the spectrum is contained in the unit disc and the point 1, see10 and that 1 is not in the spectrum, then in view of27, Proposition 3.6 , the set

λ−1²R

λ−1², I−T

: |λ|1, λ /1

4.1 isR-bounded, becauseλ−1²Rλ−1², I−Tis an analytic function in a neighborhood of the circle. TheS-boundedness assumption of the operatorT follows from maximal regularity and the fact thatI−T is invertible. In fact, we get the following estimate:

supn≥0Sn ≤ I−T⁻¹KT. 4.2

By Corollary 3.4, there exists a unique bounded solution xn of 3.1 such that Δ²xn ∈ lpZ, X. Then,Δ²xn → 0 asn→ ∞.Next, observe thatAssumption 3.1and estimate3.10 imply

f

n, x_n,Δxn_X≤f

n, x_n,Δxn

−fn,0,0_Xfn,0,0_X

≤αnxn,Δxn_X×Xfn,0,0_X

≤α_nsup_n∈Zx_n,Δxn_X×Xfn,0,0_X

≤3α_nx_∞fn,0,0_X.

4.3

Sincef·,0,0∈l1Z, Xandαn∈l1Z, we obtain thatfn, xn,Δxn→0 asn→ ∞.

Then, the result follows from the fact that 1∈ρTand3.1.

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 ∈ BH such that T < 1. Suppose that Assumption 3.1holds inH. Then, the system3.1is stable.

Proof. First, we note that each Hilbert space is UMD, and then the concept ofR-boundedness and boundedness coincide; see7 . Since T < 1, we get that T is analytic and 1 ∈ ρT.

Furthermore, for|λ|1, λ /1,the inequality λ−1²R

λ−1², I−T

λ−1² λλ−2

∞ n0

T λλ−2

n

≤ |λ−1|²

|λ−2| − T ≤ 4

1− T 4.4 shows that the set4.1is 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 fn, zis locally Lipschitz with respect toz ∈ X×X; that is for each positive numberR, for alln∈Z, andz, w∈X×X,zX×X≤R, wX×X≤R

fn, z−fn, w_X≤ln, Rz−wX×X, 5.1 where:Z×0,∞→0,∞is a nondecreasing function with respect to the second variable.

ii^∗There is a positive numberasuch that_∞

n0n, a<∞.

iii^∗f·,0,0∈1Z, X.

We need to introduce some basic notations. We denote byW^2,pm the Banach space of all sequencesV Vnbelonging to∞Z, X,such thatVn0 if 0≤n≤m, andΔ²V ∈pZ, X equipped with the norm·. For λ > 0, denote byW^2,pmλ the ballV ≤λinW^2,pm. Our main result in this section is the following local version ofTheorem 3.3.

Theorem 5.2. Suppose that the following conditions are satisfied.

a^∗Assumption 5.1holds.

b^∗T is anS-bounded operator and it has discrete maximal regularity.

Then, there are a positive constantm∈Nand a unique bounded solutionx xnof 3.1for n≥msuch thatx_n0 if 0≤n≤mand the sequenceΔ²x_nbelongs to_pZ, X. Moreover, one has

x∞Δ²x_p≤a, 5.2

whereais the constant of condition (ii)^∗.

Proof. Letβ∈0,1/3. Usingiii^∗andii^∗,there aren1andn2inNsuch that M2KT^∞

jn1

fj,0,0_X≤βa, 5.3 T:β

M2KT^∞

jn2

j, a<1

3, 5.4

whereM:sup_n∈ZSn.

LetV be a sequence in W^2,pma/3 , withm max{n1, n₂}. A short argument similar to 3.7and involvingAssumption 5.1shows that the sequence

gn:

⎧⎨

⎩

0, if 0≤n≤m,

fn, Vn,ΔVn, ifn > m, 5.5

belongs to_p. By the discrete maximal regularity, the Cauchy problem3.12withg_ndefined as in5.5has a unique solutionznsuch thatΔ²zn∈lpZ, X, which is given by

zn KV _n

⎧⎪

⎪⎨

⎪⎪

⎩

0, if 0≤n≤m,

n−1−m

k0

Skfn−1−k, Vn−1−k,ΔVn−1−k, ifn≥m1. 5.6

We will prove thatKV belongs toW^2,pma/3 . In fact, since

V_j,ΔVj_X×X≤3|V|∞≤3|V|< a, 5.7

we have byAssumption 5.1 KV _nXMⁿ⁻²

jm

f

j, Vj,ΔVj_X

≤Mⁿ⁻²

jm

f

j, V_j,ΔVj

−fj,0,0_XMⁿ⁻²

jm

fj,0,0_X

≤Mⁿ⁻²

jm

lj, aV_j,ΔVj_X×XMⁿ⁻²

jm

fj,0,0_X

≤M^∞

jm

lj, aaM^∞

jm

fj,0,0_X.

5.8

Proceeding in a way similar to3.20, we get forn≥m

Δ²KV _nf

n, Vn,ΔVn ^n−1−m

k1

I−TSkf

n−1−k, Vn−1−k,ΔVn−1−k

. 5.9

Hence,

Δ²KV _pf

m, Vm,ΔVm^p_X ^∞

nm1

Δ²KV _n^p_X1/p

≤f

m, Vm,ΔVm_X

_∞

nm1

f

n, V_n,ΔVn ^n−1−m

k1

I−TSkf

n−1−k, V_n−1−k,ΔVn−1−k^p_X1/p

≤f

m, Vm,ΔVm_X

1KT_∞

nm

f

n, Vn,ΔVn^p_X1/p

≤

2K_T^∞

nm

f

n, V_n,ΔVn_X.

5.10

Therefore, using5.8we get Δ²KV _p≤

2K_T∞

jm

lj, aa^∞

jm

fj,0,0_X

. 5.11

Then, inequalities5.8and5.11together with5.3and5.4imply KV≤

M2K_T^∞

jm

j, aa

M2K_T^∞

jm

fj,0,0_X

≤ 1

3 −β

aβa 1 3a,

5.12

proving that KV belongs to W^2,pma/3 . In an essentially similar way to the proof of Theorem 3.3, for allV andWinW^2,pma/3 , we prove that

KV −KW ∞≤3M^∞

jm

j, aV −W, 5.13

Δ²KV −Δ²KW _p≤3

1KT^∞

jm

j, aV −W, 5.14

whence

KV −KW ≤3

M1KT^∞

jm

j, aV −W3T −βV −W. 5.15

Since 3T −β<1,Kis a 3T −β-contraction. This completes the proof of the theorem.

This enables us to prove, as an application, the following corollary.

Corollary 5.3. LetB_i : X ×X → X,i 1,2,be two bounded bilinear operators,y ∈ ₁Z, X, andα, β ∈ ₁Z,R. In addition, suppose thatT is aS-bounded operator and has discrete maximal regularity. Then, there is a unique bounded solutionxsuch thatΔ²x∈lpZ, Xfor the equation

xn2−2xn1TxnynαnB1

xn, xn βnB2

Δxn,Δxn

. 5.16

Proof. Take ln, R : 2R|αn||βn|B1 B2.Then, _∞

n0n,1 < ∞.Note also that fn,0,0 ynbelongs to₁Z, X.Hence,Assumption 5.1is 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 inSection 4remains 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.

References

1 R. P. Agarwal, Difference Equations and Inequalities, vol. 155 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992.

2 A. Drozdowicz and J. Popenda, “Asymptotic behavior of the solutions of the second order difference equation,” Proceedings of the American Mathematical Society, vol. 99, no. 1, pp. 135–140, 1987.

3 S. Elaydi, An Introduction to Difference Equations, Undergraduate Texts in Mathematics, Springer, New York, NY, USA, 3rd edition, 2005.

4 V. B. Kolmanovskii, E. Castellanos-Velasco, and J. A. Torres-Mu ˜noz, “A survey: stability and bound- edness of Volterra difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 53, no. 7-8, pp. 861–928, 2003.

5 J. P. LaSalle, The Stability and Control of Discrete Processes, vol. 62 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1986.