SEMILINEAR EVOLUTION EQUATIONS OF SECOND ORDER VIA MAXIMAL REGULARITY

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.


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 x 0 0 and x 1 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 l p Z ; X spaces.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 problems see, e.g., Amann 6 ,Denk et al. 7 ,Clément 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 H ölder 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 H ölder 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 R-boundedness properties of the resolvent operator for linear secondorder 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 l p 1 < p < ∞ for the semilinear problem 1.1 by using maximal regularity and a contraction principle.We also get some a priori estimates for the solutions x n and their discrete derivatives Δx n and Δ 2 x n .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 .

Discrete maximal regularity
Let X be a Banach space.Let Z denote the set of nonnegative integer numbers and let Δ be the forward difference operator of the first order, that is, for each x : Z → X and n ∈ Z , Δx n x n 1 − x n .We consider the second-order difference equation where T ∈ B X , Δ 2 x n Δ Δx n , and f : Z → X. Denote C 0 I, the identity operator on X, and define 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 Moreover, The following definition is the natural extension of the concept of maximal regularity from the continuous case cf., 16 .
As a consequence of the definition, if T ∈ B X has discrete maximal regularity, then T has discrete l p -maximal regularity, that is, for each f n ∈ l p Z ; X we have Δ 2 x n ∈ l p Z ; X , where x n is the solution of the equation We introduce the means 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 .

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Remark 2.3.a Let S, T ⊂ B X, Y be R-bounded sets, then S T : 2.10 c Also, each subset M ⊂ B X of the form M {λI : λ ∈ Ω} is R-bounded whenever Ω ⊂ 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 L p R, X for some and then all p ∈ 1, ∞ .Here, the Hilbert transform H of a function f ∈ S R, X , the Schwartz space of rapidly decreasing X-valued functions, is defined by 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 ∈ B X is called analytic if the set 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 ∈ B X be analytic.Then, the following assertions are equivalent.
i T has discrete maximal regularity of order 2.
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 Rboundedness plays a fundamental role in recent works by Clément-Da Prato 26 , Clément et al. 22 , Weis 27, 28 , Arendt-Bu 20, 29 , and Keyantuo-Lizama 30-32 .

Semilinear second-order evolution equations
In this section, our aim is to investigate the existence of bounded solutions, whose second discrete derivative is in p for semilinear evolution equations via discrete maximal regularity.
Next, we consider the following second-order evolution equation: which is equivalent to 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 satisfy the Lipschitz condition on X × X, that is for all z, w ∈ X × X and n ∈ Z , we get f n, z − f n, w X ≤ α n z − w X×X , where α : We remark that the condition α ∈ l 1 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.

3.4
Hence, 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 x n of 3.1 such that Δ 2 x n ∈ l p Z , X .Moreover, one has the following a priori estimates for the solution:

3.6
where M : sup n∈Z S n and C > 0.

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Proof.Let V be a sequence in W 2,p 0 .Then, using Assumption 3.1 we obtain that the function where

3.8
Analogously, we have 3.9 On the other hand, Since T has discrete maximal regularity, the Cauchy problem has a unique solution z n such that Δ 2 z n ∈ l p Z , X , which is given by

3.13
We now show that the operator K : W 2,p 0 → W 2,p 0 has a unique fixed point.To verify that K is well defined, we have only to show that KV ∈ l ∞ Z , X .In fact, we use Assumption 3.1 as above and M : sup n∈Z S n to obtain

3.14
It proves that the space W 2,p 0 is invariant under K. Let V and V be in W 2,p 0 .In view of Assumption 3.1 i and M < ∞, we have initially as in 3.14

3.15
Hence, we obtain

3.16
On the other hand, using the fact that S 1 I, we observe first that

3.17
Since S 2 2I, we get

3.18
Taking into account that z n 1 S * g n is solution of 3.12 , we get the following identity: Using 3.19 , we obtain for n ≥ 1

3.22
Since K T is bounded on l p Z , X , using Assumption 3.1, we obtain

3.23
Hence, we obtain from 3.16 and 3.23 where a : 3M α 1 and b : 1 1 K T M −1 .Next, we consider the iterates of the operator K. Let V and V be in W 2,p 0 .Taking into account that S 1 I, S 0 0, and

3.26
On the other hand, from 3.15 we get

3.27
Using estimates 3.26 and 3.27 , we obtain for n ≥ 2

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Next, using KV 0 KV 1 0 and estimates 3.28 and 3.10 , we obtain Furthermore, using the identity the fact that Δ 2 K 2 V 0 f 0, 0, 0 for all V ∈ W 2,p 0 , and Lemma 3.2, we obtain

3.33
From estimates 3.30 and 3.33 , we get with a and b defined as above.Taking into account 3.26 , 3.28 , 3.29 , and 3.10 , we can infer that Next, using estimate 3.35 and Lemma 3.2, we get

3.37
Using 3.35 , we get

3.39
From estimates 3.37 and 3.39 , we get

3.40
An induction argument shows us that

3.41
Since ba n /n! < 1 for n sufficiently large, by the fixed point iteration method K has a unique fixed point V ∈ W 2,p 0 .Let V be the unique fixed point of K, then by Assumption 3.1 we have

3.43
On the other hand, we have

3.45
From 3.43 and 3.45 , we get T hen, by application of the discrete Gronwall inequality 1, Corollary 4.12, page 183 , we get

3.48
Finally, by 3.20 we obtain Hence, using the fact that Δ 2 V 0 f 0, 0, 0 and proceeding analogously as in 3.23 , we get

3.51
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 ∈ B X is an analytic S-bounded operator such that the set Then, there is a unique bounded solution x x n of 3.1 such that Δ 2 x n ∈ l p Z , X .Moreover, the a priori estimates 3.6 hold.Example 3.5.Consider the semilinear problem where f is defined and satisfies a Lipschitz condition with constant L on a Hilbert space H.In addition, suppose q n ∈ l 1 Z .Then, Assumption 3.1 is satisfied.In our case, applying the preceding result, we obtain that if T ∈ B H is an analytic S-bounded operator such that the set { λ − 1 2 R λ − 1 2 , I − T : |λ| 1, λ / 1} is bounded, then there exists a unique bounded solution x x n of 3.52 such that Δ 2 x n ∈ l p Z , H .Moreover, max sup

3.53
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 q n ∈ l 1 Z , H , then the equation has a unique bounded solution x x n such that Δ 2 x n ∈ l p 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.

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 ∈ B X is analytic and 1 ∈ ρ T .Then, the system 3.1 is stable, that is the solution x n of 3.1 is such that x n → 0 as n → ∞.
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 , 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: By Corollary 3.4, there exists a unique bounded solution x n of 3.1 such that Δ 2 x n ∈ l p Z , X .Then, Δ 2 x n → 0 as n → ∞.Next, observe that Assumption 3.1 and estimate 3.10 imply

4.3
Since f •, 0, 0 ∈ l 1 Z , X and α n ∈ l 1 Z , we obtain that f n, x n , Δx n → 0 as n → ∞.Then, the result follows from the fact that 1 ∈ ρ 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 ∈ 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 ∈ ρ T .Furthermore, for |λ| 1, λ / 1, the inequality shows that the set 4.1 is bounded .
Of course, the same result holds in the finite dimensional case.

Local perturbations
In the process of obtaining our next result, we will require the following assumption.
i * The function f n, z is locally Lipschitz with respect to z ∈ X × X; that is for each positive number R, for all n ∈ Z , and z, w where : Z × 0, ∞ → 0, ∞ is a nondecreasing function with respect to the second variable.
ii * There is a positive number a such that

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We need to introduce some basic notations.We denote by W 2,p m the Banach space of all sequences V V n belonging to ∞ Z , X , such that V n 0 if 0 ≤ n ≤ m, and Δ 2 V ∈ p Z , X equipped with the norm • .For λ > 0, denote by W belongs to p .By the discrete maximal regularity, the Cauchy problem 3.12 with g n defined as in 5.5 has a unique solution z n such that Δ 2 z n ∈ l p Z , X , which is given by

5.6
We will prove that KV belongs to W f j, 0, 0 X .

5.8
Proceeding in a way similar to 3.20 , we get for n ≥ m Hence, f n, V n , ΔV n X .

5.10
Therefore, using 5.8 we get

3 . 5 . 2 .
ball V ≤ λ in W 2,p m .Our main result in this section is the following local version of Theorem 3.Theorem 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 ∈ N and a unique bounded solution xx n of 3.1 for n ≥ m such that x n 0 if 0 ≤ n ≤ m and the sequence Δ 2 x n belongs to p Z , X .Moreover, one hasx ∞ Δ 2 x p ≤ a, 5.2where a is the constant of condition (ii) * .Proof.Let β ∈ 0, 1/3 .Using iii * and ii * , there are n 1 and n 2 in N such that sup n∈Z S n .Let V be a sequence in W 2,p m a/3 , with m max{n 1 , n 2 }.A short argument similar to 3.7 and involving Assumption 5.1 shows that the sequence g n : n ≤ m, f n, V n , ΔV n , if n > m, 5.5