## SHORT NOTE A note on the continuous dependence

In this paper, we study the continuous dependence on data for the solution u Â¼ {ui}i\$1 of the boundary value problem. uiÐ§1 2 Ð1 Ð§ uiÑui Ð§ uiui21 [ ciAui, i \$ 1,.

Journal of Difference Equations and Applications Vol. 17, No. 4, April 2011, 637–641

SHORT NOTE A note on the continuous dependence on data for second-order difference inclusions G. Apreuteseia1 and N. Apreuteseib* a Faculty of Mathematics, University ‘Al. I. Cuza’, No. 11, Bulevardul Carol I, 700506 Iasi, Romania; bDepartment of Mathematics, Technical University ‘Gh. Asachi’, No. 11, Bulevardul Carol I, 700506 Iasi, Romania

1.

Introduction

In this paper, we study the continuous dependence on data for the solution u ¼ {ui }i\$1 of the boundary value problem 8 < uiþ1 2 ð1 þ ui Þui þ ui ui21 [ ci Aui ; : u0 ¼ a; supkui k , 1:

i \$ 1; ð1:1Þ

i\$1

Here, A : DðAÞ # H ! H is a maximal monotone operator in the Hilbert space H, D(A) is the domain of A and a [ H is a given value. By k·k, we denote the norm in H. The problem we study in the present paper is the discrete variant of the results obtained in . This result complements the main theorem from , where a similar study was done on finite sets of integers i. Other properties of second-order evolution equations associated to maximal monotone operators were studied in [5,7,8], while their discrete variants are treated in [6,8,9]. A detailed study can be found in . The sequences of real numbers ðci Þi\$1 and ðui Þi\$1 are supposed to be positive and satisfy specific conditions which assure the existence of the solution for the difference inclusion (1.1). More exactly, assume either

0 , ui , 1; ; i \$ 1 and

1 X 1 ¼ 1; h i¼1 i

where hi ¼

i X

1

j¼1

ui ui21 . . . uj

;

ð1:2Þ

or

ui \$ 1; ; i \$ 1:

* Corresponding author. Email: [email protected] ISSN 1023-6198 print/ISSN 1563-5120 online q 2011 Taylor & Francis DOI: 10.1080/10236190903167983 http://www.informaworld.com

ð1:3Þ

638

G. Apreutesei and N. Apreutesei

Condition (1.2) is available if, for example, ui ¼ i=ði þ 1Þ; ; i \$ 1. Theorems 3.2 and 3.4 from  assure the existence of the solution in DðAÞ of problem (1.1), provided that either (1.2) or (1.3) holds. Our goal is to prove that the function which associates to {a,A}, the solution u ¼ {ui }i\$1 of problem (1.1) is continuous in the following sense. Consider the sequence of difference inclusions 8 n n n n n < uiþ1 2 ð1 þ ui Þ ui þ ui ui21 [ ci A ui ; i \$ 1; n n n : u0 ¼ a ; supkui k ¼ C n , 1; n \$ 1;

ð1:4Þ

i\$1

where A n : DðA n Þ # H ! H is a sequence of maximal monotone operators in H and a n [ H. Assume that a n ! a in H and A n ! A in the sense of resolvent, i.e. ðI þ lA n Þ21j ! ðI þ lAÞ21j; as n ! 1; ; l . 0; ; j [ H: In , we proved a similar result for the problem on a finite interval of i 8 < uNiþ1 2 ð1 þ ui ÞuNi þ ui uNi21 [ ci AuNi þ f i ; 1 # i # N : uN0 ¼ a; Theorem 1.1. (). Let

be the solution of (1.6) and

uNNþ1 ¼ b:

ð1:5Þ

ð1:6Þ

  u N ¼ uNi 1#i#N   u nN ¼ unN i 1#i#N

be the solution of (1.6) with An, an, bn, f ni instead of A, a, b, fi. If a n ! a, b n ! b and N f ni ! f i in H, as n ! 1, 1 # i # N and A n ! A in the sense of resolvent, then unN i ! ui in H, as n ! 1, for every i [ N, 1 # i # N. We establish in the sequel an analogous result for problem (1.1): Theorem 1.2. Let A : DðAÞ # H ! H, A n : DðA n Þ # H ! H be maximal monotone operators in H such that 0 [ DðAÞ > DðA n Þ, 0 [ A0 > A n 0, and let a, a n [ H, ci . 0, ui . 0, ;i \$ 1 be sequences satisfying (1.2) or (1.3). Denote by u ¼ {ui }i\$1 and   u n ¼ uni i\$1 ; the solutions of (1.1) and (1.4). If a n ! a in H and A n ! A in the sense of resolvent, then uni ! ui as n ! 1, uniformly on finite sets of i [ Z, i \$ 0. 2. The proof of the main result To prove Theorem 1.2, we approximate the solutions of problems (1.1) and (1.4) with the solutions of the problems on finite sets of i, 8 < uNiþ1 2 ð1 þ ui ÞuNi þ ui uNi21 [ ci AuNi ; 1 # i # N; ð2:1Þ : uN0 ¼ uNNþ1 ¼ a;

Journal of Difference Equations and Applications 8 nN nN n nN < unN iþ1 2 ð1 þ ui Þui þ ui ui21 [ ci A ui ; : unN 0

¼

unN Nþ1

n

¼a ;

639

1 # i # N;

n \$ 1:

ð2:2Þ

Here, N [ Z, N \$ 1 is a fixed number. By Theorem 6.1.2, p. 143 from , we know that these problems have unique solutions   u N ¼ uNi 1#i#N [ DðAÞN ;   [ DðA n ÞN : u nN ¼ unN i 1#i#N Moreover, we can prove the following auxiliary results. Lemma 2.1. The sequence Cn from (1.4) is bounded in R. Proof. Since 0 [ DðA n Þ and 0 [ A n 0, for all n \$ 1, it follows that uni ¼ 0 is the unique solution of problem (1.4) with un0 ¼ 0. Then, Theorems 6.2.1, p. 154 and 6.2.2, p.157 from  imply that  n u  # ka n k; ; n; i \$ 1; i

where   u n ¼ uni i\$1 is the solution of (1.4) with the initial condition un0 ¼ a n . So,   Cn ¼ supuni  i\$1

is bounded in R. Following the proofs of Theorems 3.2 and 3.4 from , we can easily show that n uNi ! ui and unN i ! ui , as N ! 1, for i belonging to a finite set of integers. We can also state some useful estimates. A n Lemma 2.2. Under the hypotheses of Theorem 1.2, there exists the limit limN!1 unN i ¼ ui , ; n \$ 1, uniformly on finite sets of i [ Z, i \$ 0. In addition, for every fixed N 0 [ Z, N 0 \$ 1 and 1 # i # N 0 , if ui verifies either (1.2) or (1.3), we have either

 nN  2ka n k u 2 un  # qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ; i i PN 1=h k k¼N 0

ð2:3Þ

 nN  u 2 u n  # i i

ð2:4Þ

or 2N 0 ka n k ; ðN 2 N 0 þ 1Þ

respectively. Analogous results are available for the solutions u and u N of problems (1.1) and (2.1), respectively. We can now prove the main result.

640

G. Apreutesei and N. Apreutesei

Proof of Theorem 1.2. Let N be fixed. We associate the auxiliary boundary value problems (2.1) and (2.2) on finite sets of i, {1; . . . ; N}. Then we have the following estimate:  n        u 2 ui  # un 2 unN  þ unN 2 uN  þ uN 2 ui ; i i i i i i

1 # i # N; n \$ 1:

Let 1 # N 0 # N be given. According to Lemma 2.2 (applied for both problems (2.1) and (2.2)), we can write  n  2ka n k þ 2kak  nN  u 2 u i  # q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ þ ui 2 uNi ; i PN k¼N 0 1=hk

1 # i # N0;

ð2:5Þ

if hypothesis (1.2) is fulfilled, or n  n    u 2 ui  # 2N 0 ðka k þ kakÞ þ unN 2 uN ; i i i N 2 N0 þ 1

1 # i # N0;

ð2:6Þ

N if hypothesis (1.3) is fulfilled. For fixed N, Theorem 1.1 assures that unN i ! ui in H, as n ! 1, for 1 # i # N. Passing to the superior limit as n ! 1 in (2.5) and (2.6), one derives that

    4kak 4N 0 kak ﬃ or lim supuni 2 ui  # lim supuni 2 ui  # qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; P N N 2 N0 þ 1 n!1 n!1 1=h k k¼N 0 1 # i # N0; if hypothesis (1.2) or (1.3) holds, respectively. In any case, letting N ! 1; we conclude that limn!1 kuni 2 ui k ¼ 0, 1 # i # N 0 . The proof is complete. A

Acknowledgements The second author’s work was supported by the project ID 342/2008, CNCSIS, Romania.

Note 1.

Email: [email protected]

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