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

(Received 21 September 2008; final version received 6 July 2009)

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 [2]. This result complements the main theorem from [1], 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 [4]. 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 [3] 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 [1], 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. ([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 [4], 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 [4] 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 supuni i$1

is bounded in R. Following the proofs of Theorems 3.2 and 3.4 from [3], 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 supuni 2 ui # lim supuni 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]

References [1] G. Apreutesei and N. Apreutesei, Continuous dependence on data for bilocal difference equations, J. Difference Equ. Appl 15(5) (2009), pp. 511– 527. [2] N. Apreutesei, Continuous dependence on data for a second-order evolution equation, Comm. Appl. Nonlin. Anal 9 (2002), pp. 75 – 84. [3] N. Apreutesei, On a class of difference equations of monotone type, J. Math. Anal. Appl 288 (2003), pp. 833– 851. [4] N. Apreutesei, Nonlinear Second-Order Evolution Equations of Monotone Type and Applications, Pushpa Publishing House, Allahabad, India, 2007. [5] B. Djafari Rouhani and H. Khatabzadeh, Asymptotic behavior of solutions to some homogeneous second-order evolution equations of monotone type, J. Inequal. Appl Art. ID 72931 (2007), p. 8. [6] B. Djafari Rouhani and H. Khatabzadeh, A note on the asymptotic behavior of solutions to a second order difference equations, J. Difference Equ. Appl 14(4) (2008), pp. 429– 432.

Journal of Difference Equations and Applications

641

[7] H. Ma and X. Xue, Second order nonlinear multivalued boundary problems in Hilbert spaces, J. Math. Anal. Appl 303(2) (2005), pp. 736– 753. [8] E. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl 113 (1986), pp. 514– 543. [9] S. Reich and I. Shafrir, An existence theorem for a difference inclusion in general Banach spaces, J. Math. Anal. Appl 160 (1991), pp. 406– 412.

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

(Received 21 September 2008; final version received 6 July 2009)

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 [2]. This result complements the main theorem from [1], 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 [4]. 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 [3] 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 [1], 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. ([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 [4], 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 [4] 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 supuni i$1

is bounded in R. Following the proofs of Theorems 3.2 and 3.4 from [3], 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 supuni 2 ui # lim supuni 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]

References [1] G. Apreutesei and N. Apreutesei, Continuous dependence on data for bilocal difference equations, J. Difference Equ. Appl 15(5) (2009), pp. 511– 527. [2] N. Apreutesei, Continuous dependence on data for a second-order evolution equation, Comm. Appl. Nonlin. Anal 9 (2002), pp. 75 – 84. [3] N. Apreutesei, On a class of difference equations of monotone type, J. Math. Anal. Appl 288 (2003), pp. 833– 851. [4] N. Apreutesei, Nonlinear Second-Order Evolution Equations of Monotone Type and Applications, Pushpa Publishing House, Allahabad, India, 2007. [5] B. Djafari Rouhani and H. Khatabzadeh, Asymptotic behavior of solutions to some homogeneous second-order evolution equations of monotone type, J. Inequal. Appl Art. ID 72931 (2007), p. 8. [6] B. Djafari Rouhani and H. Khatabzadeh, A note on the asymptotic behavior of solutions to a second order difference equations, J. Difference Equ. Appl 14(4) (2008), pp. 429– 432.

Journal of Difference Equations and Applications

641

[7] H. Ma and X. Xue, Second order nonlinear multivalued boundary problems in Hilbert spaces, J. Math. Anal. Appl 303(2) (2005), pp. 736– 753. [8] E. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl 113 (1986), pp. 514– 543. [9] S. Reich and I. Shafrir, An existence theorem for a difference inclusion in general Banach spaces, J. Math. Anal. Appl 160 (1991), pp. 406– 412.