DSpace at VNU: Asymptotic equilibrium of integro-differential equations with infinite delay

Math Sci (2015) 9:189–192
DOI 10.1007/s40096-015-0166-5

ORIGINAL RESEARCH

Asymptotic equilibrium of integro-differential equations
with infinite delay
Le Anh Minh1 • Dang Dinh Chau2

Received: 25 June 2014 / Accepted: 2 September 2015 / Published online: 21 September 2015
Ó The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract The asymptotic equilibrium problems of ordinary differential equations in a Banach space have been
considered by several authors. In this paper, we investigate
the asymptotic equilibrium of the integro-differential
equations with infinite delay in a Hilbert space.
Keywords Asymptotic equilibrium Á Integro-differential
equations Á Infinite delay

8
0
1
Zt
>
>
< dxðtÞ
¼ AðtÞ@ xðtÞ þ
kðt À hÞxðhÞdhA;
dt
>
À1

>
:
xðtÞ ¼ uðtÞ;

t > 0;
t60
ð1Þ

where AðtÞ : H ! H, u in the phase space B, and xt is
defined as
xt ðhÞ ¼ xðt þ hÞ;

À1\h 6 0:

Introduction
The asymptotic equilibrium problems of ordinary differential equations in a Banach space have been considered by
several authors, Mitchell and Mitchell [3], Bay et al. [1],
but the results for the asymptotic equilibrium of integrodifferential equations with infinite delay still is not presented. In this paper, we extend the results in [1] to a class
of integro-differential equations with infinite delay in a
Hilbert space H which has the following form:

Preliminaries
We assume that the phase space ðB; jj:jjB Þ is a seminormed
linear space of functions mapping ðÀ1; 0Š into H satisfying the following fundamental axioms (we refer reader to
[2])
(A1 )

For a [ 0, if x is a function mapping ðÀ1; aŠ into
H, such that x 2 B and x is continuous on [0, a],
then for every t 2 ½0; aŠ the following conditions

hold:
(i)
(ii)
(iii)

& Le Anh Minh

Dang Dinh Chau

1

Department of Mathematical Analysis, Hong Duc University,
Thanh Ho´a, Vietnam

2

Department of Mathematics, Hanoi University of Science,
VNU, Hanoi, Vietnam

xt belongs to B;
jjxðtÞjj 6 Gjjxt jjB ;
jjxt jjB 6 KðtÞ sups2½0;tŠ jjxðsÞjj þ MðtÞjjx0 jjB

where G is a possitive constant, K; M : ½0; 1Þ ! ½0; 1Þ,
K is continuous, M is locally bounded, and they are independent of x.
(A2 )
(A3 )

For the function x in (A1 ), xt is a B-valued
continuous function for t in [0, a].

The space B is complete.

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Math Sci (2015) 9:189–192

Example 1
(i)

(M4 )

Let BC be the space of all bounded continuous
functions from ðÀ1; 0Š to H, we define C0 :¼
fu 2 BC : limh!À1 uðhÞ ¼ 0g and C1 :¼ fu 2
BC : limh!À1 uðhÞ exists in Hg endowed with the
norm
jjujjB ¼

(ii)

sup

&
'
u 2 CððÀ1; 0Š; XÞ : lim ech uðhÞ exists in H

T

herein S(0, 1) is a unit ball in H, j ¼ LðK þ MÞ þ
1; where K, M, L are given in (c1 ), (c2 ) and (M3 ).

h!À1

Theorem 1 If (M1 ), ðM2 Þ, ðM3 Þ and ðM4 Þ are satisfied,
then Eq. (1) has an asymptotic equilibrium.
Proof We shall begin with showing that all solutions of
(1) has a finite limit at infinity. Indeed, Eq. (1) may be
rewritten as
0
1
Z0
dxðtÞ
¼ AðtÞ@ xðtÞ þ
kðÀhÞxt ðhÞdhA ;
dt
À1

endowed with the norm
jjujjB ¼

sup

ð2Þ

jjuðhÞjj

h2ðÀ1;0Š

then C0 ; C1 satisfies (A1 )–(A3 ). However, BC
satisfies (A1 ) and (A3 ), but (A2 ) is not satisfied.
For any real constant c, we define the functional
spaces Cc by
Cc ¼

There exists a constant T [ 0 such that
Z1
1
jjAðtÞhjjdt\q\ ;
sup
j
h2Sð0;1Þ

ech jjuðhÞjj:

h2ðÀ1;0Š

Then conditions (A1 )–(A3 ) are satisfied in Cc .

then for t > s > T we have
0
1
Z0
Zt
kðÀhÞxs ðhÞdhAds
xðtÞ ¼ xðsÞ þ AðsÞ@ xðsÞ þ
À1

s

Remark 1 In this paper, we use the following acceptable
hypotheses on K(t), M(t) in (A1 )(iii) which were introduced
by Hale and Kato [2] to estimate solutions as t ! 1,

and
jjxðtÞjj

*
0
1
+

Zt
Z0


kðÀhÞxs ðhÞdhAds; h 
¼ sup  xðsÞ þ AðsÞ@xðsÞ þ
h2Sð0;1Þ

s
À1


*
+

Z0

Z t 

ds
 xðsÞ þ
6 jjxðsÞjj þ sup
kðÀhÞx
ðhÞdh;AðsÞh
s


h2Sð0;1Þ


s
À1
!

(c1 ) K ¼ KðtÞ is a constant for all t > 0;
(c2 ) MðtÞ 6 M for all t > 0 and some M.
Example 2 For the functional space Cc in Example 1, the
hypotheses (c1 ) and (c2 ) are satisfied if c > 0.
Definition 1 Equation (1) has an asymptotic equilibrium
if every solution of it has a finite limit at infinity and, for
every h0 2 H, there exists a solution x(t) of it such that
xðtÞ ! h0 as t ! 1.

Main results

6 jjxðsÞjj þ q ðLK þ 1Þ sup jjxðnÞjj þ LMjjujjB
n2½0;tŠ

ð3Þ
implies
À
Á
jjjxðtÞjjj 6 jjxðsÞjj þ q ðLK þ 1ÞjjjxðtÞjjj þ LMjjujjB
or

Now, we consider the asymptotic equilibrium of Eq. (1)
which satisfies the following assumptions:
(M1 )
(M2 )
(M3 )

A(t) is a strongly continuous bounded linear
operator for each t 2 Rþ ;
A(t) is a self-adjoint operator for each t 2 Rþ ;
k satisfies
Zþ1

jkðhÞjdh ¼ L\ þ 1;

0

and

123

jjjxðtÞjjj 6

jjxðsÞjj þ qLMjjujjB
1 À qðLK þ 1Þ

where
jjjxðtÞjjj ¼ sup jjxðnÞjj:
06n6t

ð4Þ


Math Sci (2015) 9:189–192

191

Now, we conclude that x(t) is bounded since
0\q\

1
1
1
¼
\
) qðLK þ 1Þ\1
j LðK þ MÞ þ 1 LK þ 1

and by (4).
Putting

and
jjx1 ðtÞjj 6 jjh0 jjð1 þ qjÞ:

Now, we consider the functional
g2 ðt; hÞ ¼ hh0 ; hi
0
1 +
Zþ1*
Zs
À
AðsÞ@x1 ðtÞ þ
kðs À hÞx1 ðhÞdhA; h ds:

M Ã ¼ sup jjxðtÞjj;
t2R

À1

t

we have

By an argument analogous to the previous one, we get

jjxðtÞ À xðsÞjj ¼ sup j \xðtÞ À xðsÞ; h [ j
h2Sð0;1Þ



0
1

Zt 

Z0


ds
 \AðsÞ@xðsÞ þ
A
kðÀhÞx
ðhÞdh
;
h
[
6 sup
s
 ;

h2Sð0;1Þ


À1

s

Ã

6 ½M ðLK þ 1Þ þ LMjjujjB Š sup

h2Sð0;1Þ

Zt

jg2 ðt; hÞj 6 jjh0 jj½jjhjj þ qj þ ðqjÞ2 Š
and there exists an element x2 ðtÞ in H, such that
g2 ðt; hÞ ¼ hx2 ðtÞ; hi
with

jjAðsÞhjjds ! 0

jjx2 ðtÞjj 6 jjh0 jjð1 þ qj þ ðqjÞ2 Þ:

s

as t > s ! þ1. That means all solutions of (1) have a
finite limit at infinity. To complete the proof, it remains to
show that for any h0 2 H, there exists a solution x(t) of (1)
such that
lim xðtÞ ¼ h0 :

Continuing this process, we obtain the linear continuous
functional
gn ðt; hÞ ¼ hh0 ; hi
0
1 +
Zþ1*
Zs
À
AðsÞ@xnÀ1 ðtÞ þ
kðs À hÞxnÀ1 ðhÞdhA; h ds
À1

t

t!þ1

Indeed, let h0 be an arbitrary fixed element of H; we choose
the initial function u belongs to B such that uð0Þ ¼ h0 and
jjujjB 6 jjh0 jj and consider the functional
g1 ðt; hÞ ¼ hh0 ; hi
0
1 +
Z1*
Zs
À
AðsÞ@h0 þ
kðs À hÞx0 ðhÞdhA; h ds

ð5Þ
and xn ðtÞ 2 H such that
gn ðt; hÞ ¼ hxn ðtÞ; hi
satisfies the following estimate
jjxn ðtÞjj 6 ð1 þ qj þ ðqjÞ2 þ Á Á Á þ ðqjÞn Þjjh0 jj 6

À1

t

jjh0 jj
:
1 À qj

Futhermore,

We have
jg1 ðt; hÞj 6 jjh0 jjjjhjj þ

Zþ1

kxn ðtÞ À xnÀ1 ðtÞk 6 jjh0 jjðqjÞn :
kx0 ðsÞ

t

þ

Zs

kðs À hÞx0 ðhÞdhkjjAðsÞhjjds::

À1

Since x0 ðsÞ  h0 ; then
jg1 ðt; hÞj 6 jjh0 jjðjjhjj þ qjÞ:
It follows from Riesz representation theorem that there
exists an element x1 ðtÞ in H, such that
g1 ðt; hÞ ¼ hx1 ðtÞ; hi

This inequality shows that fxn ðtÞg is uniformly convergent
on ½T; þ1Þ since qj\1. Put
xðtÞ ¼ lim xn ðtÞ:
n!þ1

In (5), let n ! þ1; we have
hxðtÞ; hi ¼ hh0 ; hi
0
1 +
Zþ1*
Zs
À
AðsÞ@xðtÞ þ
kðs À hÞxðhÞdhA; h ds
t

À1

ð6Þ
and since

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Math Sci (2015) 9:189–192

jhxn ðtÞ; h0 ij\

Zþ1

kxnÀ1 ðsÞ

T

þ

Zs

creativecommons.org/licenses/by/4.0/), which permits unrestricted
use, distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
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made.

kðs À hÞxnÀ1 ðhÞdhkk AðsÞhkds

À1

or
jjh0 jjq
;
jhxn ðtÞ; h0 ij 6
1 À qj
we have xn ðtÞ ! h0 as q ! 0, which means that there
exists a solution of (1) converging to h0 . The theorem is
proved.
Open Access This article is distributed under the terms of the
Creative Commons Attribution 4.0 International License (http://

123

References
1. Bay, N.S., Hoan, N.T., Man, N.M.: On the asymptotic equilibrium

and asymptotic equivalence of differential equations in Banach
spaces. Ukr. Math. J. 60(5), 716–729 (2008)
2. Hale, J.K., Kato, J.: Phase space for retarded equations with
infinite delay. Fukcialaj Ekvacioj 21, 11–41 (1978)
3. Mitchell, A.R., Mitchell, R.W.: Asymptotic equilibrium of ordinary differential systems in a Banach space. Theory Comput. Syst.
9(3), 308–314 (1975)