CC BY
21Probl. Anal. Issues Anal. Vol. 11 (29), No 1, 2022, pp. 67-80 67
DOI: 10.15393/j3.art.2022.10350
UDC 517.9, 517.4
A. Fadili, F. Maragh
INTEGRAL RESOLVENT FOR VOLTERRA EQUATIONS AND FAVARD SPACES
Abstract. The objective of this work is to give a characterization
of the domain D (A) of A in terms of the integral resolvent family
t
of the equation x (t) = x0 + f a (t — s) Ax(s)ds, t ^ 0, where A is
0
a linear closed densely defined operator, a e Ljoc (R+) in a general Banach space X and x0 e X. Furthermore, we give a relationship between the Favard classes (temporal and frequency) for integral resolvents.
Key words: semigroups, scalar Volterra integral equations, integral resolvent families, Favard spaces
2020 Mathematical Subject Classification: 45D05, 45E05, 46B70, 47D06
1. Introduction, Definitions, and Notations. Let (X, || ||) be a
Banach space, A a linear closed operator with dense domain D (A), defined in X and a e L]oc (R+) a scalar kernel. We consider the linear Volterra equation:
t
x « = / a (t — 8) M.X* + /(i), t > 0, i1)
0
where f e C (R+,X). Denote by [D (A)] the domain of A equipped with the graph-norm and define the convolution product * of a scalar function a and a vector-valued function f by:
t
(a * f )(t):=j a (t — s) f (s) ds, t ^ 0.
0
Definition 1. [11, Definition 1.1] A function x eC (R+,X) is called: © Petrozavodsk State University, 2022
1) A strong solution of (1) if x eC (R+, [D (A)]) and (1) is satisfied.
2) Mild solution of (1) if a * x eC (R+, [D (A)]) and
x = f (t) + A [a * x](t), t ^ 0. (2)
Obviously, every strong solution of (1) is a mild solution. Conditions under which mild solutions are strong, were studied in [11].
Definition 2. [7, Definition 2.2] Equation (1) is called well-posed if for each v e D (A) there is a unique strong solution x (t,v) on R+ of
x (t,v) = a (t) v + (a * Ax) (t) ,t ^ 0, (3)
and for any sequence (vn) C D (A) ,vn ^ 0 implies x (t,vn) ^ 0 in X uniformly on compact intervals.
Definition 3. [7, Definition 2.3] Let a e C (R+) be a scalar kernel. A strongly continuous family (R (t))t^0 C C, (X); (the space of bounded linear operators in X) is called an integral resolvent for equation (1), if the following three conditions are satisfied:
(R1) R(0) = a(0)I.
(R2) R(t) commutes with A, which means R(t)(D(A)) C D(A) for all
t ^ 0, and AR(t)x = R(t)Ax for all x e D(A) and t ^ 0. (R3) For each x e D(A) and all t ^ 0 the resolvent equations holds:
t
w* = a n * + / «(t - .)
0
Note that when a(t) = 1, R(t) corresponds to a C0-semigroup. If there exists an integral resolvent for (1), then a mild solution of (1) may be obtained by the formula
t
x(t) = f (t) + A j R (t - s) f (s)ds, t ^ 0.
0
In fact, this supposes that R (t) is an integral resolvent for (1); let f eC (R+,X) and x eC (R+,X) be a mild solution for (1). Then R * f is well-defined and continuous and we obtain, from (R3) and (1):
a * x = (R — Aa * R) * x = R * x — R * Aa * x = R * f.
Hence, R * f eC (R+, [D (A)]), and from (1) we obtain
t
x(t) = f (t) + A J R (t - s) f (s)ds, t > 0. 0
The following result establishes the relation between well-posedness and existence of an integral resolvent. In what follows, % denotes the range of a given operator.
Theorem 1. [7, Theorem 2.4] Equation (1) is well-posed if and only if (1) admits an integral resolvent (R (t))t>0. If this is the case, we have, in addition, n (a * R (t)) C D (A) for all t> 0 and
t
W* = « « x + Afa it - » (4)
0
for each x e X, t > 0.
Definition 4. An integral resolvent ( R (t))t>0 is called exponentially bounded if there exist M > 0 and u e R, such that ||^^ (t)\\ ^ Me for all t > 0 and the pair ( M, u) is called the type of (R (t))t>0. The growth bound of (R (t))t>0 is
U0 :=inf {u e R, \\R (i)|| ^ Meut, t> 0,M > 0};
if u0 < 0, the integral resolvent is called exponentially stable.
Note that, contrary to the case of the C0-semigroup, an integral resolvent for (1) does not need to be exponentially bounded (see [3], [11], [7]). However, there are verifiable conditions guaranteeing that (1) possesses an exponentially bounded integral resolvent.
Remark 1. Note that if a e C~ (R+) Pi L1 (R+) with a (0) = 1 and such that a (A) admits zeros with arbitrary large real part, the problem (1) cannot admit an exponentially bounded integral resolvent (For more details see [11, Page 45-46]).
We will use the Laplace transform at times; suppose that g: R+ ^ X is measurable and there exist M > 0, u e R, such that \\g (i)|| ^ MeMt for almost all > 0; then the Laplace transform
(X
g (A)= i e-xtg (t)dt,
exists for all A g C with Re A > u.
A function a G L]oc (R+) is u (resp. w+)-exponentially bounded if
oo
f e-MS \a (s)| ds < for some u g R (resp. u > 0) (see, e. g [6, p. 113]).
The following proposition, stated in [7, Theorem 3.1], establishes the relation between an integral resolvents and the Laplace transforms:
Proposition 1. Let a G C (R+) be u-exponentially bounded and let (R (t))t>0 C £ (X) be strongly continuous and exponentially bounded,
such that the Laplace transform R (A) exists for A > u. Then (R (t))t>0 is an integral resolvent of (1) if and only if the following conditions hold:
1) a (X) = 0 and g p (A), for all X > u, where p (A) is the set
a (A)
resolvent of A.
2) K (A) : = („ I — ^ called the resolvent associated to R(t)
\a(X) /
satisfies:
o
1 ~ A-1 f- xt
I - A^ x = e-xtR (t) xdt,
\a(X)
0
for all x G X and all X > u.
Under these assumptions, the Laplace transform of R (■) is well-defined and it is given by R (X) = K (A) for all X > u.
2. The Domain of A. We have the following characterization of D (A) given in [11]:
Proposition 2. Let equation (1) admit an integral resolvent family with growth bound u (such that the Laplace transform of the resolvent exists for X > u) for u-exponentially bounded a G Lfoc (R+). Set, for 0 < 9 < 2 and £ > 0,
% Re A>w + e, |arg A| ^ öl.
[a (X) J
Then the following characterization of D (A) holds:
D (A) = < x G X: lim p,A (p,I — A)-1 x exists >
For more details concerning the proof of this proposition, see
[11, Corollary I.1.6] and the proof of the [11, Theorem 1.4].
t
Without loss of generality, we may assume that / la ( s)| ds = 0 for all
0
t > 0. Otherwise, we would have, for some t0 > 0, that a (t) = 0 for almost all t e [0, t0] and, thus, by definition of a integral resolvent, R (t) = 0 for t e [0, i0]. This implies that A is bounded, which is the trivial case with X = D (A).
In what follows, we will use the following assumption on , such that
t
J la (s)|ds = 0, for all i > 0.
0
Assumption H: There exist £a > 0 and ta > 0, such that for all 0 < t ^ t„, we have:
J a (t — s) a (s) ds > ea J la (s) Ids. 00
This is the case for functions a that satisfy a (I) C (0,1] at some interval I = [0, ta).
In fact, if a(t) e (0,1] for all t e (0,1], there exists 0 < ea ^ 1
t
0
t
f a (t — s) a (s) ds ^ £a f la (s) | ds for all
0
and ta > 0, such that we have 0 < t^ ta.
t i
For all t e (0,1], then we have J a(s) ds ^ J a (s) ds ^ 1. So,
00
t t
a (s) ds ^ — a (t — s) a (s) ds.
J Sa J
00
It is necessary and sufficient that
t
1 ^ — a (t — s) a (s) ds.
£a J 0
Then we have
t
J a (t — s) a (s) ds ^ £a. 0
t
t
1 t t Thus, — fa (t — s) a (s) ds ^ 1 ^ f a (s) ds.
£a 0 0
Hence, there exist ea > 0 and ta > 0, such that we have, for all
0 < t ^ ta:
t t / a(t — s)a (s) ds ^ £a |a (s) | ds.
For almost all reasonable functions in applications, it is easy to see that they satisfy this assumption. There are, nonetheless, examples of functions that do not.
Define the set D (A) as follows
~ , .. , ^ -,. R (t) x — a (t)x . ,
D (A) = {x eX: lim —V-. ,\J exists}.
v ; { t^o+ (a * a) (t) }
If |a (i)| is continuous and nondescreasing and
IIR WII
limsup -——— < to, t^o+ |a (i)|
then
D (A) = {x e X: lim R (t) x — a ^ X = Ax}. (5)
v ; 1 t^o+ ( a * a) (t) } v y
This was proved in [8], [7]. Note that in the case of semigroups: a (t) = 1 then ( a * a) ( ) = and in the case of cosine families: a( ) = then (a * a) (t) = , we have the well-known result that D (A) = D (A) (see [12], [4]). In what follows, we prove that this is the case in general.
Theorem 2. Under Assumption H, we have D (A) = D (A) and
R ( ) x ( ) x
lim —V-w \ ' = Ax.
t^o+ (a * a) (t)
Proof. Let x e D (A) and let there exist ea > 0 and ta > 0, such that for all 0 < t ^ ta we have
a(t — s)a (s) ds ^ ea |a (s)lds.
For an arbitrary e > 0, let t e (0, ta], such that ||R ( s) Ax — a (s) AxW < e for all s e [0, t] due to the fact
lim HR (s) — a (s) IH =0 by the strong continuity of R (■) and a (■) and the property (R1). Then
R (t) x — a (t) x
(a * a) (t)
- Ax
1
(a * a) (t)
( R (t) x — a (t) x — (a * a) (t) Ax)
1
(a * a) (t)
a (t — s) AR(s)xds — a(t — s) a (s
1
(a * a) (t)
(a * a) (t)
a (t — s) R(s)Axds — a(t — s) a (s
a(t — s) ( R(s) — a(s
<
<
l(a *a) (t)l
la (t — s )| \\( R( s) — a( s))Ax\\ds ^
l(a *a) (t)l
la (t — s)| ds
= 77-TVT7 la ( S)| (is ^ — <
l(a * a) (t)l J ^ £a '
0
due to the properties (R2), (R3) and Assumption H. This shows that D (A) CD (A).
Conversely, let x e D (A) and let
n R (t)x — a (t)x
lim -:-t^t-= y.
t^0+ (a * a) (t)
Under Assumption H, we have
t
t
t
t
t
1
t
t
a(t — s) R(s)xds — x
(a * a) (t) J 0
' 'a (t —
a (t — s) R(s)xds — a (t — s) a (s) xds^j
<
(a * a) (t)
00
« u / )() I \a (f — s R(s ) — a (s))x\\ds^
|(a * a) (t)\ J 0
« w*m\I}a {s)leds « z
t
Then ( ata](t] f a(t — s) R(s)xds ^ x as t ^ 0+. On the other hand, by
(a*a;( ; o
t
Theorem 1, we have J a (t — s) R(s)xds E D (A). Since A is closed, we
0
obtain, by
lim --— a(t — s) R(s)xds = x,
t^o+ (a *a)(t) J v ' K '
0
the following:
1 f 1
lim --rrrAi a(t — s)R(s)xds = lim ---—-(R(t)x — a(t)x) = y.
t^o+ (a * a)(t) J ' K> t^o+ (a * a)(ty K ' K> ' y
0
Finally, y = Ax. Hence, x E D (A) and
R ( ) x ( ) x
lim —y-. // = Ax.
t^o+ (a * a) (t)
□
From now on and in view of this result, we say that the pair (A, a) is a generator of an integral resolvent ( R (t))t^o.
3. The Favard class of A. The following definition, which corresponds to a natural extension, in our context, of the Favard class, is frequently
used in the approximation theory for semigroups and resolvent families (see e.g., [10], [4], [7], [1], [2], [9]).
Definition 5. Let equation (1) admit a bounded integral resolvent (R(t))t>0 on X, for an -exponentially bounded a E Ljoc (R+). We define the Favard spaces (frequency and temporal) associated to (A, a) as follows:
Fa,A :=\x EX/ sup = \x E X/ sup
1
A
(ok1 -A)
-1
a ( A) \a(X)
^-AK (A)x aa ( A
x
< oo
< oo
and
„ . v, ||R(t)x — a(t)x\\
Fa,A '■ = \ x E X/ sup - ; " < ro
o«i l(a * a) (t)|
Remark 2. The Favard class of A with kernel a (t) can be alternatively defined as the subspace of X given by
{x E X/ limsup ^ ^ A | ^ ^ I — A) x < rol. I a (A) \a (A) J J
We prove that Fa,A is stable by R (t) for any scalar kernel a.
Proposition 3. Let equation (1) admit a bounded integral resolvent (R(t))t>0 on X, for an -exponentially bounded a E L]ooc (R+). We have R (t) (F(1,a) C Fa,A, for all t > 0.
Proof. For all x E D(A) and t > 0, from (R2) we have:
A R( ) x = R( ) A x,
and, by [5, Theorem 7], ((/I — A)-1 commutes with R (t) for all / E p(A)).
Since R (t) is bounded, then, under Proposition 1, the following holds for all A > u:
a(A) = 0 and x— e p(A).
Hence, we have
a ( a)
1 I — A 1R (t) = R (t) (
.a (A)
a ( a)
A
1
1
Now, if x G Fa,A, then
sup
1
-A
a ( X) Va ( X)
1 \-i
I — A) x
< oo,
by (6), (R2) and the boundedness of R (t). We have:
sup
\>u
1 ( 1 \-i -A(—-I — A) R (t)x \a(X) /
à(X) Va (X)
^ ||R (t)\\ x sup
\>u
A
à ( X) Va ( X)
(îm1 — A) x
-1
< +œ.
Then R (t) x g Fa,A for all t ^ 0; hence, we deduce that R (t) (Fa,A) c Fa,A for all t ^ 0. □
The proof of the following proposition is immediate.
Proposition 4. The Favard classes of A with kernel a(t), Fa,A, and Fa,A are Banach spaces with respect to the norms
l|x||
Fa
||x|| + sup
\>u
1
A
à( X) \a(X)
im1 -A)
1
x
and
||x| + sup ||R<fxx|
0<t^i \(a * a) (t)\
respectively.
Now we will prove that Fa,A = Fa,A holds for all a G L/oc (R+) satisfying Assumption H.
Theorem 3. Let equation (1) admit a bounded integral resolvent family (R(t))t>0 on X, for an u+-exponentially bounded a G Lfoc (R+), and suppose that Assumption H holds. Then
Fa,A = Fa,A.
Proof. Let Assumption H hold: there exist £a > 0 and ta > 0, such that for all 0 <t < ta:
a(t — s)a (s) ds ^ £a \a (s)\ds.
1
Take x g Fa>A; let \\R(s)\\ ^ M for some M > 0 and all s g [0,t], where 0 < t ^ ta and ta > 1. We have
l(a * a) (t)l ^ £a (1 *lal)(t).
Then (—I P ( .) ^ —. By Theorem 1, Remark 2, under the stability of
I(a *a)(t)I £a Fa,A by R (t):
\\R (t)x — a (t)x\\
l(a * a) (t)l
1
^ 77-:TTT7 ^m SVLP
l(a *a)(t)l
1
l(a *a) (t)l
A a(t — s) R (s) xds
<
a(t — s)R (s) ^ . . a( ^ I — A) xds a (A) \a(A) /
1
l(a * a) (t)l
a ( ) d
lim sup
A^+œ
1(1 \-1 ^MX)1 — A) R W
aa ( A aa ( A (1 *lal)(t)
x
<
M
l(a *a) (t)l
\\R ( s )x\\FnA ^ — \\x\\
Fa
Hence, we obtain x g Fa,A. Conversely, let x G FaA and set
Write
\ R ( ) x — a ( ) x\
SUP i/-w,\| • = Jx < œ.
o<t<i l(a *a)(t)l
1(1 \-i 1 < ax1—A) = ^AK (X)
aa ( X) aa ( X)
a (A)'
for all A > u.
Using the integral representation of the resolvent (see Proposition 1), we obtain:
1
a (A)
AK (A) x = K (A) x — x^x
(a (A))2 v"'"" a (A)
K ( A) x — aa ( A) x a~**~a (A)
-\SR (s) x — a (s) xd ( * a) ( )
t
t
t
a
Since R(t) is bounded, by Theorem 1, and under Assumption H, we have: 1
â(X)
AK (X)x
<
e~Xsds
< 1 ( sup X V t>A
A a ( - ) R ( ) x 0
| R ( ) x - a ( ) x|
sup-f7-TTT1- <
t>o \(a *a)(t)\
1 (L ||x||
\(a *a) (t)\
+ Ja
U "(^ + J.)
/ U V £a /
+ Jx) < TO
with L = sup WAR (i) ||. This implies sup a^AK (X)x
>1
\>U)
< , which ends
the proof. □
Example 1. When a (t) = 1, we recall that (R (t))t>0 corresponds to a bounded C0-semigroup generated by A. In this situation, we obtain
F1
1, A
\x GX/ sup IIXA (XI — A) 1 x|| < to >
I A>0 J
and
F1,A = \ x G X/ sup >0
||T (t)x — x||
<
}
and we have Fi,a = Fi,a. This case is well-known (see e. g., [4]).
Example 2. Let a (t) = b + ct with b > 0 and c > 0 with b + c < 1, satisfying Assumption H; let equation (1) correspond to a solid in the Kelvin-Voigt model (see [11, Page 131]). Since a(\) = j + , where r denotes the Gamma function, and since (a * a) (t) = b2t + bt2 + yi3; then in this situation we obtain
Fa, A = x g X/ sup ^ A>0
A(' — ( X + ^ )A)
1
x
<
and
Fr
a, A
| R ( ) x — ( + ) x|
< x E X/ sup
I 0<t <1 b2t + bt 2 + Ç t 3
<
And we have FaA = FaA thanks to Theorem 3.
Acknowledgment. The authors wish to thank the editor and the referees for thoughtful comments and constructive suggestions.
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Received May 31, 2021. In revised form,, November 12, 2021. Accepted November 29, 2021. Published online December 23, 2021.
Ahmed Fadili
Laboratory LIMATI, Department of Mathematics and Informatics, Polydisciplinary Faculty, Sultan Moulay Slimane University, Mghila, PB 592 Beni Mellal, Morocco Email: ahmed.fadili@usms.ma
Fouad Maragh
Laboratory LAMA, Department of Mathematics, Faculty of Science, Ibn Zohr University,
Boîte Postale 32/S Agadir 80000 Souss-Massa, Morocco Email: f.maragh@uiz.ac.ma
