1. Introduction
The aim of this paper is to prove a necessary and sufficient condition in order that a given tube of a Banach space
X be viable for a semilinear differential equation with infinite delay. Namely, let
X be a real Banach space,
the infinitesimal generator of a
-semigroup
,
be a tube in
X with closed values,
. We consider the semilinear differential equation with infinite delay:
with the initial condition
where
is the phase space defined axiomatically,
defined by
for all
. We are interested to find necessary and sufficient conditions in order that
be viable for (
1), i.e., for each
and
with
, there exist a
and at least a mild solution to (
1) satisfying the initial condition
and
for
.
We recall that the function
is a mild solution to (
1) and (
2) if
,
u is continuous on
and satisfying
for
.
The viability problem for the differential equation
has been studied by many authors by using various frameworks and techniques. In this respect, we note the pioneering work of Nagumo [
1] who considered the finite dimensional case,
and
F is continuous. In this context, he showed that a necessary and sufficient condition in order that
be a viable domain for (
4) is the following tangency condition:
for each
. It is interesting to note that Nagumo’s result (or some variant of it) has been rediscovered several times, among others by Brezis [
2], Crandall [
3], Hartman [
4], and Martin [
5]. For the development in this area, we refer the readers to Ursescu [
6], Pavel [
7] and [
8], Pavel and Motreanu [
9], Cârjǎ and Marques [
10], Cârjǎ and Vrabie [
11]. Viability for fractional differential equations was also discussed in [
12,
13]. Brief reviews of the main contributions in this area can be found in [
10,
11]. We emphasize Pavel’s main contribution who was the first who formulated the corresponding tangency condition applying to the semilinear case. More precisely, Pavel [
7] showed that, whenever
A generates a compact
-semigroup and
F is continuous on
, where
D is locally closed in
X, a sufficient condition for viability is
for each
.
As for the functional differential equations, the development was initialed about existence and stability by Travis and Webb [
14,
15] and Webb [
16,
17]. Since such equations are often more realistic to describe natural phenomena than those without delay, they have been investigated in variant aspects by many authors (see, e.g., [
18,
19,
20] and references therein). Pavel and Iacob [
21] discussed the viability problem for semilinear differential equations with finite delay (the case
. They proved that, whenever
A generates a compact
semigroup and
f is continuous from
into X, a necessary and sufficient condition for viability for (
1) is
for each
, each
with
, where
is a locally closed subset in
X. Dong and Li [
22] proved the same result when
f is of Carathéodory type. Necula et al. studied the viability for delay evolution equations with nonlocal initial conditions in [
23].
The purpose of this paper is to discuss the viable problem of the semilinear differential equation with infinite delay (
1). In the study of equations with finite delay, the state space is the space of all continuous functions on
,
, endowed with the uniform norm topology. When the delay is unbounded, the selection of the state space
plays an important role in the study of both qualitative and quantitative theory. A usual choice is a semi-normed space satisfying suitable axioms, which was introduced by Hale and Kato [
24]. For a detailed discussion on the topic, we refer to the book by Hino et al. [
25]. We prove that a necessary and sufficient condition in order that
be viable for (
1) is the tangency condition. We only suppose that
f is of Carathéodory type. The difficulty is that the semi-norm on
is defined axiomatically, and the convergence of a sequence in
cannot be obtained directly. Our result extends and improves that of Pavel and Iacob [
21] who considered the case in which
f is continuous, Dong and Li [
22] for finite delay and
, and also extends the well-known existence result of Hale [
26] who considered the case in which
X is finite dimensional and
. Moreover, using a standard argument based on Zorn’s Lemma, we get the existence of noncontinuable (saturated) mild solutions.
2. Preliminaries
Throughout this paper
X will be a real Banach space,
the generator of a
semigroup
. Then
is exponentially bounded, i.e., there are constants
and
such that
Moreover, if
is a compact semigroup (i.e.,
maps bounded subsets into relatively compact subsets for
), then
is continuous in the uniformly operator topology for
(see Pazy [
27]) and
X is separable (see [
10]). For more details of semigroups of linear operators, we refer the readers to Pazy [
27].
In this paper we will employ an axiomatic definition of the space
introduced by Hale and Kato [
24]. To establish the axioms of space
, we follow the terminology used in [
25]. Thus,
will be a linear space of functions mapping
into
X endowed with a seminorm
. We will assume that
satisfies the following axioms:
(A) If
, is continuous on
and
then for every
, the following conditions hold:
- (i)
is in ;
- (ii)
;
- (iii)
,
where
is a constant;
,
K is continuous and
M is locally bounded and
and
M are independent on
.
(A1) For the function in (A), is a -valued continuous function on .
(B) The space is complete.
We first list the conditions here, for the convenience of reference.
(C1) is the infinitesimal generator of the semigroup . For , is compact.
(C2) is closed valued and for each and , there exist and such that is nonempty for all , and the mapping is closed on . Here is the closed ball centered x with radius r.
(T) (Tangency condition)
for a.e.
and all
with
, where
denotes the distance from
to the subset
.
Since the distance is non-expansive, i.e.,
by standard arguments (see [
8,
21]), Condition (T) is equivalent to
for a.e.
and all
with
.
Remark 1. If then the conditions (C2) means that D is locally closed, and the tangency condition (T) is reduced to (
6)
. We say that a function
is of Caratheodary type if
f satisfies
- (1)
for each , the function is measurable on ;
- (2)
for almost every (a.e.) , is continuous on ;
- (3)
for every
, there is a function
such that
for a.e.
and every
with
.
A Carathéodory type function has the following Scorza Dragoni property which is nothing but the special case of [
28,
29]. We denote by
λ the Lebesgue measure on
and by
, the collection of all Lebesgue measurable sets in
.
Theorem 1. Let be separable metric spaces and or . Let be a function such that is measurable for every and is continuous for almost every . Then, for each , there exists a compact subset such that and the restriction of f to is continuous.
Suppose that
,
and
u is continuous on
. Then the mapping
, from
into
is also continuous. The following result is a kind of variance of Lebesgue derivative type, which is useful in the sequel. We omit the proof since it is similar to that of [
10], Theorem 2.
Theorem 2. Assume that X is a separable Banach space, is closed valued and satisfying (C2), is a semigroup on X and is a function of Carathéodory type. Then there exists a negligible subset Z of such that, for every , one hasfor all functions with , and u is continuous on . 3. Main Result
We are now ready to state our main result of this paper, the necessary and sufficient condition of the viability for semilinear differential equations with infinite delay.
Theorem 3. Suppose that the conditions (C1) and (C2) hold, and f is of Carathéodory type. Then a necessary and sufficient condition in order that be viable for Equation (
1)
is the tangency condition (T). Proof of necessity. Let
Z be given by Theorem 2, let
. Let
such that
. By hypothesis, there exists
with
and a function
u continuous on
, satisfying (
3) with
. Since
for all
, we have
Letting , one obtains the condition (T). ☐
To prove the sufficiency, we need the following lemma.
Lemma 1. Suppose that the hypotheses of Theorem 3 hold. Given with , there exists a with , such that for every positive integer n, there exist an open subset with , an increasing sequence , and an approximate solution on in the following sense:- (i)
;
- (ii)
;
- (iii)
in case while in case for ;
- (iv)
for , where . Moreover, .
Proof. Let
with
. Due to (C2), there exist
and
such that
for
. Define
by
for
and
for
. Then
and
is continuous on
by the axiom (A1). Set
,
and
, where
K and
are the functions appeared in the axiom (A) and (
9) respectively. We may assume that
. Further, on the basis of the definition of
and the continuity of the semigroup
, we may choose
small enough such that
and
where
.
In view of Theorems 1 and 2, we may choose an open set
, with
and
, such that
f is continuous on
, where
Z is the set obtained in Theorem 2. We can also assume that for each
, (
8) and (
9) hold. Fix
. We shall construct
and
by induction. Set
. To simplify notation, we drop
n as a superscript for
etc. Suppose that
u is constructed on
. Then we define
in the following manner. If
, set
, and if
, then we define
as the following two cases.
In view of (
7) and the fact that
one can easily see that
. Choose a number
, such that
Define
. By (
16), there is
such that
Consequently,
can be written as
with
. In this case we define
u on
as
Case 2 :
. In this case we set
By (
8) we see that
. Choose
, such that
Define
. By (
20), there is
such that
Consequently,
can be written as
with
. In this case we define
u on
as
Setting
in case
and
in case
for
. Let us define the step functions
and
as
in case
,
in case
and
for
. Then
can be written as
. By the induction hypotheses,
u can be written in the form
Let us check that
. To do this, we first note that each
, there is an integer
k such that
. Due to (
23), we have
On the basis of the definition of
, (
12), (
13) and the above inequality, we have
hence
. Using (
23) again , we derive
for all
, i.e.,
for
. Thus, properties (ii), (iii) and (iv) are verified.
To prove property (i), we first note that
exists, since
is increasing and
for all
. Suppose that
, then
. We have to prove
. To do this, we first show that
also exists. In fact, let
. Using (
23) for
and
, we derive
Now given
. Since
, there is
such that
for
with
. By the existence of
, there is a positive integer
such that
for all
. Choose
with the properties: for
,
;
with ;
with .
From (
24) to (
30), we obtain that
for all
, i.e.,
is a Cauchy sequence. Therefore
exists, and
since
is closed for all
. We define
. By (iv) we have
and therefore
. Accordingly,
u is continuous on
, and hence
is continuous on
. Therefore,
.
We assert that
for sufficiently large
n. Indeed, if
, then there are only finite many
since
is closed. Therefore there is a positive integer
such that
for all
. But then
by (
15), which contradicts the fact that
for sufficiently large
n.
We now assume by contradiction that
. We choose
such that
Since
and
as
, there is a positive integer
such that
for all
. On the basis of (
19), we have
for
and
. Letting
in (
33), one obtains an inequality which contradicts (
32). Hence
, which concludes the proof.
Proof of sufficiency. Let
be a sequence of open subsets of
such that
and
for all
. Take
and a sequence of
n-approximate solutions
and
obtained in Lemma 1. Let us define
for
. Then
for all
and
can be written in the form
for all
. Set
Since the semigroup
, is compact and
is uniformly integrable on
, by a standard argument involving a compactness result, it follows that there is a
such that at least on a subsequence we have
uniformly in
. Since
for all
, it follows that
uniformly in
. Let us observe that if
, then
for sufficiently large
n, and then we have
as
. Also we have
as
for all
. Therefore
as
for a.e.
. Moreover,
implies
due to (C2). Finally, passing to limit in (
34), one obtains (
3), which completes the proof. ☐
Remark 2. In [10], the function f is defined on , and not on the whole , which is more general. Here, if we define , and let , where denotes the distance between and and is the number appeared in the proof of Lemma 1. From the proof of Lemma 1 we can see that, if f is defined on , then the result of Theorem 3 still holds. Concerning the continuation of the solution to (
1) satisfying (
2). Recall that a solution
of (
1), with
is said to be a continuation to the right of the solution
to (
1), if
for all
. A solution
u is said to be noncontinuable if it has no proper continuation. Using a standard argument based on Zorn’s Lemma, one can easily verify that, if the hypotheses of Theorem 3 hold, and
is a noncontinuable mild solution to (
1) satisfying (
2), then either
or
. Moreover, the tangency condition (T) is also necessary. Precisely, we have
Theorem 4. Under the hypotheses of Theorem 3, a necessary and sufficient condition in order that for each , and each with , there is a noncontinuable mild solution to (
1)
satisfying (
2)
is the tangency condition (T). Remark 3. Consider (
1)
with finite delay (i.e., the case ). If , then the condition (C2) reduce to “D is locally closed”. We can obtain the following result [22]. Theorem 5. Let be a locally closed subset in a general Banach space, a function of Carathéodory type, and let be the infinitesimal generator of a compact semigroup . Then a necessary and sufficient condition in order that D be a viable domain of (
1)
is the tangency condition (T). Remark 4. If D is open, then the tangency condition (T) is automatically satisfied. In this case, by Theorem 3, one obtains the locally existence result of problem (
1)
and (
2)
, which extends the well-known result of J. K. Hale [26], who considered the case in which X is finite dimensional (i.e., ) and . Theorem 6. Let X be a real Banach space X, a function of Carathéodory type, and let A be the infinitesimal generator of a compact semigroup . Then for each , and each with , the problem (
1)
and (
2)
has a locally mild solution, for some , with .