1. Introduction
The theory of differential equations involving the Stieltjes derivative [
1] instead of the usual derivative has recently developed significantly (we refer to [
2,
3,
4] and the references therein). The reason is the fact that it offers another point of view on the theory of measure differential problems [
1,
5,
6,
7,
8] and, in particular cases, it yields new results for ordinary differential equations, for difference equations, or for impulsive differential equations [
1,
2,
3]. At the same time, it allows one to assert new results for the (also recent) theories of dynamic equations on time scales and generalized differential equations [
7,
9,
10]. This is the main reason for the use of Stieltjes derivatives.
On the other hand, it is clear that passing from single-valued problems to the set-valued setting [
11,
12,
13,
14,
15], thus studying Stieltjes differential inclusions, widens the field of possible applications of theoretical results.
Aware of the significance of periodic boundary conditions when studying real life processes, the present work focuses on the analysis of a first-order differential inclusion with periodic boundary value conditions
involving the Stieltjes derivative with respect to a left-continuous non-decreasing function
and a real function
p, Lebesgue-Stieltjes integrable with respect to
g.
This problem was investigated in the single-valued framework in [
16] and then generalized to the multivalued setting in [
17] under the assumptions that
has convex, compact values and it is a Carathéodory multifunction.
Applying now an idea presented (and used in the study of initial value problems) in [
4], we prove that the existence results in [
17] can be obtained in the less restrictive case where
F is convex, compact-valued, and upper semi-continuous with respect to its second variable, except for a set (which can be dense in
, e.g., Example 3.11 in [
4]), where a condition involving the notion of contingent
g-derivative must be imposed. We also obtain the compactness (in the topology of uniform convergence) of the solution set.
The technical tool used in the proof is a successive approximation, combined with weak compactness criteria in the space of Lebesgue-integrable functions.
By taking specific nondecreasing functions
g, we then derive the existence of solutions for impulsive differential inclusions with periodic boundary assumptions, in both cases, the number of abrupt perturbations being finite or countable (i.e., allowing Zeno behavior, see [
18]) and present an example to illustrate the wide applicability of the developed study. The result provided includes already published results, see for instance [
14,
19,
20,
21] or [
13] (alternative studies for this kind of problems concern equations with variable impulsive time, e.g., [
22], fractional periodic first-order impulsive inclusions, e.g., [
23] or problems with constraints, such as [
24,
25]). The generalization is double: besides imposing convex, compact values and upper semicontinuity everywhere, in all these works the impulsive maps are single-valued, while we allow set-valued impulsive mappings.
Finally, given the equivalence (under a rather convenient hypothesis) of the theory of Stieltjes differential equations with that of measure differential equations [
1,
3] and consequently (cf. [
7], see also [
9]) with the theory of dynamic equations on time scale domains, we derive an existence result for dynamic inclusions on bounded time scales with periodic boundary conditions. We thus generalize several results in the literature, such as the multivalued results in [
10,
26,
27] (or the single-valued results in [
28,
29]) where the multifunction on the right-hand side was supposed to be convex, compact-valued, and upper semicontinuous concerning the second argument.
2. Preliminaries
A function having limits and at every points , is regulated. A collection of regulated functions is said to be equiregulated if for every and every there exists such that for all : whenever (and similarly at the right).
In [
30] (Theorem 5.1) it was proved the following Ascoli-type theorem.
Theorem 1. If an equiregulated sequence of maps converges pointwise, then the convergence is uniform.
Let be the set of the discontinuity points of the nondecreasing left-continuous function g (which is at most countable). Then the measurability w.r.t. the -algebra defined by g will be called g-measurability, stands for the Stieltjes measure generated by g and the Lebesgue-Stieltjes (shortly, LS-) integrability w.r.t. g means the abstract Lebesgue integrability w.r.t. the Stieltjes measure . Let be the space of real functions LS-integrable w.r.t. g on . It is well known that if , the primitive is a function of bounded variation, therefore it is regulated.
Let us see an auxiliary result.
Lemma 1. Let be a collection of -valued functions on , LS-integrable w.r.t. g. If there exists satisfyingthen the set of primitives is equiregulated. Proof. Let
and
. Then there is
such that
whenever
.
It immediately follows that, for every
,
for all
.
As the inequality for the left limit can be proved in the same way, the equiregulatedness is verified. □
In our study, the usual derivative will be replaced by a notion of Stieltjes derivative ([
1], see also [
31]) which has already found significant applications in real life problems (e.g., [
2,
32,
33]). The Stieltjes or
g-derivative of a function
with respect to
g at a point
is (in the case of existence of limits)
Note that if
, it suffices that
u have right limit
in order to have
The particular cases of the theory of Stieltjes differential equations are coming from taking particular non-decreasing functions g. More precisely: when we get ordinary differential equations, in the case where g is a sum of step functions we retrieve difference equations, while for the situation where g is a sum of the identical function with step functions we arrive at impulsive problems. This is the advantage of using Stieltjes derivatives: we are studying various problems under one roof.
The following result concerning the
g-derivative of a limit of a sequence of
g-absolutely continuous functions will be useful in the main proof. A function
is
g-absolutely continuous (see [
34] or [
1]) if for every
there is
such that
for any set
of non-overlapping subintervals of
with
.
Lemma 2. ([
4] (Lemma 3.8)).
Let be a sequence of g-absolutely continuous -valued functions on , pointwisely convergent to some . If there exists such that for all ,then u is also g-absolutely continuous and We rely on an existence result provided in [
17] for the single-valued linear periodic problem
under the nonresonance assumption:
An important matter is to realize that the sign of
has to be considered. As in [
2], the set
is finite and, if we denote its elements by
(and, for the sake of convenience, take
and
), we define
and
Theorem 2. ([
17] (Theorem 2, Remarks 2 and 3)).
Let . Then the problem (2)
has a unique solution, where Let us remind [
16] of the fact that the mapping
is bounded, say by
Moreover there are two positive constants
such that
whence
As we place ourselves in the multivalued framework, some basic notions of set-valued analysis are needed.
A compact, convex-valued mapping is upper semicontinuous at if, for every , there exists satisfying the inclusion , for any .
Remind the reader of a concept which generalizes the Bouligand’s contingent cone to a given set ([
11], page 177) to multifunctions: the contingent derivative [
35] of
at
. It is the set-valued map
defined as follows:
contains all
for which there exist a sequence
convergent to
and a sequence
with
A single-valued function
satisfying
almost everywhere is called a selection of
. Finally, we denote, for a set
, by
3. Main Result
Consider, as announced, the set-valued Stieltjes differential problem with periodic boundary condition
The concept of solution we are searching for is given in [
17] (Definition 4):
is a solution of problem (1) if it is
g-absolutely continuous and satisfies
with
, together with the equality
.
In [
17] (Theorem 4) the existence of solutions of (1) is proved under the hypotheses that
F is convex, compact-valued and upper semicontinuous with respect to its second argument.
We now prove that these assumptions can be relaxed, by using the notion of contingent
g-derivative, recently introduced in [
4].
Definition 1. Let . The contingent g-derivative of at is the set containing all for which there exists a sequence convergent to 0, such that and In the single-valued case, one falls on the concept of
g-derivative (see [
4]), while in the case where
one gets the usual contingent derivative for
,
.
Indeed, if
, one can find a sequence
convergent to
and a sequence
with
Conversely, if
, there exists a sequence
convergent to 0 such that
and
So there exist
convergent to
and
such that
whence
.
Theorem 3. Let satisfy the following hypotheses.
Hypothesis 1. for every regulated function , has g-measurable selections;
Hypothesis 2. - (i)
for every , is convex and compact-valued and it is upper semicontinuous;
- (ii)
for -a.e. , is convex compact-valued and upper semicontinuous on , where the set is empty or there exists a countable family of maps , , such thatand whenever ,
Hypothesis 3. there is a function ϕ, LS-integrable w.r.t. g, such thatfor -almost all and all . Then the periodic Stieltjes differential inclusion (1) admits solutions. Besides, the solution set is compact in the space of g-absolutely continuous functions with respect to the uniform convergence topology.
Proof. Let
and choose a
g-measurable selection
of
. By Hypothesis 3,
and so, Theorem 2 yields the existence of a unique solution
of the periodic Stieltjes differential problem
namely
Next, let
be a
g-measurable (therefore, LS-integrable w.r.t.
g) selection of
. By Theorem 2, there exists a unique solution
of problem
given by
and we continue the process: for each
, choose a selection
of
. There exists a unique solution
of problem
given by
The sequence
is relatively compact in the weak topology of
by (Hypothesis 3), therefore, one can find a subsequence (not relabelled) convergent in this topology to some
. As for all
we derive that, for each
,
So,
tends pointwise to
defined by
Again by Theorem 2,
u is the unique solution of
Let us check that it is a solution of our differential inclusion (1). By basic properties of Lebesgue-Stieltjes integrals, it is g-absolutely continuous.
Since the sequence is relatively weakly compact, one can find a sequence of convex combinations of -a.e. convergent to f, so let us consider a -null set , such that, for every , and the specified sequence of convex combinations of is convergent, at the point t, to .
Define for each
the set
As in [
4],
. Then the set
is of
-null measure and suppose Hypothesis 2.(ii) is verified except for
.
Take now
and consider, in order to prove that
the following possible situations.
Case I.
or (
and
). Then
is convex and compact and
is upper semicontinuous at
. It means that for every
there is
such that
Consequently, for every
one can find
such that
Since is a limit of convex combinations of it follows that, in this case, .
Case II.
and
. There exists some
such that
. Obviously,
and so, there exists a sequence
convergent to 0 such that
for all
k and, as
u is
g-differentiable at
t, by Definition 1,
At the same time, Lemma 2 implies that
For every
, there exists
(depending on
t) such that
and, as
it is the limit, as
, of a sequence
such that for each
,
. For each
m one can thus find
and
with
and also
such that
But
whence
and, as
,
Using Hypothesis 2.(ii) one gets
and the fact that
u is a solution is achieved.
To prove the compactness of the solution set, let us choose a sequence
of solutions. For each
, one can find a selection
of
satisfying:
In the same way as before,
is, by Hypothesis 3, relatively weakly compact in
, thus one can find a subsequence (labelled with the same indexes) convergent in this topology to some
and one gets the pointwise convergence of
towards
, which is the unique solution of
On the other hand, the sequences of the primitives involved when explicitly writing each
, namely
are, due to the inequality (4) combined with Lemma 1, equi-regulated. It follows, by Theorem 1, that the convergence is uniform.
In other words, tends uniformly to .
Looking at the proof of the similar step of the existence part, it is clear that, similarly, it can be shown that u is a solution of (1) and the compactness of the solution set is verified. □
4. Existence Results for Periodic Impulsive Inclusions without Upper Semicontinuity
As already discussed, the theory of Stieltjes differential equations contains, in a particular case, the theory of impulsive equations. In what follows we shall deduce from our main Theorem 3 the existence of solutions for impulsive differential inclusions with periodic boundary conditions, without upper semicontinuity assumption on the multifunction on the right-hand side.
Consider thus a first order periodic differential inclusion
with multivalued jumps
,
, where
and
.
This problem can be seen as a measure differential problem of type (1) with
as a consequence of the definition of the
g-derivative:
Traditionally, a function is a solution of the specified impulsive problem if it is continuous at every , left continuous at every , it has right limit at every point , and, of course, it satisfies the conditions in (5).
We derive from Theorem 3 the following existence result.
Theorem 4. Let satisfy the following hypotheses.
Hypothesis 4. For every regulated function , has measurable selections.
Hypothesis 5. - (i)
for every , is convex, compact valued and it is upper semicontinuous; besides, there exist some constants such that for each - (ii)
for a.e. , is convex, compact-valued and upper semicontinuous on , the set being empty or the union where each satisfies condition (3) with instead of .
Hypothesis 6. There exists a function , Lebesgue-integrable on , such thatfor almost all and all . Then the periodic differential inclusion (5) admits solutions.
Proof. Since all the other hypotheses in Theorem 3 are easy to check, let us only note that
, defined as
satisfies Hypothesis 6. □
But, what is more, from our main result we can deduce the existence of solutions for the first order differential inclusions with periodic boundary conditions when infinitely many instantaneous modifications occur in the behavior of the process, more precisely when there are countably many impulses. We thus significantly generalize the known results, since for periodic impulsive boundary value problems this case is not, as far as the authors know, covered in the literature.
We are thus concerned with
with multivalued jumps
,
, where
and
. Suppose that the set of points where
accumulate is of null Lebesgue measure.
We can write it as a measure differential problem of type (1) with
In this framework, we can deduce the following result.
Theorem 5. Let satisfy the following hypotheses.
Hypothesis 7. For every regulated function , has measurable selections.
Hypothesis 8. - (i)
for every , is convex, compact valued and it is upper semicontinuous; besides, there exist some constants with such that for each - (ii)
for a.e. , is convex and compact for all and is upper semicontinuous on , the set being empty or the union where , satisfy the condition (3) with replacing .
Hypothesis 9. There exists a function , Lebesgue-integrable on , such thatfor almost all and all . Then the periodic differential inclusion (6) admits solutions.
Proof. It is enough to note that, when defining the set E in the proof of Theorem 3, we can suppose that it includes the set of accumulation points of which are not impulsive points (this is -null), while, at the accumulation points belonging to , the upper semicontinuity and the convexity and compactness of the values are checked. By the expression on g in this case, obviously at any t which is not a point where accumulate, . □
Let us study the following periodic impulsive problem without upper semicontinuity.
Example 1. Consider the problemwhereand let be the multifunction given in [4] (Example 3.11), which is not upper semicontinuous, nor convex compact-valued on a dense subset of , but satisfies the Hypotheses 7, 8(ii) and 9 in Theorem 5. If each is defined byit is easy to see that it satisfies the Hypothesis 8(i) so the existence of solutions for the considered problem is guaranteed by Theorem 5. Let us notice that there are countably many impulse moments that accumulate at the middle of the interval, so, it is a hybrid system with a Zeno behavior, which cannot be studied using the classical theory of impulsive differential inclusions. Remark 1. It is easy to construct from here an example of Stieltjes differential problem of type (1) for which Theorem 3 applies, by considering g and F given by (7).
5. Existence Result for Periodic Dynamic Inclusions on Time Scales without Upper Semicontinuity
We prove in this section that our Theorem 3 also yields an existence result for dynamic inclusions on time scales with periodic boundary conditions (for a comprehensive introduction in the time scales theory, see [
36]).
Let be a bounded time scale (i.e., a nonempty and closed subset of , with the standard topology inherited from ). Suppose . For two points in , let be the time scales interval.
The forward jump operator is defined by . By convention, .
A point
is called right-dense, resp. right-scattered, if
, resp.
. It is known (Lemma 3.1 in [
37]) that the set of right-scattered points of
is at most countable; let
be its elements. Suppose that the set of right-dense points where the right-scattered points accumulate is of null Lebesgue measure (it happens, for instance, if
is a typical time scales domain, see [
2]).
The Lebesgue measure on
,
, was introduced and studied in [
37], while for the Lebesgue
-integral we refer to [
36,
37,
38,
39] or [
40].
Let us recall that, by [
37] (Theorem 5.2), if
, for any
,
On the other hand, following [
41], let
be defined by
which is left-continuous and nondecreasing and satisfies
It follows, by the definition of the Lebesgue-Stieltjes integral, that
Definition 2. The function is said to be Δ
-differentiable at if there is an element (the Δ
-derivative of f at t) such that for any there is a neighborhood of t such that In [
39] (Theorem 1.3) it was proved that:
(i) f is continuous at any point where it is -differentiable;
(ii) at a right-scattered point where
f is left-continuous, it is
-differentiable and
In this section, the focus is on a dynamic inclusion on time scales with periodic boundary condition
where
is Lebesgue
-integrable and
.
We say that
is a solution for the considered problem if it is
-absolutely continuous, there exists a selection
f of
Lebesgue
-integrable with
,
-a.e. and
. For completeness, let us recall [
10] that
u is
-absolutely continuous if for every
there is
such that
for any set
of non-overlapping intervals with
and
For any function
we define its Slavik extension [
41]
Let us also extend, in the same way, the multifunction in order to get , .
Lemma 3. A function is a solution of problem (9)
if and only if is a solution ofwhich is a problem of type (1).
Proof. Let
u be a solution of (9). Then one can find a selection
f of
, Lebesgue
-integrable on
, such that
Since if
u is left-continuous at the right-scattered points one can see (as in [
2]) that, for every
,
so it follows that
for every
. At the same time, the
g-derivative of
does not make sense on
as
g is constant, but
, therefore
and on the set where the equality holds,
since
on
.
Also, and , so is indeed a solution of (10).
Conversely, let
y be a solution of (10). Then the function
given by
for every
is a solution on the dynamic problem (9) since
□
This characterization allows us to deduce the existence of solutions of (9) from Theorem 3.
Theorem 6. Let satisfy the assumptions below.
Hypothesis 10. For every regulated function , has measurable selections.
Hypothesis 11. - (i)
for every right-scattered , is convex, compact valued and it is upper semicontinuous;
- (ii)
for -a.e. right-dense , is convex compact-valued and upper semicontinuous on , the set being empty or the union where for each , satisfies the inclusion (3) with instead of ;
Hypothesis 12. there exists a function ϕ, Lebesgue Δ
-integrable on , such that for -almost all and all .
Then the dynamic inclusion with periodic boundary conditions (9) has solutions.
Proof. When satisfies our assumptions, satisfies the hypotheses of Theorem 3. To see this, it suffices to note that, when defining the set E in the proof of Theorem 3, we may consider that it contains the right-dense points where accumulate (this is a -null set). At the accumulation points belonging to we have upper semicontinuity and convex compact values. Finally, at any t which is not a point where accumulate, .
It follows that the Stieltjes differential problem (10) has at least one solution whence, by Lemma 3, the dynamic problem on time scales (9) has solutions. □
Following the same idea as in Example 1, let us see a periodic dynamic problem on time scales without upper semicontinuity for which the existence of solutions can be provided.
Example 2. Consider the time scale domainand the impulsive problemwhere and is the multifunction given in [4] (Example 3.11) for , respectively The set of right-dense points where the right-scattered points accumulate is of null Lebesgue measure (it consists in ).
F obviously satisfies the Hypotheses 10 and 12 in Theorem 6.
Besides, for any right-scattered point (i.e., for any ), is compact convex-valued and constant, thus upper semicontinuous.
At almost any right-dense point (i.e., almost everywhere in ), is not upper semicontinuous, nor convex compact-valued on a dense subset of , but satisfies the Hypothesis 11(ii), thus the existence of solutions for the considered problem is a consequence of Theorem 6.
6. Conclusions
In this work, an existence theory was developed for first-order differential inclusions with periodic boundary conditions, leading to a very general result (Theorem 3) due to the substitution of the usual derivative of the state with the Stieltjes derivative related to a nondecreasing function. For a particular non-decreasing function, the existence of solutions of impulsive differential problems with finitely or countably many impulsive moments and set-valued impulses was derived (Theorems 4, respectively 5). Furthermore, being aware of the tight connection between the theory of Stieltjes differential equations and that of dynamic equations on time scales, an existing result for dynamic inclusions on time scales with periodic conditions on the boundary was provided (Theorem 6). Future directions connected to this work could focus on generalized differential equations with similar conditions on the boundary.
Besides the generality of our outcome following from the remark that the measure differential equations encompass a large number of classical problems (such as ordinary differential equations, difference equations, impulsive and generalized differential equations), we highlight that this is the first study of differential inclusions with periodic boundary conditions involving Stieltjes derivative where the hypotheses on the right-hand side to have convex compact values and to be upper semicontinuous are relaxed.