In this paper we continue our research from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) on continuous dependence on a parameter k of solutions to linear integral equations of the form , , , where , X is a Banach space, is the Banach space of linear bounded operators on X, , have bounded variations on , are regulated on . The integrals are understood as the abstract Kurzweil-Stieltjes integral and the studied equations are usually called generalized linear differential equations (in the sense of Kurzweil, cf. (Kurzweil in Czechoslov. Math. J. 7(82):418-449, 1957) or (Kurzweil in Generalized Ordinary Differential Equations: Not Absolutely Continuous Solutions, 2012)). In particular, we are interested in the situation when the variations need not be uniformly bounded. Our main goal here is the extension of Theorem 4.2 from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283-303, 2013) to the nonhomogeneous case. Applications to second-order systems and to dynamic equations on time scales are included as well.
MSC: 45A05, 34A30, 34N05.
Keywords:abstract generalized differential equation; continuous dependence; time scale dynamics
In the theory of differential equations it is always desirable to ensure that their solutions depend continuously on the input data. In other words to ensure that small changes of the input data causes also small changes of the corresponding solutions. For ordinary differential equations, in some sense a final result on the continuous dependence was delivered by Kurzweil and Vorel in their paper  from 1957. In fact, it was a response to the averaging method introduced few years before by Krasnoselskij and Krein . The extension of the averaging method and the problem of the continuous dependence of solutions on input data were the main motivations for Kurzweil to introduce his notion of generalized differential equations in .
By generalized linear differential equations we understand linear integral equations of the form
where , X is a Banach space, is the Banach space of linear bounded operators on X, , has bounded variation on , is regulated on and the integrals are understood in the Kurzweil-Stieltjes sense. By a solution of (1.1) we understand a function such that exists and (1.1) is true for all .
For , such equations are special cases of equations introduced in 1957 by Kurzweil (see ) in connection with the advanced study of continuous dependence properties of ordinary differential equations (see also ). In this connection, we want to highlight the recent monograph  bringing a new insight into the topic. Linear equations of the form (1.1) have been in the finite-dimensional case thoroughly treated by Schwabik, Tvrdý and Ashordia (see e.g.[5,6] and ).
Basic theory of the abstract Kurzweil-Stieltjes integral (called also abstract Perron-Stieltjes or simply gauge-Stieltjes integral) and generalized linear differential equations in a general Banach space has been established by Schwabik in a series of papers [8-10] written between 1996 and 2000. Some of the needed complements have been added in our paper .
Proposition 1.1Let have a bounded variation on . Then, (1.1) possesses a unique solutionxon for every and every function regulated on if and only if
where stands for the identity operator onX. In such a casexis regulated on , has a bounded variation on and
Primarily we are concerned with the continuous dependence of solutions of generalized linear differential equations on a parameter. In particular, we assume that the given equation (1.1) has a unique solution x for each f regulated on and each and we consider a sequence of equations depending on a parameter ,
where have bounded variation on , are regulated on and for . We are looking for conditions ensuring that (1.4) has a unique solution for each k large enough and the sequence tends uniformly on to x, i.e.
In  we proved the following two theorems. The first one deals with the case that the variations of are uniformly bounded.
Proposition 1.2 [, Theorem 3.4]
Let have bounded variation on , be regulated on and for . Furthermore, assume (1.2),
Then (1.1) has a unique solutionxon . Furthermore, for each sufficiently large there is a unique solution on to (1.4) and (1.5) holds.
Proposition 1.3 [, Theorem 4.2]
Let have bounded variation on and let for . Furthermore, assume (1.2), (1.9) and
Then the equation
has a unique solutionxon . Moreover, for each sufficiently large, the equation
has a unique solution on and (1.5) holds.
Let us recall the following observation.
Lemma 1.4Let have bounded variation on and let (1.10) be satisfied. Then (1.7) is true as well.
Proof The proof follows from the obvious inequality
The only known result (cf. [, Corollary 4.4]) concerning nonhomogeneous equations (1.1), (1.4) and the case when (1.6) is not satisfied requires that X is a finite-dimensional space. The aim of this paper is to fill this gap.
For a more detailed list of related references, see .
Throughout these notes X is a Banach space and is the Banach space of bounded linear operators on X. By we denote the norm in X. Similarly, denotes the usual operator norm in .
Assume that and denotes the corresponding closed interval. A set with is said to be a division of if . The set of all divisions of is denoted by .
A function is called a finite step function on if there exists a division of such that f is constant on every open interval , .
For an arbitrary function we set and
is the variation of f over . If , we say that f is a function of bounded variation on . denotes the Banach space of functions of bounded variation on equipped with the norm .
The function is called regulated on if for each there is such that and for each there is such that . By we denote the Banach space of regulated functions equipped with the norm . For , we put and . Recall that cf.e.g. the assertion contained in Section 1.5 of .
In what follows, by an integral we mean the Kurzweil-Stieltjes integral. Let us recall its definition. As usual, a partition of is a tagged system, i.e., a couple where , and holds for . Furthermore, any positive function is called a gauge on . Given a gauge δ on , the partition P is called δ-fine if holds for all . We remark that for an arbitrary gauge δ on there always exists a δ-fine partition of . It is stated by the Cousin lemma (see e.g. [, Lemma 1.4]).
For given functions and and a partition of , where , , we define
We say that is the Kurzweil-Stieltjes integral (or shortly KS-integral) of g with respect to F on and denote if for every there exists a gauge δ on such that
Analogously, we define the integral using sums of the form
Proposition 2.1Let and .
(i) If and , then exists and
(ii) If and , then exists and
3 Main result
Our main result is based on the following lemma which is an analog of the assertion formulated for ODEs by Kiguradze in [, Lemma 2.5]. Its variant was used also in the study of FDEs by Hakl, Lomtatidze and Stavrolaukis in [, Lemma 3.5].
Lemma 3.1Let for and assume that (1.2) and (1.10) hold.
Then there exist and such that
Proof Assume that (3.1) is not true, i.e. assume that for each there are and such that
We will prove that (3.2) leads to a contradiction. To this aim, first, rewrite inequality (3.2) as
Then, by (3.3) and (3.4) we can immediately see that for all . Hence,
By (3.3) we have
and, in particular,
Moreover, the equalities (3.6) and (3.7) yield
Now, let be fixed. We have
We claim that
Indeed, by (3.10) and Proposition 2.1(ii) we have
follows due to (1.10). Moreover, using Proposition 2.1(i) and (3.8), we get
Now, (3.13) follows immediately from (3.14) and (3.15).
Finally, having in mind Proposition 1.1 (cf. (1.3)) and (3.11), (3.5), and (3.13), we conclude that
This, together with (3.7) and (3.9), implies that , which is impossible as for all . The assertion of the lemma is true. □
Theorem 3.2Let , , and for . Assume (1.2), (1.9), (1.10), and
Then (1.1) has a unique solution on . Moreover, for each sufficiently large, (1.4) has a unique solution on and (1.5) is true.
Proof First, recall that, by Lemma 1.4 our assumption (1.10) implies that (1.7) is true, as well. Therefore, by [, Lemma 4.2], there is such that
and (1.4) has a unique solution for each (cf. Proposition 1.1). By Lemma 3.1 we may choose and in such way that (3.1) holds.
Put for . Then and
for and . Using (3.1) we deduce that the inequality
holds for all . Thus, due to (1.9), (1.10) and (3.17), we have , wherefrom (1.5) immediately follows. The proof of the theorem has been completed. □
Remark 3.3 The proof of Theorem 3.2 could be substantially simplified and also extended to the case if the following assertion was true.
Let for and
holds for each .
Unfortunately, this is in general not true even in the scalar case as shown by the following example that was communicated to us by Ivo Vrkoč.
Example 3.4 Let . For puta
It is easy to verify that
for all . In particular, (3.18) is true. However, if
then f is regulated, and (3.19) is not valid since
where the right-hand side evidently tends to ∞ for .
Moreover, the functions (3.20) and (3.21) provide us with the argument explaining that the condition in Theorem 3.2 cannot be extended to . Indeed, consider the equations
where for and . Obviously, is a solution to (3.23) on and, for any , (3.24) possesses a solution on . Furthermore, conditions (1.10) and (3.17) are satisfied. However, as we will see, does not converge to x.
Let be fixed. It is not difficult to verify that the solution to (3.24) on is given by
for . Furthermore, since
Similarly, for we have
From these formulas we can deduce that
if m is even, while for m odd and we get
In particular, for . Using the above relations and the definition of f, we get
if is even, and
if is odd.
On the other hand, like in (3.22), we have
where the right-hand side tends to ∞ when . Consequently, the sequence cannot have a finite limit for .
Remark 3.5 Reasonable examples of sequences that tend to a function f of bounded variation are provided e.g. by sequences of the form , where tends to and tends to 0.
Remark 3.6 For and , define
Then is said to be the semi-variation ofF on (cf.e.g.).b It is clear that if then F has bounded semi-variation on while the reversed implication is not true in general (cf. [, Theorem 2]). By  and , the Kurzweil-Stieltjes integral is well defined when both functions, A and x, are regulated and A has bounded semi-variation. Therefore, the study of generalized linear differential equations has a good sense also when A is regulated and has bounded semi-variation instead of having , cf. and . However, the possible extension of Theorem 3.1 to such a case remains open.
Analogously to operator valued functions, the semi-variation of a function could be defined using
However, it may be shown that, in this case, f has a bounded semi-variation if and only . Therefore, the possible replacement of the condition in Theorem 3.1 by the requirement that f has a bounded semi-variation is not interesting.
4 Some applications
Second-order measure equations
Let Y be a Banach space, , and . Consider the following system of generalized linear differential equations:
Put and for and define functions and by
and system (4.1) can be reformulated as (1.1), where and is a function with values in X. One can verify that condition (1.2) is satisfied whenever one of the following conditions is true:
where stands for the identity operator on Y.
Indeed, assume e.g. that (4.3) holds and let for some and . Then
By (4.3) the latter equality can happen only if . Consequently , and hence , as well. Similarly, we would show that implies also in the case that (4.4) is satisfied. This shows that the operator is injective.
To prove its surjectivity, assume first (4.3) and let be given. Put
that is, for . Similarly, we can show that for each there is such that also in the case that (4.4) is satisfied. The operator is surjective. To summarize, according to the Banach theorem, the operator possesses a bounded .
Now, consider the systems
where , , and . Assume that (4.3) or (4.4) is true and
Define and for like A and f in (4.2) (however, replace P, Q, g, and h by , , and , respectively). It is easy to see that then the assumptions of Theorem 3.2 are satisfied. Therefore, we can state the following assertion.
Corollary 4.1Assume that (4.3) or (4.4) holds and that (4.8)-(4.10) are satisfied. Then system (4.1) has a unique solution on . Moreover, for each sufficiently large, the system (4.7) has a unique solution on and
In , Meng and Zhang investigated the continuous dependence on a parameter k for second-order linear measure differential equations of the form
where are normalized measures on (generated by functions of bounded variation on and right-continuous in ), and stands for the generalized right-derivative of y. The main result of  is Theorem 1.1, which states that the weak∗ convergence implies the uniform convergence of the corresponding solutions, the weak∗ convergence and the ending velocity convergence .
Notice that our systems (4.7) reduce to (4.11) when , , and for and both and are constant [, Definition 3.1]. Similarly, if, in addition, and for and both g and h are constant, then system (4.1) reduces to the second-order linear measure differential equation of the form
where μ is a normalized measure on and . Obviously, both existence conditions (4.3) and (4.4) are now satisfied. In view of this, assuming that μ and have a bounded variation on and
it follows from our Corollary 4.1 that
holds for the corresponding solutions of (4.11) and (4.12).
Thus, in comparison with Theorem 1.1 in , our convergence assumptions are partially stronger. The reason is that our result includes also the uniform convergence of the sequence . On the other hand, the weak∗ convergence which appears in  includes the uniform boundedness of the variations (cf.e.g. [, Lemma 2.4] or [, Section 26]) which is not required in our case.
Linear dynamic equations on time scales
Let us recall some basics of the theory of dynamic equations on time scales. A nonempty closed subset of ℝ is called time scale. For given , we put . For , we define
The point is said to be right-dense if , while it is left-dense if . A function is rd-continuous in if f is continuous at every right-dense point of and there exists for every left-dense point (see e.g.).
Let us consider the linear dynamic equation
where and , are rd-continuous functions and stands for the Δ-derivative. By a solution of (4.13) we understand a function satisfying the integral equation
where the integral is the Riemann Δ-integral defined e.g. in .
As noticed by Slavík (see [, Theorem 5]), the Riemann Δ-integral can be regarded as a special case of the Kurzweil-Stieltjes integral. More precisely:
Let be an rd-continuous function and
Then holds for .
As a consequence, a relationship between the solutions of (4.13) and generalized linear differential equations can be deduced.
Proposition 4.2 [, Theorem 12]
If is a solution of (4.13) then
is a solution of (1.1), where
Symmetrically, if is a solution of (1.1), withAandfgiven by (4.14), then defined by for is a solution of (4.13).
It is important to mention that, thanks to the properties of , the functions and given by (4.14) are well defined, left-continuous and of bounded variation on .
Using the correspondence stated in Proposition 4.2 and Theorem 3.2 we obtain the following result.
Theorem 4.3Let , for be rd-continuous functions in and let , , be given. Assume that
Then initial value problem (4.13) has a solutiony, the initial value problems
have solutions for all , and
Proof For each and , define
It is not difficult to see that, if , then
On the other hand,
These estimates, together with (4.15) and (4.16) imply that the assumptions of Theorem 3.2 are satisfied. Therefore, the uniform convergence of solutions of equation (1.5) to the solution x of (1.1) follows. Since by Proposition 4.2 the solutions of (4.13) and (4.17) are, respectively, obtained as the restriction of x and to , the proof is complete. □
Remark 4.4 It is worth to mention that Theorem 4.3 given above encompasses Theorem 5.5 from . This is due to the fact that the weighted convergence assumptions in [, Theorem 5.5] involves not only the supremum , but also .
The authors declare that they have no competing interests.
The authors contributed equally to the manuscript and read and approved the final draft.
GA Monteiro has bee supported by the Institutional Research Plan No. AV0Z10190503 and by the Academic Human Resource Program of the Academy of Sciences of the Czech Republic and M Tvrdý has been supported by the grant No. 14-06958S of the Grant Agency of the Czech Republic and by the Institutional Research Plan No. AV0Z10190503. The authors sincerely thank Ivo Vrkoč for his valuable contribution to this paper.
stands, as usual, for the integer part of the nonnegative real number x.
Sometimes it is called also the ℬ-variation ofF on (with respect to the bilinear triple , cf.e.g.).
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