Abstract
In this paper we continue our research from (Monteiro and Tvrdý in Discrete Contin. Dyn. Syst. 33(1):283303, 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 KurzweilStieltjes 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):418449, 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):283303, 2013) to the nonhomogeneous case. Applications to secondorder 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 dynamics1 Introduction
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 [1] from 1957. In fact, it was a response to the averaging method introduced few years before by Krasnoselskij and Krein [2]. 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 [3].
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 KurzweilStieltjes 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 [3]) in connection with the advanced study of continuous dependence properties of ordinary differential equations (see also [1]). In this connection, we want to highlight the recent monograph [4] bringing a new insight into the topic. Linear equations of the form (1.1) have been in the finitedimensional case thoroughly treated by Schwabik, Tvrdý and Ashordia (see e.g.[5,6] and [7]).
Basic theory of the abstract KurzweilStieltjes integral (called also abstract PerronStieltjes or simply gaugeStieltjes integral) and generalized linear differential equations in a general Banach space has been established by Schwabik in a series of papers [810] written between 1996 and 2000. Some of the needed complements have been added in our paper [11].
Taking into account the closing remark in [9], we can see that the following basic existence result is a particular case of [[9], Proposition 2.10].
Proposition 1.1Lethave a bounded variation on. Then, (1.1) possesses a unique solutionxonfor everyand every functionregulated onif and only if
wherestands for the identity operator onX. In such a casexis regulated on, has a bounded variation onand
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 [12] we proved the following two theorems. The first one deals with the case that the variations of are uniformly bounded.
Proposition 1.2 [[12], Theorem 3.4]
Lethave bounded variation on, be regulated onandfor. Furthermore, assume (1.2),
Then (1.1) has a unique solutionxon. Furthermore, for eachsufficiently large there is a unique solutiononto (1.4) and (1.5) holds.
The second result from [12], inspired by Opial’s paper [13], concerns the situation when variations of (1.6) need not be uniformly bounded and (1.1) and (1.4) reduce to homogeneous equations.
Proposition 1.3 [[12], Theorem 4.2]
Lethave bounded variation onand letfor. Furthermore, assume (1.2), (1.9) and
Then the equation
has a unique solutionxon. Moreover, for eachsufficiently large, the equation
has a unique solutiononand (1.5) holds.
Let us recall the following observation.
Lemma 1.4Lethave bounded variation onand 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. [[12], Corollary 4.4]) concerning nonhomogeneous equations (1.1), (1.4) and the case when (1.6) is not satisfied requires that X is a finitedimensional space. The aim of this paper is to fill this gap.
For a more detailed list of related references, see [12].
2 Preliminaries
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 [9].
In what follows, by an integral we mean the KurzweilStieltjes 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 δfine if holds for all . We remark that for an arbitrary gauge δ on there always exists a δfine partition of . It is stated by the Cousin lemma (see e.g. [[5], Lemma 1.4]).
For given functions and and a partition of , where , , we define
We say that is the KurzweilStieltjes integral (or shortly KSintegral) 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
Some basic estimates for the KSintegrals are summarized in the following proposition. For the proofs, see [[12], Proposition 2.1] and [[11], Lemma 2.2].
For more details concerning the abstract KSintegration and further references, see [810,14] and [11].
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 [[15], Lemma 2.5]. Its variant was used also in the study of FDEs by Hakl, Lomtatidze and Stavrolaukis in [[16], Lemma 3.5].
Lemma 3.1Letforand assume that (1.2) and (1.10) hold.
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
where
Then, by (3.3) and (3.4) we can immediately see that for all . Hence,
Now, denote
and
By (3.3) we have
and, in particular,
Moreover, the equalities (3.6) and (3.7) yield
i.e.
where
We claim that
Indeed, by (3.10) and Proposition 2.1(ii) we have
wherefrom
follows due to (1.10). Moreover, using Proposition 2.1(i) and (3.8), we get
and, hence,
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
i.e.
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, , andfor. Assume (1.2), (1.9), (1.10), and
Then (1.1) has a unique solutionon. Moreover, for eachsufficiently large, (1.4) has a unique solutiononand (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 [[12], 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.
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.
Then
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.
and define
and
It is easy to verify that
and
for all . In particular, (3.18) is true. However, if
then f is regulated, and (3.19) is not valid since
where the righthand 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
and
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
where
and
we have
and hence
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
and
On the other hand, like in (3.22), we have
where the righthand 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.
and
Then is said to be the semivariation ofF on (cf.e.g.[17]).^{b} It is clear that if then F has bounded semivariation on while the reversed implication is not true in general (cf. [[18], Theorem 2]). By [8] and [11], the KurzweilStieltjes integral is well defined when both functions, A and x, are regulated and A has bounded semivariation. Therefore, the study of generalized linear differential equations has a good sense also when A is regulated and has bounded semivariation instead of having , cf.[9] and [10]. However, the possible extension of Theorem 3.1 to such a case remains open.
Analogously to operator valued functions, the semivariation of a function could be defined using
However, it may be shown that, in this case, f has a bounded semivariation if and only . Therefore, the possible replacement of the condition in Theorem 3.1 by the requirement that f has a bounded semivariation is not interesting.
4 Some applications
Secondorder 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
Clearly,
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
i.e.
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
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 solutionon. Moreover, for eachsufficiently large, the system (4.7) has a unique solutiononand
In [19], Meng and Zhang investigated the continuous dependence on a parameter k for secondorder linear measure differential equations of the form
where are normalized measures on (generated by functions of bounded variation on and rightcontinuous in ), and stands for the generalized rightderivative of y. The main result of [19] 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 [[19], Definition 3.1]. Similarly, if, in addition, and for and both g and h are constant, then system (4.1) reduces to the secondorder 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 [19], 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 [19] includes the uniform boundedness of the variations (cf.e.g. [[20], Lemma 2.4] or [[21], 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 rightdense if , while it is leftdense if . A function is rdcontinuous in if f is continuous at every rightdense point of and there exists for every leftdense point (see e.g.[22]).
Let us consider the linear dynamic equation
where and , are rdcontinuous functions and stands for the Δderivative. By a solution of (4.13) we understand a function satisfying the integral equation
where the integral is the Riemann Δintegral defined e.g. in [22].
As noticed by Slavík (see [[23], Theorem 5]), the Riemann Δintegral can be regarded as a special case of the KurzweilStieltjes integral. More precisely:
Letbe an rdcontinuous function and
and
As a consequence, a relationship between the solutions of (4.13) and generalized linear differential equations can be deduced.
Proposition 4.2 [[23], Theorem 12]
Ifis a solution of (4.13) then
is a solution of (1.1), where
Symmetrically, ifis a solution of (1.1), withAandfgiven by (4.14), thendefined byforis 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, leftcontinuous 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, forbe rdcontinuous functions inand let, , be given. Assume that
Then initial value problem (4.13) has a solutiony, the initial value problems
It is not difficult to see that, if , then
and, consequently,
On the other hand,
and, analogously,
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 [12]. This is due to the fact that the weighted convergence assumptions in [[12], Theorem 5.5] involves not only the supremum , but also .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally to the manuscript and read and approved the final draft.
Acknowledgements
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. 1406958S 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.
End notes

stands, as usual, for the integer part of the nonnegative real number x.

Sometimes it is called also the ℬvariation ofF on (with respect to the bilinear triple , cf.e.g.[8]).
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