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 3.1 Let for and assume that (1.2) and (1.10) hold.
Then there exist
Assume that (3.1) is not true, i.e.
assume that for each
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
By (3.3) we have
and, in particular,
Moreover, the equalities (3.6) and (3.7) yield
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 3.2 Let
, and for
(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.
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
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.
. 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.
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.
It is easy to verify that
. In particular, (3.18) is true. However, if
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.
be fixed. It is not difficult to verify that the solution to (3.24) on
is given by
. Furthermore, since
From these formulas we can deduce that
is even, while for m
. Using the above relations and the definition of f
, we get
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.
Then is said to be the semi-variation of F 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.