Continuous dependence of solutions of abstract generalized linear differential equations with potential converging uniformly with a weight

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 x(t) = x̃k + ∫ t a d[Ak]x + fk(t) – fk(a), t ∈ [a,b], k ∈ N, where –∞ < a < b <∞, X is a Banach space, L(X) is the Banach space of linear bounded operators on X , x̃k ∈ X , Ak : [a,b] → L(X) have bounded variations on [a,b], fk : [a,b] → X are regulated on [a,b]. 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 vara Ak 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: Primary 45A05; secondary 34A30; 34N05

where −∞ < a < b < ∞, X is a Banach space, L(X) is the Banach space of linear bounded operators on X, x k ∈ X, A k : [a, b ] → L(X) have bounded variations on [a, b ], f k : [a, b ] → X are regulated on [a, b ]. 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 J. Kurzweil, cf. [8] or [9]).
In particular, we are interested in the situation when the variations var b a A k need not be uniformly bounded. Our main goal here is the extension of Theorem 4.2 from [14] to the nonhomogeneous case. Applications to second order systems and to dynamic equations on time scales are included as well.

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 J. Kurzweil and Z. Vorel in their paper [10] from 1957. In fact, it was a response to the averaging method introduced few years before by M. Krasnoselskij and S.G. Krein [7]. The extension of the averaging method and the problem of the continuous dependence of solutions on input data were the main motivations for J. Kurzweil to introduce his notion of generalized differential equations in [8].
By generalized linear differential equations we understand linear integral equations of the form For X = R m , such equations are special cases of equations introduced in 1957 by J. Kurzweil (see [8]) in connection with the advanced study of continuous dependence properties of ordinary differential equations (see also [10]). Linear equations of the form (1.1) have been in the finite dimensional case thoroughly treated byŠ. Schwabik, M. Tvrdý and M. Ashordia (see e.g. [17], [21] and [1]).
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 byŠ. Schwabik in a series of papers [18]- [20] written between 1996 and 2000. Some of the needed complements have been added in our paper [13].
Taking into account the closing remark in [19], we can see that the following basic existence result is a particular case of [

2)
where I X stands for the identity operator on X. In such a case x is regulated on [a, b ], x − f has a bounded variation on [a, b ] 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 [a, b ] and each x ∈ X and we consider a sequence of equations depending on a parameter k ∈ N In [14] we proved the following two theorems. The former one deals with the case that the variations of A k are uniformly bounded.
Then the equation has a unique solution x on [a, b ]. Moreover, for each k ∈ N sufficiently large, the equation has a unique solution x k on [a, b ] and (1.5) holds.
Let us recall the following observation. Proof follows from the obvious inequality The only known result (cf. [14,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 more detailed list of related references, see [14].

Preliminaries
Throughout these notes X is a Banach space and L(X) is the Banach space of bounded linear operators on X. By · X we denote the norm in X. Similarly, · L(X) denotes the usual operator norm in L(X).
Assume that −∞ < a < b < ∞ and [a, b ] denotes the corresponding closed interval.
In what follows, by an integral we mean the Kurzweil-Stieltjes integral. Let us recall its definition. As usual, a partition of [a, b ] is a tagged system, i.e., a cou- We say that J ∈ X is the Kurzweil-Stieltjes integral (or shortly KS-integral) of g with Analogously, we define the integral b a F d[g] using sums of the form

Main result
Our main result is based on the following lemma which is an analogue of the assertion formulated for ODEs by Kiguradze in [6, Lemma 2.5]. Its variant was used also in the study of FDEs by Hakl, Lomtatidze and Stavrolaukis in [5, Lemma 3.5].

. Lemma. Let
Then there exist r * > 0 and k 0 ∈ N such that Proof. Assume that (3.1) is not true, i.e. assume that for each n ∈ N there are k n ∈ N and y n ∈ G([a, b ], X) 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 u n (a) X < 1 n for all n ∈ N. Hence, and and, in particular, lim Moreover, the equalities (3.6) and (3.7) yield Consequently, Now, let n ∈ N be fixed. We have We claim that lim Indeed, by (3.10) and Proposition 2.
for t ∈ [a, b ] and k ≥ k 0 . Using (3.1) we deduce that the inequality holds for each f ∈ G ([a, b ], X). 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 for all k ∈ N. In particular, (3.18) is true. However, if where the right hand side evidently tends to ∞ for k → ∞.

Moreover, the functions (3.20) and (3.21) provide us with the argument explaining that the condition f ∈ BV ([a, b ], X) in Theorem 3.2 can not be extended to f ∈ G([a, b ], X).
Indeed, consider the equations and Let k ∈ N be fixed. It is not difficult to verify that the solution to (3.24) on [0, 1] is given by for m = 1, . . . , n k −1. Since Similarly, for m = 2, 3, . . . , n k −1 we have and hence From these formulas we can deduce that if m is even, while for m odd and m > 1 we get In particular, x k (1) = c n k −1,k exp(−4/k) for k ∈ N. Using the above relations and the definition of f , we get if n k is even, and On the other hand, like in (3.22), we have where the right-hand side tends to ∞ when k → ∞. Consequently, the sequence x k (1) cannot have a finite limit for k → ∞. Then SV b a (F ) is said to be the semi-variation of F on [a, b ] (cf. e.g. [4]). 2 It is clear that if F ∈ BV ([a, b ], L(X)) then F has bounded semi-variation on [a, b ] while the reversed implication is not true in general (cf. [22,Theorem 2], [12]). By [18] and [13], the Kurzweil- x 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 ([a, b ], X), cf. [19] and [20]. 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 f : [a, b ] → X could be defined using However, it may be shown (cf. [12]) that, in this case, f has a bounded semi-variation if and only f ∈ BV ([a, b ], X). Therefore, the possible replacement of the condition f ∈ BV ([a, b ], X) in Theorem 3.1 by the requirement that f has a bounded semi-variation is not interesting.

Some applications
Second order measure equations.
Let Y be a Banach space, y, z ∈ Y , P, Q ∈ BV ([a, b ], L(Y )) and g, h ∈ BV ([a, b ], Y ). Consider the following system of generalized linear differential equations Put X = Y × Y and (y, z) X = y Y + z Y for (y, z) ∈ X and define functions A : [a, b ] → L(X) and f : [a, b ] → X by A(t)(y, z) = P (t) z, Q(t) y ∈ X and f (t) = (g(t), h(t)) ∈ X for y, z ∈ Y and t ∈ [a, b ]. (4.2) Clearly, and system (4.1) can be reformulated as equation (1.1), where x = ( y, z) and x = (y, z) 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 I Y stands for the identity operator on Y. Indeed, assume e.g. that To prove its surjectivity, assume first (4.3) and let (u, v) ∈ X be given. Put Then, y − ∆ − P (t)z = u and To summarize, according to the Banach theorem, the operator Now, consider the systems , Y ) and k ∈ N. Assume that (4.3) or (4.4) is true and and 2) (however replace P, Q, g, and h by P k , Q k , g k and h k , respectively). It is easy to see that then the assumptions of Theorem 3.2 are satisfied. Therefore, we can state the following assertion.
In [11], Meng and Zhang investigated continuous dependence on a parameter k for secondorder linear measure differential equations of the form where µ k are normalized measures on [0, 1] (generated by functions of bounded variation on [0, 1] and right-continuous in (0, 1)), y, z ∈ R and y • stands for the generalized right-derivative of y. The main result of [11] is Theorem 1.1 which states that the weak* convergence µ k → µ implies the uniform convergence y k ⇒ y of the corresponding solutions, the weak* convergence y • k → y • and the ending velocity convergence y • k (1) → y • (1). Notice that our systems (4.5) reduce to (4.9) when [a, b ] = [0, 1], X = R, P k (t) = t and Q k (t) = µ k (t) for t ∈ [0, 1] and both g k and h k are constant ( [11,Definition 3.1]). Similarly, if, in addition, P (t) = t and Q(t) = µ(t) for t ∈ [0, 1] and both g and h are constant, then system (4.1) reduces to the second-order linear measure differential equation of the form it follows from our Corollary 4.1 holds for the corresponding solutions of (4.9) and (4.10). Thus, in comparison with Theorem 1.1 in [11], our convergence assumptions are partially stronger. The reason is that our result includes also the uniform convergence of the sequence {y • k }. On the other hand, the weak* convergence which appears in [11] is equivalent to the uniform boundedness of the variations var 1 0 µ k which is not required in our case.
where y ∈ R m and P : [a, b ] T → L(R m ), h : [a, b ] T → R m are rd-continuous functions and y ∆ stands for the ∆-derivative. By a solution of (4.11) we understand a function y : [a, b ] T → R m satisfying the integral equation where the integral is the Riemann ∆-integral defined e.g. in [2]. As noticed by A. Slavík (see [16,Theorem 5]), the Riemann ∆-integral can be regarded as a special case of the Kurzweil-Stieltjes integral. More precisely, Let f : [a, b ] T → R m be an rd-continuous function and Then As a consequence, a relationship between the solutions of (4.11) and generalized linear differential equations can be deduced. ([16,Theorem 12]). If y : [a, b ] T → R m is a solution of (4.11) then Then initial value problem (4.11) has a solution y, initial value problems y ∆ k (t) = P k (t) y k (t) + h k (t) , y k (a) = y k , t ∈ [a, b ] T It is not difficult to see that, if a ≤ c < d ≤ b, then

. Proposition
and, consequently, On the other hand, and, analogously, These estimates, together with (4.13) and (4.14) imply that the assumptions of Theorem 3.2 are satisfied. Therefore, the uniform convergence of solutions x k of equations (1.5) to the solution x of (1.1) follows. Since by Proposition 4.2 the solutions of (4.11) and (4.15) are respectively obtained as the restriction of x and x k to [a, b ] T , the proof is complete.

. Remark.
It is worth to mention that Theorem 4.3 given above encompasses Theorem 5.5 from [14]. This is due to the fact that the weighted convergence assumptions in [14,Theorem 5.5] involves not only the supremum sup