Generalized linear differential equations in a Banach space: Continuous . . .

This paper deals with integral equations of the form 
 \begin{eqnarray*} x(t)=\tilde{x}+∫_a^td[A]x+f(t)-f(a), t∈[a,b], 
\end{eqnarray*} 
in a Banach space $X,$ where $-\infty\ < a < b < \infty$, $\tilde{x}∈ X,$ 
$f:[a,b]→X$ is regulated on [a,b] and $A(t)$ is for each $t∈[a,b], 
$ a linear 
bounded operator on $X,$ while the mapping $A:[a,b]→L(X)$ has a bounded variation 
on [a,b] Such equations are called generalized linear differential equations. Our aim 
is to present new results on the continuous dependence of solutions of such equations 
on a parameter. Furthermore, an application of these results to dynamic equations on time 
scales is given.

in a Banach space X, where −∞ < a < b < ∞, x ∈ X, f : [a, b ] → X is regulated on [a, b ], and A(t) is for each t ∈ [a, b ] a linear bounded operator on X, while the mapping A: [a, b ]→L(X) has a bounded variation on [a, b ]. Such equations are called generalized linear differential equations. Our aim is to present new results on the continuous dependence of solutions of such equations on a parameter. Furthermore, an application of these results to dynamic equations on time scales is given.

Introduction
The theory of generalized differential equations in Banach spaces enables the investigation of continuous and discrete systems, including the equations on time scales and the functional differential equations with impulses, from the common standpoint. This fact can be observed in several papers related to special kinds of equations, such as e.g. those by Imaz and Vorel [14], Oliva and Vorel [25], Federson and Schwabik [7], Schwabik [27] or Slavík [33]. This paper is devoted to generalized linear differential equations of the form (1.1) in a Banach space X. A complete theory for the case when X = R m can be found, for instance, in the monographs by Schwabik [27] or Schwabik, Tvrdý and Vejvoda [32]. See also the pioneering paper by Hildebrandt [12]. Concerning integral equations in a general Banach space, it is worth to highlight the monograph by Hönig [13] having as a background the interior (Dushnik) integral. On the other hand, dealing with the Kurzweil-Stieltjes integral, the contributions by Schwabik in [29] and [30] are essential for this paper.
In the case X = R m (i.e. for ordinary differential equations), fundamental results on the continuous dependence of solutions on a parameter based on the averaging principle have been delivered by Krasnoselskii and Krejn [16], Kurzweil and Vorel [18], Kurzweil [19], Opial [26] and Kiguradze [15]. In particular, the problem of continuous dependence gave an inspiration to Kurzweil to introduce the notion of generalized differential equation in the papers [19] and [20]. For linear ordinary differential equations, the most general result seems to be that given by Opial. An interesting observation is contained in the fundamental paper by Artstein [2]. A different approach can be found in the papers [21]- [23] by Meng Gang and Zhang Meirong dealing also with measure differential analogues of Sturm-Liouville equations and, in particular, describing the weak and weak*continuous dependence of related Dirichlet or Neumann eigenvalues on a potential. After Kurzweil, the problem of continuous dependence on a parameter for generalized differential equations has been treated by several authors, see e.g. Schwabik [27], Ashordia [3], Fraňková [8], Tvrdý [35], [36], Halas [9], Halas and Tvrdý [11]. Up to now, to our knowledge, only Federson and Schwabik [7] (cf. also Appendix to ABFS) dealt with the case of a general Banach space X. Our aim is to prove new results valid also for infinite dimensional spaces. In particular, in Sections 3 and 4 we give sufficient conditions ensuring that the sequence {x n } of solutions of the generalized linear differential equations tends to the solution x of (1.1). The crucial assumptions of Section 3 are the uniform boundedness of the variations var b a A n of A n and uniform convergence of A n to A. In Section 4, we present the extension of the classical result by Opial to the case X = R m , where we do not require the uniform boundedness of var b a A n and the uniform convergence is replaced by a properly stronger concept. Finally in Section 5, we apply the obtained results to dynamic equations on time scales.

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.
For an arbitrary function f : [29, 1.5]. Moreover, it is known that regulated function are uniform limits of finite step functions (see [13,Theorem I.3.1 ]) and that they can have at most a countable number of points of discontinuity (see [13,Corollary 3 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- (D) and α j−1 ≤ ξ j ≤ α j holds for j = 1, 2, . . . , ν(D) . Furthermore, any positive function δ : [a, b ]→(0, ∞) is called a gauge on [a, b ]. Given a gauge δ, the partition P is called We say that I ∈ X is the Kurzweil-Stieltjes integral (or shortly KS-integral) of g with respect to F on [a, b ] and denote Analogously, we define the integral b a F d[g] using sums of the form For the reader's convenience some of the further properties of the KS-integral needed later are summarized in the following proposition.
, L(X)) and g is bounded on [a, b ] ). This proves the assertion (i In addition, we need the following convergence result. Then Proof. Let ε > 0 be given. By [13,Theorem I.3.1 ], we can choose a finite step function g : [a, b ] → X such that g − g ∞ < ε. Furthermore, let n 0 ∈ N be such that For a fixed t ∈ [a, b ], by Proposition 2.1 (i) and (ii), we obtain for n ≥ n 0

Uniformly bounded variations
Given A ∈ BV ([a, b ], L(X)), f ∈ G([a, b ], X) and x ∈ X, consider the integral equation x exists and x satisfies the equality (3.1) for each t ∈ [a, b ].
For our purposes the following property is crucial In particular, taking into account the closing remark in [29] we can see that the following result is a particular case of [29, Proposition 2.10].

. Proposition. Let
In addition, the following two important auxiliary assertions are true: and has at most finitely many elements. As c A = 0 is impossible, this proves (3.4). iii) Now, let x be a solution of (3.1). Put B(a) = A(a) and and (cf. Proposition 2.1 (i))

. Lemma. Let
for all t ∈ (a, b ] and all n ∈ N sufficiently large. Moreover, there is µ * ∈ (0, ∞) such that for all n ∈ N sufficiently large.
} has at most a finite number of elements. Let c A be defined as in (3.4). Then, as by (3.6) lim Thus, By [34, Lemma 4.1-C], this implies that Furthermore, due to (3.2), the relation

is invertible if and only if I−T n (t) is
invertible. Now, let t ∈ D and n ≥ n 0 be given. Then, due to (3.4) and (3.9), we have T n (t) L(X) < 1 4 . Consequently, by [34, Lemma 4.1-C], I − T n (t) and therefore also [I−∆ − A n (t)] are invertible. Moreover, taking into account (3.4) and (3.10), we can see that To summarize, there exists n 0 ∈ N such that for all t ∈ (a, b ] and n ≥ n 0 . This completes the proof. The main result of this section is the following Theorem, which generalizes in a linear case the recent results by Federson and Schwabik [7]) and covers the results known for generalized linear differential equations in the case X = R m . Unlike [3], to prove it we do not utilize the variation-of-constants formula. Therefore it is not necessary to assume the additional and lim First, notice that according to (3.12) and ( Having in mind (3.6), we can see that the relation By (3.11) and by Lemmas 3.2 and 3.3 we have Consequently, using (3.17) and (3.18) we deduce that lim n→∞ w n X = 0. Finally, by (3.12) and (3.16), we conclude that (3.15) is true. We will close this section by a comparison of Theorem 3.4 with two similar available results: Proposition 8.3 in [1] (see also [27,Theorem 8.2] where dim X < ∞)) and Theorem 8.8 from [27]. We will use the usual notation The former result can be for the linear case reformulated as follows.
hold. Moreover, let a nondecreasing function h : [a, b ] → R and a continuous increasing func- Finally, let x n be solutions of (3.14) for n ∈ N and let x: .
Similarly, when restricted to the linear case, Theorem 8.8 from [27] reduces to If x is the solution of (3.1) then, for any n ∈ N sufficiently large, equation (3.14) has a unique solution x n on [a, b ] and (3.15) holds.

. Remark.
Notice that the proof of Theorem 3.6 as given in [27] cannot be extended to the case of a general Banach space since it relies on the Helly's Choice Theorem.

. Proposition. Let
and i) The relation (3.11) follows immediately from (3.28). In particular, ii) Notice that (3.28) and (3.19) imply that and iii) Let ε > 0 and t ∈ (a, b ] be given and let us choose s 0 ∈ (a, t) and n 0 ∈ N so that Then, by (3.29) and (3.31), This means that iv) Now, suppose that (3.6) is not valid. Then there is ε > 0 such that for any ∈ N there exist m ≥ and t ∈ [a, b ] such that We may assume that m +1 > m for any ∈ N and Let t 0 ∈ (a, b] and assume that the set of those ∈ N for which t ∈ (a, t 0 ) has infinitely many elements, i.e. there is a sequence { k } ⊂ N such that t k ∈ (a, t 0 ) for all k ∈ N and lim k→∞ t k = t 0 .
Denote s k = t k and B k = A m k for k ∈ N. Then, in view of (3.34), we have (3.37)

By (3.29), we get
for k ∈ N. Therefore, by (3.32) and since lim k→∞ (h(t 0 −) − h(s k )) = 0 due to (3.36), we can As a consequence, we get finally by (3.37) If t 0 ∈ [a, b) and the set of those ∈ N for which t ∈ (a, t 0 ) has only finitely many elements, then there is a sequence { k } ⊂ N such that t k ∈ (t 0 , b) for all k ∈ N and lim k→∞ t k = t 0 . As before, let s k = t k and B k = A m k for k ∈ N and notice that s k ∈ (t 0 , b) for k ∈ N, lim k→∞ s k = t 0 and (3.37) are true. Arguing similarly as before we get that there is k 0 ∈ N such that a contradiction. Thus, (3.6) is satisfied.
To obtain (3.12) we would use the inequalities in (3.27) and follow the steps (ii)-(iv).

. Proposition. Let
and Hence, in view of (3.38), for any n ∈ N we have This proves (3.11). Suppose that (3.6) does not hold. Then there is ε > 0 such that for any ∈ N there exist m ≥ and t ∈ [a, b ] such that m +1 > m for ∈ N and the relations (3.34) and (3.35) are true.
Let t 0 ∈ (a, b) and let an arbitrary ε > 0 be given. Since h is continuous, we may choose η > 0 in such a way that t 0 − η, t 0 + η ∈ [a, b ] and and by (3.26), (3.38) and (3.40) there is 2 ∈ N, 2 ≥ 1 , such that The relations (3.19) and (3.42) imply immediately that Finally, let 3 ∈ N be such that 3 ≥ 2 and Hence, choosing ε < 1 5 ε and making use of (3.34), we get ε > A m (t ) − A(t ) L(X) ≥ ε, a contradiction. This proves that (3.6) is satisfied. The modification of the proof in the cases t 0 = a or t 0 = b is obvious.

Variations bounded with a weight
The main result of this section deals with the homogeneous generalized linear differential equation where, as before, A ∈ BV ([a, b ], L(X)) and x ∈ X. Further, it extends the result cf.[26, Theorem 1] obtained by Z. Opial for the case X = R m for some m ∈ N, and A, A n absolutely continuous on [a, b ]. As in the previous section we will assume that the fundamental existence assumption (3.2) is satisfied. To our aim, we need the following estimate well known in the case dim X < ∞ .
Furthermore, for each k = 1, 2, . . . , n, choose δ k > 0 in such a way that , and a+δ 0 < t 1 and b−δ 0 > t n . It follows that holds for any n ∈ N. To summarize, for any n ∈ N we have Therefore S n ≤ ε + (var b a g) for each n ∈ N and ε > 0. Thus, S n ≤ var b a g for all n ∈ N , wherefrom the desired estimate immediately follows. A, A n ∈ BV ([a, b ], L(X)) and x, x n ∈ X for n ∈ N. Assume (3.2), (3.13) and lim

Theorem. Let
Then (4.1) has a unique solution x on [a, b ]. Moreover, for each n ∈ N sufficiently large, the equation has a unique solution x n on [a, b ] and (3.15) holds.
Proof. First, notice that, since (4.2) implies (3.6). Therefore, by Lemma 3.3, there is n 0 ∈ N such that (3.7) holds for each t ∈ (a, b ] and each n ≥ n 0 . Assume n ≥ n 0 . Let x and x n be the solutions on [a, b ] of (4.1) and (4.3), respectively. Then where By Lemma 3.2 we have Thus, in view of the assumption (3.13), to prove the assertion of the theorem, we have to show that lim n→∞ h n ∞ = 0.
To this aim, we integrate by parts (cf. Proposition 2.1 (iii)) in the right-hand side of (4.5) and use Substitution Formula (cf. Proposition 2.1 (iv)). Then we get into the right-hand side of (4.8) and using Lemma 4.1, we obtain the estimates where α n = A n −A ∞ 2+3 var b a A n . Note that, due to (4.2), we have lim n→∞ α n = 0. (4.10) We can see that to show that lim n→∞ h n ∞ = 0, it is sufficient to prove that the sequence { x n ∞ } is bounded. By (4.6) and (4.9) we have Similarly, let B n (t) = A n (t) f n (t) 0 0 and y n = x n 1 for t ∈ [a, b ] and n ∈ N .
It is easy to check that equations (3.1) and (3.14) are respectively equivalent to the equations and y n (t) = y n + t a d[B n ] y n , n ∈ N (4.12) in the following sense: if x is a solution to (3.1) and y(t) = x(t) 1 , then y is a solution to (4.11). Conversely, if y is a solution to (4.11) and x is formed by its first m-components then x is a solution to (3.1), where x ∈ R m is formed by the first m-components of y. An analogous relationship holds also between equations (3.14) and (4.12), of course. Having this in mind, we can see that the following assertion is true.
, and x, x n ∈ R m for n ∈ N. Assume (3.2), (3.13) and where the integral is the Riemann ∆-integral defined e.g. in [5]. Slavík proved in [33] that this ∆-integral corresponds to a special case of the Kurzweil-Stieltjes integral. In addition, in [33] the relationship between dynamic equations on time scale and generalized differential equations is described. For the reader's convenience, we summarize the needed results from [33] in the following proposition.

. Proposition.
(i) [33,Theorem 5] Let f : [a, b ] T → R m be a rd-continuous function. Define Then (ii) [33,Theorem 12] If y : [a, b ] T → R m is a solution of (5.1) then x = y • σ is a solution of (3.1), where T is a solution of (5.1). for each n ∈ N, that is,

. Remark. Note that
. Analogously, holds for each n ∈ N, wherefrom, by (5.4) and Remark 5.2, the estimate follows. Hence, the assumption (3.11) of Theorem 3.4 is satisfied, as well. Consequently, we can use Theorem 3.4 to prove that (3.15) holds. By Proposition 5.1 (ii), the functions y, y n : [a, b ] T → R m , n ∈ N, obtained as the restriction of x and x n to [a, b ] T , respectively, are solutions to (5.1) and (5.6). Therefore, thanks to (3.15), (5.7) is also true, which completes the proof.

. Remark.
Two results on the continuous dependence of solutions to nonlinear dynamic equations have been recently delivered by A. Slavík, cf. [33,Theorems 14 and 16]. To prove them, it was sufficient to apply Proposition 5.1 and Theorems 8.2 and 8.7 from [27]. So, with respect to our Propositions 3.8 and 3.9, we can see that the above Theorem 5.3 provides for the linear case more general result than both Theorem 14 and Theorem 16 in [33].
Making use of Corollary 4.4 we obtain the following assertion.   However, for systems of the dimension ≥ 2, result analogous to that mentioned above for ODE's is not true, as shown by the following example: Put This is caused by the fact that the convergence A n → A is not uniform on [0, 1].
2. -Emphatic convergence. If the condition (3.6) is violated, the situation is rather more complicated. When dim X < ∞, then some results based on the notion of the emphatic convergence can be found e.g. in [20], [8], [27,Chapter 9], [9] (cf. also [10]). We suppose to treat the case when X is a general Banach space later.
3. -Linear functional differential equations with impulses. In view of the observations from Federson and Schwabik [7], we can see that the results of this paper can be applied also to linear functional differential equations with impulses. More details will be given later elsewhere.