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

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Abstract

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)= x ˜ k + a t d[ A k ]x+ f k (t) f k (a), t[a,b], kN, 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 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 var a b A k 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.

1 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

x(t)= x ˜ + a t d[A]x+f(t)f(a),
(1.1)

where <a<b<, X is a Banach space, L(X) is the Banach space of linear bounded operators on X, x ˜ X, A:[a,b]L(X) has bounded variation on [a,b], f:[a,b]X is regulated on [a,b] and the integrals are understood in the Kurzweil-Stieltjes sense. By a solution of (1.1) we understand a function x:[a,b]X such that a b d[A]x exists and (1.1) is true for all t[a,b].

For X= R m , 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 finite-dimensional case thoroughly treated by Schwabik, Tvrdý and Ashordia (see e.g. [5, 6] and [7]).

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 [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 1.1 Let A:[a,b]L(X) have a bounded variation on [a,b]. Then, (1.1) possesses a unique solution x on [a,b] for every x ˜ X and every function f:[a,b]X regulated on [a,b] if and only if

[ I X Δ A ( t ) ] 1 L(X)for all t(a,b],
(1.2)

where I X stands for the identity operator on X. In such a case x is regulated on [a,b], xf has a bounded variation on [a,b] and

x ( t ) X c A ( x ˜ X + 2 f ) exp ( c A var a t A ) ,t[a,b],
(1.3)

where 0< c A := sup t ( a , b ] [ I X Δ A ( t ) ] 1 L ( X ) <.

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 kN,

x k (t)= x ˜ k + a t d[ A k ] x k + f k (t) f k (a),
(1.4)

where A k :[a,b]L(X) have bounded variation on [a,b], f k :[a,b]X are regulated on [a,b] and x ˜ k X for kN. We are looking for conditions ensuring that (1.4) has a unique solution x k for each k large enough and the sequence { x k } tends uniformly on [a,b] to x, i.e.

lim n x k x =0.
(1.5)

In [12] we proved the following two theorems. The first one deals with the case that the variations of A k are uniformly bounded.

Proposition 1.2 [[12], Theorem 3.4]

Let A, A k :[a,b]L(X) have bounded variation on [a,b], f, f k :[a,b]X be regulated on [a,b] and x ˜ , x ˜ k X for kN. Furthermore, assume (1.2),

α := sup k N ( var a b A k ) <,
(1.6)
lim k A k A =0,
(1.7)
lim k f k f =0,
(1.8)
lim k x ˜ k x ˜ X =0.
(1.9)

Then (1.1) has a unique solution x on [a,b]. Furthermore, for each kN sufficiently large there is a unique solution x k on [a,b] to (1.4) and (1.5) holds.

The second result from [12], inspired by Opial’s paper [13], concerns the situation when variations of A k (1.6) need not be uniformly bounded and (1.1) and (1.4) reduce to homogeneous equations.

Proposition 1.3 [[12], Theorem 4.2]

Let A, A k :[a,b]L(X) have bounded variation on [a,b] and let x ˜ , x ˜ n X for kN. Furthermore, assume (1.2), (1.9) and

lim k ( 1 + var a b A k ) A k A =0.
(1.10)

Then the equation

x(t)= x ˜ + a t d[A]x,t[a,b],
(1.11)

has a unique solution x on [a,b]. Moreover, for each kN sufficiently large, the equation

x k (t)= x ˜ k + a t d[ A k ] x k
(1.12)

has a unique solution x k on [a,b] and (1.5) holds.

Let us recall the following observation.

Lemma 1.4 Let A, A k :[a,b]L(X) have bounded variation on [a,b] and let (1.10) be satisfied. Then (1.7) is true as well.

Proof The proof follows from the obvious inequality

A k A ( 1 + var a b A k ) A k A for all kN.

 □

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 finite-dimensional 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 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. A set D={ α 0 , α 1 ,, α ν ( D ) }[a,b] with ν(D)N is said to be a division of [a,b] if a= α 0 < α 1 << α ν ( D ) =b. The set of all divisions of [a,b] is denoted by D[a,b].

A function f:[a,b]X is called a finite step function on [a,b] if there exists a division D={ α 0 , α 1 ,, α ν ( D ) } of [a,b] such that f is constant on every open interval ( α j 1 , α j ), j=1,2,,ν(D).

For an arbitrary function f:[a,b]X we set f = sup t [ a , b ] f ( t ) X and

var a b f= sup D D [ a , b ] j = 1 ν ( D ) f ( α j ) f ( α j 1 ) X

is the variation of f over [a,b]. If var a b f<, we say that f is a function of bounded variation on [a,b]. BV([a,b],X) denotes the Banach space of functions f:[a,b]X of bounded variation on [a,b] equipped with the norm f BV = f ( a ) X + var a b f.

The function f:[a,b]X is called regulated on [a,b] if for each t[a,b) there is f(t+)X such that lim s t + f ( s ) f ( t + ) X =0 and for each t(a,b] there is f(t)X such that lim s t f ( s ) f ( t ) X =0. By G([a,b],X) we denote the Banach space of regulated functions f:[a,b]X equipped with the norm f . For t[a,b), s(a,b] we put Δ + f(t)=f(t+)f(t) and Δ f(s)=f(s)f(s). Recall that BV([a,b],X)G([a,b],X) cf. e.g. the assertion contained in Section 1.5 of [9].

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 couple P=(D,ξ) where D={ α 0 , α 1 ,, α ν ( D ) }D[a,b], ξ=( ξ 1 ,, ξ ν ( D ) ) [ a , b ] ν ( 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 on [a,b], the partition P is called δ-fine if [ α j 1 , α j ]( ξ j δ( ξ j ), ξ j +δ( ξ j )) holds for all j=1,2,,ν(D). We remark that for an arbitrary gauge δ on [a,b] there always exists a δ-fine partition of [a,b]. It is stated by the Cousin lemma (see e.g. [[5], Lemma 1.4]).

For given functions F:[a,b]L(X) and g:[a,b]X and a partition P=(D,ξ) of [a,b], where D={ α 0 , α 1 ,, α ν ( D ) }, ξ={ ξ 1 ,, ξ ν ( D ) }, we define

S(dF,g,P)= j = 1 ν ( D ) [ F ( α j ) F ( α j 1 ) ] g( ξ j ).

We say that JX is the Kurzweil-Stieltjes integral (or shortly KS-integral) of g with respect to F on [a,b] and denote J= a b d[F]g if for every ε>0 there exists a gauge δ on [a,b] such that

S ( d F , g , P ) J X <εfor all δ-fine partitions P of [a,b].

Analogously, we define the integral a b Fd[g] using sums of the form

S(F,dg,P)= j = 1 ν ( D ) F( ξ j ) [ g ( α j ) g ( α j 1 ) ] .

Some basic estimates for the KS-integrals are summarized in the following proposition. For the proofs, see [[12], Proposition 2.1] and [[11], Lemma 2.2].

Proposition 2.1 Let F:[a,b]L(X) and g:[a,b]X.

  1. (i)

    If FBV([a,b],L(X)) and gG([a,b],X), then a b d[F]g exists and

    a b d [ F ] g X ( var a b F ) g .
  2. (ii)

    If FG([a,b],L(X)) and gBV([a,b],X), then a b d[F]g exists and

    a b d [ F ] g X 2 F g BV .

For more details concerning the abstract KS-integration 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 3.1 Let A, A k BV([a,b],L(X)) for kN and assume that (1.2) and (1.10) hold.

Then there exist r >0 and k 0 N such that

y r ( y ( a ) X + ( 1 + var a b A k ) sup t [ a , b ] y ( t ) y ( a ) a t d [ A k ] y X ) for all  y G ( [ a , b ] , X )  and  k k 0 . }
(3.1)

Proof Assume that (3.1) is not true, i.e. assume that for each nN there are k n N and y n G([a,b],X) such that

y n >n ( y n ( a ) X + ( 1 + var a b A k n ) sup t [ a , b ] y n ( t ) y n ( a ) a t d [ A k n ] y n X ) .
(3.2)

We will prove that (3.2) leads to a contradiction. To this aim, first, rewrite inequality (3.2) as

1 n > u n ( a ) X + ( 1 + var a b A k n ) sup t [ a , b ] u n ( t ) u n ( a ) a t d [ A k n ] u n X ,
(3.3)

where

u n (t)= y n ( t ) y n for t[a,b] and nN.
(3.4)

Then, by (3.3) and (3.4) we can immediately see that u n ( a ) X < 1 n for all nN. Hence,

lim n u n ( a ) X =0.
(3.5)

Now, denote

v n (t)= u n (t) u n (a) a t d[ A k n ] u n for t[a,b] and nN
(3.6)

and

z n (t)= u n (t) v n (t)for t[a,b] and nN.
(3.7)

By (3.3) we have

v n = 1 y n ( sup t [ a , b ] y n ( t ) y n ( a ) a t d [ A k n ] y n X ) v n < 1 n ( 1 + var a b A k n ) 1 n for  n N , }
(3.8)

and, in particular,

lim n v n =0.
(3.9)

Moreover, the equalities (3.6) and (3.7) yield

z n (t)= u n (a)+ a t d[ A k n ] u n for t[a,b] and nN.

Consequently, z n BV([a,b],X),

z n (a)= u n (a)and z n BV 1+ var a b A k n for nN.
(3.10)

Now, let nN be fixed. We have

z n ( t ) = z n ( a ) + a t d [ A k n ] z n + a t d [ A k n ] v n = z n ( a ) + a t d [ A ] z n + a t d [ A k n A ] z n + a t d [ A k n ] v n for  t [ a , b ] ,

i.e.

z n (t)= z n (a)+ a t d[A] z n + h n (t)for t[a,b],
(3.11)

where

h n (t)= a t d[ A k n A] z n + a t d[ A k n ] v n for t[a,b].
(3.12)

We claim that

lim n h n =0.
(3.13)

Indeed, by (3.10) and Proposition 2.1(ii) we have

sup t [ a , b ] a t d [ A k n A ] z n X 2 A k n A z n BV 2 A k n A ( 1 + var a b A k n ) ,

wherefrom

lim n sup t [ a , b ] a t d [ A k n A ] z n X =0
(3.14)

follows due to (1.10). Moreover, using Proposition 2.1(i) and (3.8), we get

sup t [ a , b ] a t d [ A k n ] v n ( var a b A k n ) v n 1 n var a b A k n ( 1 + var a b A k n ) 1 n

and, hence,

lim n sup t [ a , b ] a t d [ A k n ] v n X =0.
(3.15)

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

lim n z n lim n c A ( z n ( a ) X + 2 h n ) exp ( c A var a b A ) =0,

i.e.

lim n z n =0.
(3.16)

This, together with (3.7) and (3.9), implies that lim n u n =0, which is impossible as u n =1 for all nN. The assertion of the lemma is true. □

Theorem 3.2 Let A, A k BV([a,b],L(X)), fBV([a,b],X), f k G([a,b],X) and x ˜ , x ˜ k X for kN. Assume (1.2), (1.9), (1.10), and

lim k ( 1 + var a b A k ) ( f k f ) =0.
(3.17)

Then (1.1) has a unique solution xBV([a,b],X) on [a,b]. Moreover, for each kN sufficiently large, (1.4) has a unique solution x k G([a,b],X) on [a,b] 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 [[12], Lemma 4.2], there is k 0 N such that

[ I X Δ A k ( t ) ] 1 L(X)for all t(a,b] and k k 0

and (1.4) has a unique solution x k G([a,b],X) for each k k 0 (cf. Proposition 1.1). By Lemma 3.1 we may choose k 0 N and r (0,) in such way that (3.1) holds.

Put u k = x k x for k k 0 . Then u k (a)= x ˜ k x ˜ and

u k (t) a t d[ A k ] u k ( x ˜ k x ˜ )= a t d[ A k A]x+ ( f k ( t ) f ( t ) ) ( f k ( a ) f ( a ) )

for t[a,b] and k k 0 . Using (3.1) we deduce that the inequality

u k r ( x ˜ k x ˜ X + ( 1 + var a b A k ) ( 2 A k A x BV + 2 f k f ) )

holds for all k k 0 . Thus, due to (1.9), (1.10) and (3.17), we have lim k u k =0, 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 fG([a,b],X) if the following assertion was true.

Let A, A k BV([a,b],L(X)) for kN and

lim k ( 1 + var a b A k ) A k A =0.
(3.18)

Then

lim k ( sup t [ a , b ] a t d [ A k ] f a t d [ A ] f X ) =0
(3.19)

holds for each fG([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.

Example 3.4 Let [a,b]=[0,1]. For kN puta

n k = [ k 3 / 2 ] + 1 , τ m , k = 1 2 n k m if  m { 0 , 1 , , n k } , a 0 , k = 2 n k k ( 1 ) n k , b 0 , k = 1 k ( 1 ) n k 1 , a m , k = 2 n k m + 1 k ( 1 ) n k m , b m , k = 3 k ( 1 ) n k m + 1 if  m { 1 , 2 , , n k 1 }

and define

A k (t)={ 0 if  t [ 0 , τ 0 , k ] , a m , k t + b m , k if  t [ τ m , k , τ m + 1 , k ]  and  m { 0 , 1 , , n k 1 }
(3.20)

and

A(t)=0for t[0,1].

It is easy to verify that

var 0 1 A k 1 k + 2 ( n k 1 ) k 1 k +2 k <

and

( 1 + var 0 1 A k ) A k A ( 1 + 2 n k 1 k ) 1 k 1 k + 2 k + 1 k 2

for all kN. In particular, (3.18) is true. However, if

f(t)={ ( 1 ) n n 4 if  t ( 2 n , 2 ( n 1 ) ]  for some  n N , 0 if  t = 0 ,
(3.21)

then f is regulated, var 0 1 f= and (3.19) is not valid since

0 1 d[ A k ]f 2 k m = 1 n k 1 1 m 4 > 2 k 1 n k 1 t 4 dt= 8 3 k ( ( n k ) 3 4 1 ) ,
(3.22)

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 fBV([a,b],X) in Theorem 3.2 cannot be extended to fG([a,b],X). Indeed, consider the equations

x(t)= 0 t d[A]x+f(t),t[0,1]
(3.23)

and

x k (t)= 0 t d[ A k ] x k + f k (t),t[0,1],kN,
(3.24)

where f k (t)=f(t) for t[0,1] and kN. Obviously, x=f is a solution to (3.23) on [0,1] and, for any kN, (3.24) possesses a solution x k on [0,1]. Furthermore, conditions (1.10) and (3.17) are satisfied. However, as we will see, x k does not converge to x.

Let kN be fixed. It is not difficult to verify that the solution to (3.24) on [0,1] is given by

x k (t)={ f ( t ) if  t [ 0 , τ 0 , k ] , c m , k exp ( a m , k t ) if  t ( τ m , k , τ m + 1 , k ]  and  m { 0 , 1 , , n k 1 } ,

where

c 0 , k =f( τ 1 , k )exp( a 0 , k τ 0 , k )=f( τ 1 , k )exp ( 1 k ( 1 ) n k + 1 )

and

c m , k = c m 1 , k exp ( ( a m 1 , k a m , k ) τ m , k ) + ( f ( τ m + 1 , k ) f ( τ m , k ) ) exp( a m , k τ m , k )

for m=1,, n k 1. Furthermore, since

( a 0 , k a 1 , k ) τ 1 , k = 4 k ( 1 ) n k and a 1 , k τ 1 , k = 2 k ( 1 ) n k ,

we have

c 1 , k = c 0 , k exp ( 4 k ( 1 ) n k ) + ( f ( τ 2 , k ) f ( τ 1 , k ) ) exp ( 2 k ( 1 ) n k ) .

Similarly, for m=2,3,, n k 1 we have

( a m 1 , k a m , k ) τ m , k = 6 k ( 1 ) n k m + 1 and τ m , k a m , k = 2 k ( 1 ) n k m ,

and hence

c m , k = c m 1 , k exp ( 6 k ( 1 ) n k m + 1 ) + ( f ( τ m + 1 , k ) f ( τ m , k ) ) exp ( 2 k ( 1 ) n k m + 1 ) .

From these formulas we can deduce that

c m , k = exp ( 2 k ( 1 ) n k + 1 ) ( c 0 , k + j = 2 m + 1 ( 1 ) j + 1 f ( τ j , k ) ) + exp ( 4 k ( 1 ) n k + 1 ) j = 1 m ( 1 ) j f ( τ j , k )

if m is even, while for m odd and m>1 we get

c m , k = exp ( 4 k ( 1 ) n k ) ( c 0 , k + j = 2 m ( 1 ) j + 1 f ( τ j , k ) ) + exp ( 2 k ( 1 ) n k ) j = 1 m + 1 ( 1 ) j f ( τ j , k ) .

In particular, x k (1)= c n k 1 , k exp(4/k) for kN. Using the above relations and the definition of f, we get

c n k 1 , k = exp ( 4 k ) ( c 0 , k + j = 2 n k 1 ( 1 ) j + 1 ( 1 ) n k j + 1 n k j + 1 4 ) + exp ( 2 k ) j = 1 n k ( 1 ) j ( 1 ) n k j + 1 n k j + 1 4 = exp ( 4 k ) ( c 0 , k + m = 2 n k 1 1 m 4 ) exp ( 2 k ) m = 1 n k 1 m 4 = exp ( 2 k ) ( c 0 , k exp ( 2 k ) 1 ) + exp ( 2 k ) ( exp ( 2 k ) 1 ) m = 2 n k 1 1 m 4 exp ( 2 k ) 1 n k 4

if n k is even, and

c n k 1 , k = exp ( 2 k ) ( c 0 , k m = 1 n k 1 1 m 4 ) + exp ( 4 k ) m = 2 n k 1 m 4 = exp ( 2 k ) ( c 0 , k 1 ) + exp ( 2 k ) ( exp ( 2 k ) 1 ) m = 2 n k 1 1 m 4 + exp ( 4 k ) 1 n k 4

if n k is odd.

Clearly, lim k c 0 , k =0,

lim k exp ( 2 k ) ( c 0 , k exp ( 2 k ) 1 ) = lim k exp ( 2 k ) ( c 0 , k 1)=1

and

lim k exp ( 2 k ) 1 n k 4 = lim k exp ( 4 k ) 1 n k 4 =0.

On the other hand, like in (3.22), we have

exp ( 2 / k ) ( exp ( 2 / k ) 1 ) m = 2 n k 1 1 m 4 = exp ( 2 / k ) exp ( 2 / k ) 1 2 / k 2 k m = 2 n k 1 1 m 4 > exp ( 2 / k ) exp ( 2 / k ) 1 2 / k 2 k 2 n k 1 t 4 d t = exp ( 2 / k ) exp ( 2 / k ) 1 2 / k 8 3 k ( ( n k ) 3 4 2 3 4 ) ,

where the right-hand side tends to ∞ when k. Consequently, the sequence x k (1) cannot have a finite limit for k.

Remark 3.5 Reasonable examples of sequences { f k }G([a,b],X) that tend to a function f of bounded variation are provided e.g. by sequences of the form f k = g k + h k , where { g k }BV([a,b],X) tends to fBV([a,b],X) and { h k }G([a,b],X) tends to 0.

Remark 3.6 For F:[a,b]L(X) and D={ α 0 , α 1 ,, α m }D[a,b], define

V a b (F,D)=sup { j = 1 m [ F ( α j ) F ( α j 1 ) ] y j X : y j X , y j X 1 , j = 1 , 2 , , m }

and

SV a b (F)=sup { V a b ( F , D ) ; D D [ a , b ] } .

Then SV a b (F) is said to be the semi-variation of F on [a,b] (cf. e.g. [17]).b It is clear that if FBV([a,b],L(X)) then F has bounded semi-variation on [a,b] while the reversed implication is not true in general (cf. [[18], Theorem 2]). By [8] and [11], the Kurzweil-Stieltjes integral a b d[A]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 ABV([a,b],X), cf. [9] and [10]. 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

V a b ( f , D ) = sup { j = 1 m F j [ f ( α j ) f ( α j 1 ) ] X : F j L ( X ) , F j L ( X ) 1 , j = 1 , 2 , , m } .

However, it may be shown that, in this case, f has a bounded semi-variation if and only fBV([a,b],X). Therefore, the possible replacement of the condition fBV([a,b],X) 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, y ˜ , z ˜ Y, P,QBV([a,b],L(Y)) and g,hBV([a,b],Y). Consider the following system of generalized linear differential equations:

y ( t ) = y ˜ + a t d [ P ] z + g ( t ) g ( a ) , t [ a , b ] , z ( t ) = z ˜ + a t d [ Q ] y + h ( t ) h ( a ) , t [ a , b ] . }
(4.1)

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,

A ( t ) L ( X ) = P ( t ) L ( Y ) + Q ( t ) L ( Y ) for t[a,b], var a b A var a b P+ var a b Q

and system (4.1) can be reformulated as (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:

[ I Y Δ Q ( t ) Δ P ( t ) ] 1 L(Y)for t(a,b],
(4.3)
[ I Y y Δ P ( t ) Δ Q ( t ) ] 1 L(Y)for t(a,b],
(4.4)

where I Y stands for the identity operator on Y.

Indeed, assume e.g. that (4.3) holds and let [ I X Δ A(t)]x=0 for some x=(y,z)X and t(a,b]. Then

y Δ P(t)z=0andz Δ Q(t)y=0,
(4.5)

i.e.

y= Δ P(t)zand [ I Y Δ Q ( t ) Δ P ( t ) ] z=0.
(4.6)

By (4.3) the latter equality can happen only if z=0. Consequently y=0, and hence x=0, as well. Similarly, we would show that [ I X Δ A(t)]x=0 implies x=0 also in the case that (4.4) is satisfied. This shows that the operator I X Δ A(t) is injective.

To prove its surjectivity, assume first (4.3) and let (u,v)X be given. Put

z= ( [ I Y Δ Q ( t ) Δ P ( t ) ] 1 ) ( v + Δ Q ( t ) u ) andy=u+ Δ P(t)z.

Then, y Δ P(t)z=u and

z Δ Q ( t ) y = z Δ Q ( t ) Δ P ( t ) z Δ Q ( t ) u = [ I Y Δ Q ( t ) Δ P ( t ) ] z Δ Q ( t ) u = v ,

that is, [ I X Δ A(t)]x=(u,v) for x=(y,z). Similarly, we can show that for each (u,v)X there is xX such that [ I X Δ A(t)]x=(u,v) also in the case that (4.4) is satisfied. The operator I X Δ A(t) is surjective. To summarize, according to the Banach theorem, the operator I X Δ A(t) possesses a bounded [ I X Δ A ( t ) ] 1 .

Now, consider the systems

y k ( t ) = y ˜ k + a t d [ P k ] z k + g k ( t ) g k ( a ) , t [ a , b ] , k N , z k ( t ) = z ˜ k + a t d [ Q k ] y k + h k ( t ) h k ( a ) , t [ a , b ] , k N , }
(4.7)

where y ˜ k , z ˜ k Y, P k , Q k BV([a,b],L(Y)), g k , h k G([a,b],Y) and kN. Assume that (4.3) or (4.4) is true and

lim k y ˜ k y ˜ Y =0, lim k z ˜ k z ˜ Y =0,
(4.8)
lim k ( 1 + var a b P k + var a b Q k ) ( P k P + Q k Q ) =0
(4.9)

and

lim k ( 1 + var a b P k + var a b Q k ) ( g k g + h k h ) =0.
(4.10)

Define A k :[a,b]L(X) and f k :[a,b]X for kN like A and f in (4.2) (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.

Corollary 4.1 Assume that (4.3) or (4.4) holds and that (4.8)-(4.10) are satisfied. Then system (4.1) has a unique solution (y,z)BV([a,b],Y×Y) on [a,b]. Moreover, for each kN sufficiently large, the system (4.7) has a unique solution ( y k , z k )G([a,b],Y×Y) on [a,b] and

lim k ( y k y + z k z ) =0.

In [19], Meng and Zhang investigated the continuous dependence on a parameter k for second-order linear measure differential equations of the form

d y +d [ μ k ( t ) ] y=0,t[0,1],y(0)= y ˜ , y (0)= z ˜ ,kN,
(4.11)

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 [19] 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 (4.7) reduce to (4.11) 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 [[19], 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

d y +d [ μ ( t ) ] y=0,t[0,1],y(0)= y ˜ , y (0)= z ˜ ,kN,
(4.12)

where μ is a normalized measure on [0,1] and y ˜ , z ˜ R. Obviously, both existence conditions (4.3) and (4.4) are now satisfied. In view of this, assuming that μ and μ k have a bounded variation on [0,1] and

lim k ( 1 + var 0 1 μ k ) μ k μ =0,

it follows from our Corollary 4.1 that

lim k ( y k y + y k y ) =0

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 { y k }. On the other hand, the weak convergence which appears in [19] includes the uniform boundedness of the variations var 0 1 μ k (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 T of is called time scale. For given a,bT, we put [ a , b ] T =[a,b]T. For tT, we define

ρ(t):=sup[a,t)Tandσ(t):=inf(t,b]T.

The point tT is said to be right-dense if σ(t)=t, while it is left-dense if ρ(t)=t. A function f: [ a , b ] T R m is rd-continuous in [ a , b ] T if f is continuous at every right-dense point of [ a , b ] T and there exists f(t) for every left-dense point t [ a , b ] T (see e.g. [22]).

Let us consider the linear dynamic equation

y Δ (t)=P(t)y(t)+h(t),y(a)= y ˜ ,t [ a , b ] T ,
(4.13)

where y ˜ R m and P: [ a , b ] T L( R m ), h: [ a , b ] T R m are rd-continuous functions and y Δ stands for the Δ-derivative. By a solution of (4.13) we understand a function y: [ a , b ] T R m satisfying the integral equation

y(t)= y ˜ + a t [ P ( s ) y ( s ) + h ( s ) ] Δs,t [ a , b ] T ,

where the integral is the Riemann Δ-integral defined e.g. in [22].

As noticed by Slavík (see [[23], Theorem 5]), the Riemann Δ-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

σ ˜ ( t ) : = inf [ t , b ] T for  t [ a , b ] , F 1 ( t ) = a t f ( s ) Δ s for  t [ a , b ] T

and

F 2 (t)= a t f ( σ ˜ ( s ) ) d [ σ ˜ ( s ) ] for t[a,b].

Then F 2 (t)= F 1 ( σ ˜ (t)) holds for t[a,b].

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]

If y: [ a , b ] T R m is a solution of (4.13) then

x=y σ ˜ y:[a,b] R m

is a solution of (1.1), where

A ( t ) = a t P ( σ ˜ ( s ) ) d [ σ ˜ ( s ) ] for  t [ a , b ] and f ( t ) = a t h ( σ ˜ ( s ) ) d [ σ ˜ ( s ) ] for  t [ a , b ] . }
(4.14)

Symmetrically, if x:[a,b] R m is a solution of (1.1), with A and f given by (4.14), then y: [ a , b ] T R m defined by y(t)=x(t) for t [ a , b ] T is a solution of (4.13).

It is important to mention that, thanks to the properties of σ ˜ :[a,b] [ a , b ] T , the functions A:[a,b]L( R n ) and f:[a,b] R n given by (4.14) are well defined, left-continuous and of bounded variation on [a,b].

Using the correspondence stated in Proposition 4.2 and Theorem 3.2 we obtain the following result.

Theorem 4.3 Let P, P k : [ a , b ] T L( R m ), h, h k : [ a , b ] T R m for kN be rd-continuous functions in [ a , b ] T and let y ˜ , y ˜ k R m , kN, be given. Assume that

lim k y ˜ k y ˜ R m =0,
(4.15)
lim k [ 1 + sup t [ a , b ] T P k ( t ) L ( R m ) ] sup t [ a , b ] T a t ( P k ( s ) P ( s ) ) Δ s L ( R m ) = 0 , lim k [ 1 + sup t [ a , b ] T P k ( t ) L ( R m ) ] sup t [ a , b ] T a t ( h k ( s ) h ( s ) ) Δ s R m = 0 . }
(4.16)

Then initial value problem (4.13) has a solution y, the initial value problems

y k Δ (t)= P k (t) y k (t)+ h k (t), y k (a)= y ˜ k ,t [ a , b ] T
(4.17)

have solutions y k for all kN, and

lim k sup t [ a , b ] T y k ( t ) y ( t ) R m =0.

Proof For each kN and t[a,b], define

A k (t)= a t P k ( σ ˜ ( s ) ) d [ σ ˜ ( s ) ] and f k (t)= a t h k ( σ ˜ ( s ) ) d [ σ ˜ ( s ) ] .
(4.18)

It is not difficult to see that, if ac<db, then

A k ( d ) A k ( c ) L ( R m ) = c d P k ( σ ˜ ( s ) ) d [ σ ˜ ( s ) ] L ( R m ) ( sup t [ a , b ] T P k ( t ) L ( R m ) ) ( var c d σ ˜ ) ,

and, consequently,

var a b A k ( sup t [ a , b ] T P k ( t ) L ( R m ) ) ( var a b σ ˜ ) ,kN.

On the other hand,

A k A = sup t [ a , b ] a σ ˜ ( t ) ( P k ( s ) P ( s ) ) Δ s L ( R m ) sup t [ a , b ] T a t ( P k ( s ) P ( s ) ) Δ s L ( R m )

and, analogously,

f k f sup t [ a , b ] T a t ( h k ( s ) h ( s ) ) Δ s R m .

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 x k 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 x k to [ a , b ] T , 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 sup t [ a , b ] T P k ( t ) L ( R m ) , but also sup t [ a , b ] T h k ( t ) R m .

Endnotes

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

b Sometimes it is called also the -variation of F on [a,b] (with respect to the bilinear triple B=(L(X),X,X), cf. e.g. [8]).

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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. 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.

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Correspondence to Milan Tvrdý.

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Keywords

  • abstract generalized differential equation
  • continuous dependence
  • time scale dynamics