R E S E A R C H
Open Access
Continuous dependence of solutions of
abstract generalized linear differential
equations with potential converging
uniformly with a weight
Giselle Antunes Monteiro
1and Milan Tvrdý
2*Dedicated to Professor Ivan Kiguradze for his merits in mathematical sciences.
*Correspondence: [email protected]
2Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic
Full list of author information is available at the end of the article
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 parameterkof solutions to linear integral equations of the formx(t) =xk+
t
ad[Ak]x+fk(t) –fk(a),
t∈[a,b],k∈N, where –∞<a<b<∞,Xis a Banach space,L(X) is the Banach space of linear bounded operators onX,xk∈X,Ak: [a,b]→L(X) have bounded variations on [a,b],fk: [a,b]→Xare 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 varb
aAkneed 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
Keywords: abstract generalized differential equation; continuous dependence; time scale dynamics
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 differ-ential equations, in some sense a final result on the continuous dependence was delivered by Kurzweil and Vorel in their paper [] from . In fact, it was a response to the aver-aging method introduced few years before by Krasnoselskij and Krein []. 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 [].
By generalized linear differential equations we understand linear integral equations of the form
x(t) =x+
t
a
d[A]x+f(t) –f(a), (.)
where –∞<a<b<∞,Xis a Banach space,L(X) is the Banach space of linear bounded operators onX,x∈X,A: [a,b]→L(X) has bounded variation on [a,b],f : [a,b]→Xis regulated on [a,b] and the integrals are understood in the Kurzweil-Stieltjes sense. By a
solutionof (.) we understand a functionx: [a,b]→Xsuch thatabd[A]xexists and (.) is true for allt∈[a,b].
For X=Rm, such equations are special cases of equations introduced in by
Kurzweil (see []) in connection with the advanced study of continuous dependence prop-erties of ordinary differential equations (see also []). In this connection, we want to high-light the recent monograph [] bringing a new insight into the topic. Linear equations of the form (.) have been in the finite-dimensional case thoroughly treated by Schwabik, Tvrdý and Ashordia (seee.g.[, ] and []).
Basic theory of the abstract Kurzweil-Stieltjes integral (called also abstract Perron-Stieltjes or simply gauge-Perron-Stieltjes integral) and generalized linear differential equations in a general Banach space has been established by Schwabik in a series of papers [–] written between and . Some of the needed complements have been added in our paper [].
Taking into account the closing remark in [], we can see that the following basic exis-tence result is a particular case of [, Proposition .].
Proposition . Let A: [a,b]→L(X)have a bounded variation on[a,b].Then, (.) pos-sesses a unique solution x on[a,b]for everyx∈X and every function f : [a,b]→X regu-lated on[a,b]if and only if
IX––A(t)
–
∈L(X) for all t∈(a,b], (.)
where IXstands 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
x(t)X≤cA
xX+ f∞ expcAvartaA, t∈[a,b], (.)
where <cA:=supt∈(a,b][IX––A(t)]–L(X)<∞.
Primarily we are concerned with the continuous dependence of solutions of general-ized linear differential equations on a parameter. In particular, we assume that the given equation (.) has a unique solutionxfor eachf regulated on [a,b] and eachx∈Xand we consider a sequence of equations depending on a parameterk∈N,
xk(t) =xk+
t
a
d[Ak]xk+fk(t) –fk(a), (.)
whereAk: [a,b]→L(X) have bounded variation on [a,b],fk: [a,b]→Xare regulated on
solutionxkfor eachklarge enough and the sequence{xk}tends uniformly on [a,b] tox,i.e. lim
n→∞xk–x∞= . (.)
In [] we proved the following two theorems. The first one deals with the case that the variations ofAkare uniformly bounded.
Proposition .[, Theorem .] Let A,Ak: [a,b]→L(X)have bounded variation on
[a,b],f,fk : [a,b]→X be regulated on [a,b]andx,xk∈X for k∈N. Furthermore, as-sume(.),
α∗:=sup k∈N
varbaAk <∞, (.) lim
k→∞Ak–A∞= , (.) lim
k→∞fk–f∞= , (.) lim
k→∞xk–xX= . (.) Then(.)has a unique solution x on[a,b].Furthermore,for each k∈Nsufficiently large there is a unique solution xkon[a,b]to(.)and(.)holds.
The second result from [], inspired by Opial’s paper [], concerns the situation when variations ofAk(.) need not be uniformly bounded and (.) and (.) reduce to
homo-geneous equations.
Proposition .[, Theorem .] Let A,Ak: [a,b]→L(X)have bounded variation on
[a,b]and letx,xn∈X for k∈N.Furthermore,assume(.), (.)and lim
k→∞
+varbaAk Ak–A∞= . (.)
Then the equation
x(t) =x+
t
a
d[A]x, t∈[a,b], (.)
has a unique solution x on[a,b].Moreover,for each k∈Nsufficiently large,the equation
xk(t) =xk+
t
a
d[Ak]xk (.)
has a unique solution xkon[a,b]and(.)holds.
Let us recall the following observation.
Lemma . Let A,Ak: [a,b]→L(X)have bounded variation on[a,b]and let (.)be satisfied.Then(.)is true as well.
Proof The proof follows from the obvious inequality
The only known result (cf.[, Corollary .]) concerning nonhomogeneous equations (.), (.) and the case when (.) is not satisfied requires thatXis a finite-dimensional space. The aim of this paper is to fill this gap.
For a more detailed list of related references, see [].
2 Preliminaries
Throughout these notesXis a Banach space andL(X) is the Banach space of bounded linear operators onX. By · Xwe denote the norm inX. Similarly, · L(X)denotes the
usual operator norm inL(X).
Assume that –∞<a<b<∞and [a,b] denotes the corresponding closed interval. A set
D={α,α, . . . ,αν(D)} ⊂[a,b] withν(D)∈Nis said to be a division of [a,b] ifa=α<α< · · ·<αν(D)=b. The set of all divisions of [a,b] is denoted byD[a,b].
A functionf : [a,b]→Xis called afinite step functionon [a,b] if there exists a division
D={α,α, . . . ,αν(D)}of [a,b] such thatf is constant on every open interval (αj–,αj),j=
, , . . . ,ν(D).
For an arbitrary functionf: [a,b]→Xwe setf∞=supt∈[a,b]f(t)Xand
varbaf = sup D∈D[a,b]
ν(D)
j=
f(αj) –f(αj–)X
is the variation off over [a,b]. Ifvarb
af <∞, we say thatf is a functionof bounded variation
on [a,b].BV([a,b],X) denotes the Banach space of functionsf : [a,b]→Xof bounded variation on [a,b] equipped with the normfBV=f(a)X+varbaf.
The functionf : [a,b]→X is calledregulatedon [a,b] if for eacht∈[a,b) there is
f(t+)∈Xsuch thatlims→t+f(s) –f(t+)X= and for eacht∈(a,b] there isf(t–)∈X
such thatlims→t–f(s) –f(t–)X= . ByG([a,b],X) we denote the Banach space of
reg-ulated functionsf : [a,b]→Xequipped with the normf∞. Fort∈[a,b),s∈(a,b] we
put+f(t) =f(t+) –f(t) and–f(s) =f(s) –f(s–). Recall thatBV([a,b],X)⊂G([a,b],X)
cf. e.g.the assertion contained in Section . of [].
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 coupleP= (D,ξ) whereD={α,α, . . . ,αν(D)} ∈D[a,b],ξ= (ξ, . . . ,ξν(D))∈[a,b]ν(D)andαj–≤ξj≤αjholds
for j= , , . . . ,ν(D). Furthermore, any positive function δ : [a,b]→(,∞) is called a
gauge on [a,b]. Given a gaugeδ on [a,b], the partitionP is calledδ-fineif [αj–,αj]⊂
(ξj–δ(ξj),ξj+δ(ξj)) holds for allj= , , . . . ,ν(D). We remark that for an arbitrary gauge
δon [a,b] there always exists aδ-fine partition of [a,b]. It is stated by the Cousin lemma (seee.g.[, Lemma .]).
For given functionsF: [a,b]→L(X) andg: [a,b]→Xand a partitionP= (D,ξ) of [a,b], whereD={α,α, . . . ,αν(D)},ξ={ξ, . . . ,ξν(D)}, we define
S(dF,g,P) =
ν(D)
j=
F(αj) –F(αj–)
g(ξj).
that
S(dF,g,P) –JX<ε for allδ-fine partitionsPof [a,b].
Analogously, we define the integralabFd[g] using sums of the form
S(F, dg,P) =
ν(D)
j= F(ξj)
g(αj) –g(αj–)
.
Some basic estimates for the KS-integrals are summarized in the following proposition. For the proofs, see [, Proposition .] and [, Lemma .].
Proposition . Let F: [a,b]→L(X)and g: [a,b]→X.
(i) IfF∈BV([a,b],L(X))andg∈G([a,b],X),thenabd[F]gexists and
abd[F]g X
≤varbaF g∞.
(ii) IfF∈G([a,b],L(X))andg∈BV([a,b],X),thenabd[F]gexists and
b
a
d[F]g X
≤F∞gBV.
For more details concerning the abstract KS-integration and further references, see [– , ] and [].
3 Main result
Our main result is based on the following lemma which is an analog of the assertion for-mulated for ODEs by Kiguradze in [, Lemma .]. Its variant was used also in the study of FDEs by Hakl, Lomtatidze and Stavrolaukis in [, Lemma .].
Lemma . Let A,Ak∈BV([a,b],L(X))for k∈Nand assume that(.)and(.)hold. Then there exist r∗> and k∈Nsuch that
y∞≤r∗y(a)X+ +varbaAk sup t∈[a,b]
y(t) –y(a) –
t
a
d[Ak]y
X
for all y∈G[a,b],X and k≥k.
⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭
(.)
Proof Assume that (.) is not true,i.e.assume that for eachn∈Nthere arekn∈Nand yn∈G([a,b],X) such that
yn∞>nyn(a)X+
+varbaAkn sup
t∈[a,b]
yn(t) –yn(a) –
t
a
d[Akn]yn
X
. (.)
We will prove that (.) leads to a contradiction. To this aim, first, rewrite inequality (.) as
n>un(a)X+
+varbaAkn sup
t∈[a,b]
un(t) –un(a) –
t
a
d[Akn]un
X
where
un(t) = yn(t)
yn∞ fort∈[a,b] andn∈N. (.)
Then, by (.) and (.) we can immediately see thatun(a)X<nfor alln∈N. Hence,
lim
n→∞un(a)X= . (.)
Now, denote
vn(t) =un(t) –un(a) –
t
a
d[Akn]un fort∈[a,b] andn∈N (.)
and
zn(t) =un(t) –vn(t) fort∈[a,b] andn∈N. (.)
By (.) we have
vn∞=
yn∞
sup t∈[a,b]
yn(t) –yn(a) –
t
a
d[Akn]yn
X
<
n( +varb aAkn)
≤
n forn∈N,
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(.)
and, in particular,
lim
n→∞vn∞= . (.)
Moreover, the equalities (.) and (.) yield
zn(t) =un(a) +
t
a
d[Akn]un fort∈[a,b] andn∈N.
Consequently,zn∈BV([a,b],X),
zn(a) =un(a) and znBV≤ +varbaAkn forn∈N. (.)
Now, letn∈Nbe fixed. We have
zn(t) =zn(a) +
t
a
d[Akn]zn+ t
a
d[Akn]vn
=zn(a) +
t
a
d[A]zn+
t
a
d[Akn–A]zn+
t
a
d[Akn]vn fort∈[a,b],
i.e.
zn(t) =zn(a) +
t
a
where
hn(t) =
t
a
d[Akn–A]zn+
t
a
d[Akn]vn fort∈[a,b]. (.)
We claim that
lim
n→∞hn∞= . (.)
Indeed, by (.) and Proposition .(ii) we have
sup t∈[a,b]
t
a
d[Akn–A]zn
X
≤Akn–A∞znBV
≤Akn–A∞
+varbaAkn ,
wherefrom
lim n→∞t∈sup[a,b]
atd[Akn–A]zn
X
= (.)
follows due to (.). Moreover, using Proposition .(i) and (.), we get
sup t∈[a,b]
t
a
d[Akn]vn
∞≤varbaAkn vn∞≤
n varb
aAkn
( +varb aAkn)
≤ n
and, hence,
lim n→∞t∈sup[a,b]
atd[Akn]vn
X
= . (.)
Now, (.) follows immediately from (.) and (.).
Finally, having in mind Proposition . (cf.(.)) and (.), (.), and (.), we conclude that
lim
n→∞zn∞≤nlim→∞cAzn(a)X+ hn∞ exp
cAvarbaA = , i.e.
lim
n→∞zn∞= . (.)
This, together with (.) and (.), implies thatlimn→∞un∞= , which is impossible as un∞= for alln∈N. The assertion of the lemma is true. Theorem . Let A,Ak∈BV([a,b],L(X)),f∈BV([a,b],X),fk∈G([a,b],X)andx,xk∈X for k∈N.Assume(.), (.), (.),and
lim k→∞
+varbaAk fk–f∞ = . (.)
Proof First, recall that, by Lemma . our assumption (.) implies that (.) is true, as well. Therefore, by [, Lemma .], there isk∈Nsuch that
IX––Ak(t)
–
∈L(X) for allt∈(a,b] andk≥k
and (.) has a unique solutionxk∈G([a,b],X) for eachk≥k(cf.Proposition .). By
Lemma . we may choosek∈Nandr∗∈(,∞) in such way that (.) holds. Putuk=xk–xfork≥k. Thenuk(a) =xk–xand
uk(t) –
t
a
d[Ak]uk– (xk–x) =
t
a
d[Ak–A]x+
fk(t) –f(t) –
fk(a) –f(a)
fort∈[a,b] andk≥k. Using (.) we deduce that the inequality
uk∞≤r∗
xk–xX+
+varbaAk Ak–A∞xBV+ fk–f∞
holds for allk≥k. Thus, due to (.), (.) and (.), we havelimk→∞uk∞= ,
where-from (.) immediately follows. The proof of the theorem has been completed.
Remark . The proof of Theorem . could be substantially simplified and also extended to the casef ∈G([a,b],X) if the following assertion was true.
Let A,Ak∈BV([a,b],L(X))for k∈Nand
lim k→∞
+varbaAk Ak–A∞= . (.)
Then
lim k→∞
sup t∈[a,b]
atd[Ak]f –
t
a
d[A]f X
= (.)
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.
Example . Let [a,b] = [, ]. Fork∈Nputa
nk=
k/+ , τm,k=
nk–m ifm∈ {, , . . . ,nk},
a,k=
nk
k (–)
nk, b,
k=
k(–) nk–,
am,k=
nk–m+
k (–)
nk–m, b
m,k=
k(–)
nk–m+ ifm∈ {, , . . . ,n
k– }
and define
Ak(t) =
⎧ ⎨ ⎩
ift∈[,τ,k],
am,kt+bm,k ift∈[τm,k,τm+,k] andm∈ {, , . . . ,nk– }
and
A(t) = fort∈[, ]. It is easy to verify that
varAk≤
k+
(nk– ) k ≤
k+ √
k<∞ and
+varAk Ak–A∞≤
+nk–
k
k≤
k+
√ k+
k
for allk∈N. In particular, (.) is true. However, if
f(t) =
⎧ ⎨ ⎩
(–)n
√
n ift∈(
–n, –(n–)] for somen∈N,
ift= , (.)
thenf is regulated,varf =∞and (.) is not valid since
d[Ak]f ≥
k nk–
m=
√ m>
k
nk
√ tdt=
k
(nk)– , (.)
where the right-hand side evidently tends to∞fork→ ∞.
Moreover, the functions (.) and (.) provide us with the argument explaining that the conditionf ∈BV([a,b],X) in Theorem . cannot be extended tof ∈G([a,b],X). In-deed, consider the equations
x(t) =
t
d[A]x+f(t), t∈[, ] (.)
and
xk(t) =
t
d[Ak]xk+fk(t), t∈[, ],k∈N, (.)
wherefk(t) =f(t) fort∈[, ] andk∈N. Obviously,x=f is a solution to (.) on [, ]
and, for anyk∈N, (.) possesses a solutionxkon [, ]. Furthermore, conditions (.)
and (.) are satisfied. However, as we will see,xkdoes not converge tox.
Letk∈Nbe fixed. It is not difficult to verify that the solution to (.) on [, ] is given by
xk(t) =
⎧ ⎨ ⎩
f(t) ift∈[,τ,k],
cm,kexp(am,kt) ift∈(τm,k,τm+,k] andm∈ {, , . . . ,nk– },
where
c,k=f(τ,k)exp(–a,kτ,k) =f(τ,k)exp
k(–) nk+
and
cm,k=cm–,kexp
(am–,k–am,k)τm,k +
f(τm+,k) –f(τm,k) exp(–am,kτm,k)
form= , . . . ,nk– . Furthermore, since
(a,k–a,k)τ,k=
k(–)
nk and a,
kτ,k= –
k(–) nk,
we have
c,k=c,kexp
k(–) nk
+f(τ,k) –f(τ,k) exp
k(–) nk .
Similarly, form= , , . . . ,nk– we have
(am–,k–am,k)τm,k=
k(–)
nk–m+ and τ
m,kam,k=
k(–) nk–m,
and hence
cm,k=cm–,kexp
k(–) nk–m+
+f(τm+,k) –f(τm,k) exp
k(–) nk–m+
.
From these formulas we can deduce that
cm,k=exp
k(–) nk+
c,k+ m+
j=
(–)j+f(τj,k)
+exp
k(–) nk+
m
j=
(–)jf(τj,k)
ifmis even, while formodd andm> we get
cm,k=exp
k(–) nk
c,k+ m
j=
(–)j+f(τj,k)
+exp k(–) nk m+ j=
(–)jf(τ j,k).
In particular,xk() =cnk–,kexp(–/k) fork∈N. Using the above relations and the
defini-tion off, we get
cnk–,k=exp
k
c,k+ nk–
j=
(–)j+ (–)
nk–j+
nk–j+
+exp k nk j=
(–)j (–)
nk–j+
=exp
k
c,k+ nk– m= √ m –exp k nk m= √ m =exp k
c,kexp
k – +exp k exp k –
nk–
m= √ m –exp k √ nk
ifnkis even, and
cnk–,k=exp
k
c,k– nk–
m= √ m +exp k nk m= √ m =exp k
(c,k– ) +exp
k exp k –
nk–
m=
√ m+exp
k √ nk
ifnkis odd.
Clearly,limk→∞c,k= ,
lim k→∞exp
k
c,kexp
k – = lim k→∞exp
k
(c,k– ) = –
and
lim k→∞exp
k √ nk = lim k→∞exp
k √ nk = .
On the other hand, like in (.), we have
exp(/k)exp(/k) –
nk–
m=
√
m=exp(/k)
exp(/k) – /k
k nk– m= √ m
>exp(/k)exp(/k) – /k k nk √ tdt
=exp(/k)exp(/k) – /k
k
(nk)–
√
,
where the right-hand side tends to∞whenk→ ∞. Consequently, the sequencexk()
cannot have a finite limit fork→ ∞.
Remark . Reasonable examples of sequences{fk} ⊂G([a,b],X) that tend to a function f of bounded variation are providede.g.by sequences of the formfk=gk+hk, where{gk} ⊂ BV([a,b],X) tends tof∈BV([a,b],X) and{hk} ⊂G([a,b],X) tends to .
Remark . ForF: [a,b]→L(X) andD={α,α, . . . ,αm} ∈D[a,b], define
Vab(F,D) =sup
m j=
F(αj) –F(αj–)
yj X
:yj∈X,yjX≤,j= , , . . . ,m
and
SVba(F) =supVab(F,D);D∈D[a,b].
ThenSVba(F) is said to be thesemi-variation of F on [a,b] (cf. e.g.[]).bIt is clear that ifF∈BV([a,b],L(X)) thenFhas bounded semi-variation on [a,b] while the reversed im-plication is not true in general (cf.[, Theorem ]). By [] and [], the Kurzweil-Stieltjes integralabd[A]xis well defined when both functions,Aandx, are regulated andAhas bounded semi-variation. Therefore, the study of generalized linear differential equations has a good sense also whenAis regulated and has bounded semi-variation instead of hav-ingA∈BV([a,b],X),cf.[] and []. However, the possible extension of Theorem . to such a case remains open.
Analogously to operator valued functions, the semi-variation of a functionf : [a,b]→X
could be defined using
Vab(f,D)
=sup
m
j= Fj
f(αj) –f(αj–)
X
:Fj∈L(X),FjL(X)≤,j= , , . . . ,m
.
However, it may be shown that, in this case,f has a bounded semi-variation if and only
f ∈BV([a,b],X). Therefore, the possible replacement of the conditionf ∈BV([a,b],X) in Theorem . by the requirement thatf has a bounded semi-variation is not interesting.
4 Some applications
Second-order measure equations
LetYbe a Banach space,y,z∈Y,P,Q∈BV([a,b],L(Y)) andg,h∈BV([a,b],Y). Consider the following system of generalized linear differential equations:
y(t) =y+
t
a
d[P]z+g(t) –g(a), t∈[a,b],
z(t) =z+
t
a
d[Q]y+h(t) –h(a), t∈[a,b].
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(.)
PutX=Y×Yand(y,z)X=yY+zYfor (y,z)∈Xand define functionsA: [a,b]→ L(X) andf : [a,b]→Xby
A(t)(y,z) =P(t)z,Q(t)y ∈X and f(t) =g(t),h(t) ∈X
fory,z∈Yandt∈[a,b].
⎫ ⎬
⎭ (.)
Clearly,
A(t)L(X)=P(t)L(Y)+Q(t)L(Y) fort∈[a,b], varbaA≤varbaP+varbaQ
conditions is true:
IY––Q(t)–P(t)
–∈
L(Y) fort∈(a,b], (.)
IY–y–P(t)–Q(t)
–∈
L(Y) fort∈(a,b], (.)
whereIY stands for the identity operator onY.
Indeed, assumee.g.that (.) holds and let [IX––A(t)]x= for somex= (y,z)∈Xand t∈(a,b]. Then
y––P(t)z= and z––Q(t)y= , (.)
i.e.
y=–P(t)z and IY––Q(t)–P(t)
z= . (.)
By (.) the latter equality can happen only ifz= . Consequentlyy= , and hencex= , as well. Similarly, we would show that [IX––A(t)]x= impliesx= also in the case that
(.) is satisfied. This shows that the operatorIX––A(t) is injective.
To prove its surjectivity, assume first (.) and let (u,v)∈Xbe given. Put
z=IY––Q(t)–P(t)
–
v+–Q(t)u and y=u+–P(t)z.
Then,y––P(t)z=uand
z––Q(t)y=z––Q(t)–P(t)z––Q(t)u
=IY––Q(t)–P(t)
z––Q(t)u=v,
that is, [IX––A(t)]x= (u,v) forx= (y,z). Similarly, we can show that for each (u,v)∈X
there isx∈Xsuch that [IX––A(t)]x= (u,v) also in the case that (.) is satisfied. The
operatorIX––A(t) is surjective. To summarize, according to the Banach theorem, the
operatorIX––A(t) possesses a bounded [IX––A(t)]–.
Now, consider the systems
yk(t) =yk+
t
a
d[Pk]zk+gk(t) –gk(a), t∈[a,b],k∈N,
zk(t) =zk+
t
a
d[Qk]yk+hk(t) –hk(a), t∈[a,b],k∈N,
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
(.)
whereyk,zk∈Y,Pk,Qk∈BV([a,b],L(Y)),gk,hk∈G([a,b],Y) andk∈N. Assume that (.)
or (.) is true and
lim
k→∞yk–yY = , klim→∞zk–zY= , (.) lim
k→∞
and
lim k→∞
+varbaPk+varbaQk gk–g∞+hk–h∞ = . (.)
DefineAk: [a,b]→L(X) andfk: [a,b]→Xfork∈NlikeAandfin (.) (however, replace P,Q,g, andhbyPk,Qk,gkandhk, respectively). It is easy to see that then the assumptions
of Theorem . are satisfied. Therefore, we can state the following assertion.
Corollary . Assume that (.)or(.)holds and that(.)-(.)are satisfied.Then system(.) has a unique solution(y,z)∈BV([a,b],Y×Y)on[a,b]. Moreover,for each k∈Nsufficiently large,the system(.)has a unique solution(yk,zk)∈G([a,b],Y×Y)on
[a,b]and lim k→∞
yk–y∞+zk–z∞ = .
In [], Meng and Zhang investigated the continuous dependence on a parameterkfor second-order linear measure differential equations of the form
dy•+ dμk(t)
y= , t∈[, ], y() =y, y•() =z, k∈N, (.) whereμkare normalized measures on [, ] (generated by functions of bounded variation
on [, ] and right-continuous in (, )),y,z∈Randy• stands for the generalized right-derivative ofy. The main result of [] is Theorem ., which states that the weak∗ conver-genceμk→μimplies the uniform convergenceyk⇒yof the corresponding solutions,
the weak∗convergencey•k→y•and the ending velocity convergencey•k()→y•(). Notice that our systems (.) reduce to (.) when [a,b] = [, ],X=R,Pk(t) =tand Qk(t) =μk(t) fort∈[, ] and bothgkandhkare constant [, Definition .]. Similarly,
if, in addition,P(t) =tandQ(t) =μ(t) fort∈[, ] and bothg andhare constant, then system (.) reduces to the second-order linear measure differential equation of the form dy•+ dμ(t)y= , t∈[, ], y() =y, y•() =z, k∈N, (.) whereμis a normalized measure on [, ] andy,z∈R. Obviously, both existence con-ditions (.) and (.) are now satisfied. In view of this, assuming thatμandμk have a
bounded variation on [, ] and
lim k→∞
+varμk μk–μ∞= ,
it follows from our Corollary . that
lim k→∞
yk–y∞+y•k–y•∞ =
holds for the corresponding solutions of (.) and (.).
Thus, in comparison with Theorem . in [], our convergence assumptions are par-tially 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 [] includes the uniform boundedness of the variationsvar
μk (cf. e.g.[, Lemma .] or [,
Linear dynamic equations on time scales
Let us recall some basics of the theory of dynamic equations on time scales. A nonempty closed subsetTofRis calledtime scale. For givena,b∈T, we put [a,b]T= [a,b]∩T. For
t∈T, we define
ρ(t) :=sup[a,t)∩T and σ(t) :=inf(t,b]∩T.
The pointt∈Tis said to beright-denseifσ(t) =t, while it isleft-denseifρ(t) =t. A func-tionf: [a,b]T→Rmisrd-continuousin [a,b]Tiffis continuous at every right-dense point
of [a,b]Tand there existsf(t–) for every left-dense pointt∈[a,b]T(seee.g.[]).
Let us consider the linear dynamic equation
y(t) =P(t)y(t) +h(t), y(a) =y, t∈[a,b]T, (.)
wherey∈Rm andP : [a,b]
T→L(Rm), h: [a,b]T→Rm are rd-continuous functions
and y stands for the-derivative. By a solution of (.) we understand a function
y: [a,b]T→Rmsatisfying the integral equation
y(t) =y+
t
a
P(s)y(s) +h(s)s, t∈[a,b]T,
where the integral is the Riemann-integral definede.g.in [].
As noticed by Slavík (see [, Theorem ]), the Riemann-integral can be regarded as a special case of the Kurzweil-Stieltjes integral. More precisely:
Let f : [a,b]T→Rmbe an rd-continuous function and
σ(t) :=inf[t,b]∩T for t∈[a,b],
F(t) =
t
a
f(s)s for t∈[a,b]T
and
F(t) =
t
a
fσ(s) dσ(s) for t∈[a,b].
Then F(t) =F(σ(t))holds for t∈[a,b].
As a consequence, a relationship between the solutions of (.) and generalized linear differential equations can be deduced.
Proposition .[, Theorem ] If y: [a,b]T→Rmis a solution of (.)then
x=y◦σy: [a,b]→Rm
is a solution of (.),where
A(t) =
t
a
Pσ(s) dσ(s) for t∈[a,b] and
f(t) =
t
a
hσ(s) dσ(s) for t∈[a,b].
⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭
Symmetrically,if x: [a,b]→Rm is a solution of (.),with A and f given by(.),then y: [a,b]T→Rmdefined by y(t) =x(t)for t∈[a,b]Tis a solution of (.).
It is important to mention that, thanks to the properties ofσ: [a,b]→[a,b]T, the
func-tionsA: [a,b]→L(Rn) andf : [a,b]→Rngiven by (.) are well defined, left-continuous
and of bounded variation on [a,b].
Using the correspondence stated in Proposition . and Theorem . we obtain the fol-lowing result.
Theorem . Let P,Pk: [a,b]T→L(Rm),h,hk: [a,b]T→Rmfor k∈Nbe rd-continuous functions in[a,b]Tand lety,yk∈Rm,k∈N,be given.Assume that
lim
k→∞yk–yRm= , (.) lim
k→∞
+ sup t∈[a,b]T
Pk(t)L(Rm)
sup t∈[a,b]T
t
a
Pk(s) –P(s) s
L(Rm)
= ,
lim k→∞
+ sup t∈[a,b]T
Pk(t)L(Rm)
sup t∈[a,b]T
athk(s) –h(s) s
Rm = . ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (.)
Then initial value problem(.)has a solution y,the initial value problems
yk(t) =Pk(t)yk(t) +hk(t), yk(a) =yk, t∈[a,b]T (.)
have solutions ykfor all k∈N,and lim
k→∞t∈sup[a,b]Tyk(t) –y(t)Rm= . Proof For eachk∈Nandt∈[a,b], define
Ak(t) =
t
a Pk
σ(s) dσ(s) and fk(t) =
t
a hk
σ(s) dσ(s). (.)
It is not difficult to see that, ifa≤c<d≤b, then
Ak(d) –Ak(c)L(Rm)= d
c Pk
σ(s) dσ(s)
L(Rm)≤
sup t∈[a,b]T
Pk(t)L(Rm)
vardcσ ,
and, consequently,
varbaAk≤
sup t∈[a,b]T
Pk(t)L(Rm)
varbaσ , k∈N.
On the other hand,
Ak–A∞= sup t∈[a,b]
σ(t)
a
Pk(s) –P(s) s
L(Rm)≤
sup t∈[a,b]T
t
a
Pk(s) –P(s)s
L(Rm)
and, analogously,
fk–f∞≤ sup t∈[a,b]T
athk(s) –h(s) s
These estimates, together with (.) and (.) imply that the assumptions of Theo-rem . are satisfied. Therefore, the uniform convergence of solutionsxkof equation (.)
to the solutionxof (.) follows. Since by Proposition . the solutions of (.) and (.) are, respectively, obtained as the restriction ofxandxkto [a,b]T, the proof is complete.
Remark . It is worth to mention that Theorem . given above encompasses Theo-rem . from []. This is due to the fact that the weighted convergence assumptions in [, Theorem .] involves not only the supremum supt∈[a,b]TPk(t)L(Rm), but also
supt∈[a,b]Thk(t)Rm.
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.
Author details
1Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic.2Mathematical Institute, Academy of Sciences of Czech Republic, Žitná 25, Praha, 115 67, Czech Republic.
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 Vrkoˇc for his valuable contribution to this paper.
Endnotes
a [x] stands, as usual, for the integer part of the nonnegative real numberx.
b Sometimes it is called also theB-variation of Fon [a,b] (with respect to the bilinear tripleB= (L(X),X,X),cf. e.g.[8]).
Received: 20 January 2014 Accepted: 13 March 2014 Published:26 Mar 2014
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10.1186/1687-2770-2014-71
