VIVERO Received 18 January 2006; Accepted 22 January 2006 We study the theory of Sobolev’s spaces of functions defined on a closed subinterval of an arbitrary time scale endowed with the
Trang 1TIME SCALES
RAVI P AGARWAL, VICTORIA OTERO–ESPINAR, KANISHKA PERERA,
AND DOLORES R VIVERO
Received 18 January 2006; Accepted 22 January 2006
We study the theory of Sobolev’s spaces of functions defined on a closed subinterval of an arbitrary time scale endowed with the LebesgueΔ-measure; analogous properties to that valid for Sobolev’s spaces of functions defined on an arbitrary open interval of the real numbers are derived
Copyright © 2006 Ravi P Agarwal et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
1 Introduction
Sobolev’s spaces are a fundamental tool in real analysis, for instance, in the use of vari-ational methods to solve boundary value problems in ordinary and partial differential equations and difference equations In spite of this, theory for functions defined on an arbitrary bounded open interval of the real numbers is well known, see [2], and for func-tions defined on an arbitrary bounded subset of the natural numbers is trivial, as far as
we know, for functions defined on an arbitrary time scale, it has not been studied before The aim of this paper is to give an introduction to Sobolev’s spaces of functions defined
on a closed interval [a, b] ∩ Tof an arbitrary time scaleTendowed with the Lebesgue Δ-measure InSection 2, we gather together the concepts one needs to read this paper, such
as theL pspaces linked to the LebesgueΔ-measure and absolutely continuous functions
on an arbitrary closed interval ofT The most important part of this paper isSection 3
where we define the first-order Sobolev’s spaces as the space ofLΔp([a, b) ∩ T) functions whose generalizedΔ-derivative belongs to L p
Δ([a, b) ∩ T), moreover, we study some of their properties by establishing an equivalence between them and the usual Sobolev’s spaces defined on an open interval of the real numbers.Section 4is devoted to the gener-alization of Sobolev’s spaces to ordern ≥2
2 Preliminaries
The LebesgueΔ-measure μΔ was defined in [1, Section 5.7] or in [5, Section 5] as the Carath´eodory extension of a set function and it may be characterized in terms of
Hindawi Publishing Corporation
Advances in Di fference Equations
Volume 2006, Article ID 38121, Pages 1 14
DOI 10.1155/ADE/2006/38121
Trang 2well-known measures as the following result shows; we refer the reader to [6–8] for a broad introduction to measure and integration theory
Proposition 2.1 The Lebesgue Δ-measure is defined over the Lebesgue measurable subsets
ofT; moreover, it satisfies the following equality:
μΔ=
⎧
⎪
⎪
λ +
i ∈ I
σ
t i
− t i
· δ t i+μ M, if M ∈ T,
λ +
i ∈ I
σ
t i
− t i
where { t i } i ∈ I , I ⊂ N , is the set of all right-scattered points ofT, M is the supremum ofT, λ
is the Lebesgue measure, δ t i is the Dirac measure concentrate at t i , and μ M is a degenerate measure defined as μ M( A) = 0 if M ∈ A and μ M( A) =+∞ if M ∈ A.
Proof From properties of measure, one can deduce relation (2.1) for the outer measures linked to these measures which plainly yields to (2.1)
As a straightforward consequence of equality (2.1), one can deduce the following for-mula to calculate the LebesgueΔ-integral; this formula was proved in [4], nevertheless,
we remark that this argument is more simple than that
Proposition 2.2 Let E ⊂ T be a Δ-measurable set If f : T → R is Δ- integrable on E, then
E f (s) Δs =
E f (s)ds +
i ∈ I E
σ
t i
− t i
· f
t i +r( f , E), (2.2)
where
r( f , E) =
⎧
⎨
⎩
μ M( E) · f (M), if M ∈ T,
I E:= { i ∈ I : t i ∈ E } and { t i } i ∈ I , I ⊂ N , is the set of all right-scattered points ofT.
Definition 2.3 Let A ⊂ T.A is called Δ-null set if μΔ(A) =0 Say that a propertyP holds Δ-almost everywhere (Δ-a.e.) on A, or for Δ-almost all (Δ-a.a.) t ∈ A if there is aΔ-null setE ⊂ A such that P holds for all t ∈ A \ E.
Definition 2.4 Let E ⊂ Tbe aΔ-measurable set and let p ∈ R ≡¯ [−∞, +∞] be such that
p ≥1 and letf : E →R¯ be aΔ-measurable function Say that f belongs to L p
Δ(E) provided
that either
E | f | p(s) Δs < ∞ if p ∈ R, (2.4)
or there exists a constantC ∈ Rsuch that
Note that equality (2.2) guarantees that in order for f : T → Rto belong toLΔp(T),
p ∈ R, andTbounded from above, it is necessary that f (M) =0 We will work with the
Trang 3LΔp(J o) spaces, whereJ =[a, b] ∩ T,a, b ∈ T,a < b, is an arbitrary closed subinterval ofT andJ o =[a, b) ∩ T; we state some of their properties whose proofs can be found in [6–8]
Theorem 2.5 Let p ∈R¯ be such that p ≥ 1 Then, the set L pΔ(J o ) is a Banach space together with the norm defined for every f ∈ LΔp(J o ) as
f L p
Δ:=
⎧
⎪
⎪ J o | f | p(s) Δs 1/ p, if p ∈ R, inf
C ∈ R:| f | ≤ C Δ-a.e on J o , if p =+∞
(2.6)
Moreover, L2
Δ(J o ) is a Hilbert space together with the inner product given for every ( f , g) ∈
L2
Δ(J o)× L2
Δ(J o ) by
(f , g) L2
Δ:=
Proposition 2.6 Suppose p ∈R¯ and p ≥ 1 Let p ∈R¯ be such that 1/ p + 1/ p = 1 Then, if f ∈ LΔp(J o ) and g ∈ LΔp (J o ), then f · g ∈ L1
Δ(J o ) and
f · g L1
Δ≤ f L p
Δ· g L p
This expression is called H¨older’s inequality and Cauchy-Schwarz’s inequality whenever
p = 2.
Proposition 2.7 If p ∈ R and p ≥ 1, then, the set C c( J o ) of all continuous functions on J o with compact support in J o is dense in LΔp(J o ).
As a consequence ofProposition 2.2, one can establish the following equivalence be-tween theLΔp(J o) spaces and the usualL p([a, b]) spaces linked to the Lebesgue measure Corollary 2.8 Let p ∈R¯ with p ≥ 1, let f : J →R¯, and let f : [a, b] →R¯ be the extension
of f to [a, b] defined as
f (t) : =
⎧
⎨
⎩
f (t), if t ∈ J,
f (t i), if t ∈t i, σ
t i
, for some i ∈ I J, (2.9)
with I J:= { i ∈ I : t i ∈ J } and { t i } i ∈ I , I ⊂ N , is the set of all right-scattered points ofT Then, f ∈ LΔp(J o ) if and only if f ∈ L p([a, b]) In this case,
f L p
As we know from general theory of Sobolev’s spaces, another important class of func-tions is just the absolutely continuous funcfunc-tions
Definition 2.9 A function f : J → Ris said to be absolutely continuous onJ, f ∈ AC(J),
if for every ε > 0, there exists a δ > 0 such that if {[a k,b k)∩ T} n
k =1, witha k,b k ∈ J, is
a finite pairwise disjoint family of subintervals ofJ satisfyingn
k =1(b k − a k) < δ, then
n
k =1| f (b k) − f (a k)| < ε.
Trang 4These functions are precisely that for which the fundamental theorem of Calculus holds
Theorem 2.10 [3, Theorem 4.1] A function f : J → R is absolutely continuous on J if and only if f is Δ-differentiable Δ-a.e on J o , fΔ∈ L1
Δ(J o ) and
f (t) = f (a) +
[a,t) ∩T fΔ(s) Δs, ∀ t ∈ J. (2.11) Absolutely continuous functions onTverify the integration by parts formula
Theorem 2.11 If f , g : J → R are absolutely continuous functions on J, then f · g is abso-lutely continuous on J and the following equality is valid:
J o
fΔg + f σ gΔ
(s) Δs = f (b) g(b) − f (a) g(a) =
J o
f gΔ+ fΔg σ
(s) Δs. (2.12) They are linked to the class of absolutely continuous functions on [a, b] as the
follow-ing property shows
Corollary 2.12 [3, Corollary 3.1] Assume that f : J → R and define ¯ f : [a, b] → R as
¯f(t) : =
⎧
⎪
⎪
f
t i
+ f
σ
t i
− f
t i
σ
t i
− t i
t − t i
, if t ∈t i, σ
t i
, for some i ∈ I J, (2.13)
with I J:= { i ∈ I : t i ∈ J } and { t i } i ∈ I , I ⊂ N , is the set of all right-scattered points ofT Then, f is absolutely continuous on J if and only if ¯f is absolutely continuous on [a, b].
Moreover, for everyn ∈ N,n ≥1, we will denote as
AC n(J) : = x ∈ AC(J) : xΔj ∈ AC
J κ j
∀ j ∈ {1, , n } , (2.14) where for every j ∈ N, ≥1,J κ j
=[a, ρ j(b)] ∩ T
3 First-order Sobolev’s spaces
The aim of this section is to study the first-order Sobolev’s spaces onJ equipped with the
LebesgueΔ-measure
Definition 3.1 Let p ∈R¯ be such thatp ≥1 andu : J →R Say that¯ u belongs to WΔ1,p(J)
if and only ifu ∈ LΔp(J o) and there existsg : J κ →R¯ such thatg ∈ LΔp(J o) and
J o
u · ϕΔ (s) Δs = −
J o
g · ϕ σ (s) Δs ∀ ϕ ∈ C1
0,rd
J κ
(3.1) with
C1
0,rd
J κ := f : J −→ R: f ∈ C1
rd
J κ , f (a) =0= f (b) (3.2) andC1
rd(J κ) is the set of all continuous functions onJ such that they areΔ-differentiable
onJ κand theirΔ-derivatives are rd-continuous on J κ
Trang 5The integration by parts formula for absolutely continuous functions onJ establishes
that the relation
VΔ1,p(J) : = x ∈ AC(J) : xΔ∈ LΔp
J o ⊂ WΔ1,p(J) (3.3)
is true for everyp ∈R¯ withp ≥1 We will show that both sets are, as class of functions, equivalent; for this purpose, we need the following lemmas
Lemma 3.2 Let f ∈ L1Δ(J o ) be such that the following equality is true:
J o(f · u)(s) Δs =0, ∀ u ∈ C c
J o
then
Proof Fix ε > 0, the density of C c( J o) inL1
Δ(J o) guarantees the existence of f1∈ C c( J o) such that f − f1 L1
Δ< ε, and so, by (3.4), we deduce that for everyu ∈ C c( J o), it is true that
J o
f1· u (s) Δs
≤ u C(J o)·f − f1
L1
Δ< ε u C(J o). (3.6) Because the sets
A1:= s ∈ J o: f1(s) ≥ ε , A2:= s ∈ J o: 1(s) ≤ − ε (3.7) are compact and disjoint subsets ofJ o, Urysohn’s lemma allows to construct a function
u0:J o → Rwhich belongs toC c( J o) and it verifies
u0≡
⎧
⎨
⎩
1; onA1,
−1; onA2,
u0 ≤1 on J o; (3.8)
so that, by definingA : = A1∪ A2, we have that
J o
f1(s) Δs =
J o
f1· u0
(s) Δs −
J o \ A
f1· u0
(s) Δs
+
J o \ A
f1(s) Δs ≤ ε + 2ε(b − a). (3.9)
As a consequence of the arbitrary choice ofε > 0, we achieve (3.5)
Lemma 3.3 Let f ∈ L1
Δ(J o ) Then, a necessary and su fficient condition for the validity of the equality
J o
f · ϕΔ (s) Δs =0, for every ϕ ∈ C0,1rd
J κ
is the existence of a constant c ∈ R such that
Trang 6Proof The necessary condition is consequence of the fundamental theorem of Calculus.
Conversely, fixu ∈ C c( J o) arbitrary; by definingh, ϕ : J → Ras
h(t) : =
⎧
⎪
⎨
⎪
⎩
u(t) −
J o u(r) Δr
b − a , ift ∈ J o,
−
J o u(r) Δr
b − a , ift = b, ϕ(t) : =
[a,t) ∩T h(s) Δs, ∀ t ∈ J,
(3.12)
the fundamental theorem of Calculus establishes thatϕ ∈ C10,rd(J κ) and so, equality (3.10) yields to
0=
J o f ·
u −
J o u(r) Δr
b − a
(s) Δs
=
J o f −
J o f (r) Δr
b − a
· u
(s) Δs.
(3.13)
Therefore,Lemma 3.2allows to deduce (3.11) withc =J o f (r) Δr/(b − a). Now, we are able to prove the characterization of functions inWΔ1,p(J) in terms of
functions inVΔ1,p(J).
Theorem 3.4 Suppose that u ∈ WΔ1,p(J) for some p ∈R¯ with p ≥ 1 and that ( 3.1 ) holds for g ∈ L pΔ(J o ) Then, there exists a unique function x ∈ VΔ1,p(J) such that the equalities
are satisfied.
Moreover, if g ∈ C rd(J κ ), then there exists a unique function x ∈ C1
rd(J κ ) such that
x = u Δ-a.e on J o, xΔ= g on J κ (3.15)
Proof Define v : J → Ras
v(t) : =
[a,t) ∩T g(s) Δs, ∀ t ∈ J; (3.16) the fundamental theorem of Calculus guarantees thatv ∈ VΔ1,p(J) and by the integration
by parts formula, we have that for everyϕ ∈ C1
0,rd(J κ),
J o
(v − u) · ϕΔ
(s) Δs = −
J o
vΔ− g
· ϕ σ (s) Δs =0; (3.17)
so that,Lemma 3.3ensures the existence of a constantc ∈ Rsuch thatv − u ≡ cΔ-almost everywhere onJ o As a consequence of the fundamental theorem of Calculus we conclude that functionx : J → Rdefined asx(t) : = v(t) − c for all t ∈ J is the unique function in
VΔ1,p(J) for which (3.14) is valid
Trang 7Furthermore, ifg ∈ C rd( J κ), then the fundamental theorem of Calculus establishes
By identifying every function inWΔ1,p(J) with its absolutely continuous representative
inVΔ1,p(J) for which (3.14) holds, the setWΔ1,p(J) can be endowed with the structure of
Banach space
Theorem 3.5 Assume p ∈R¯ and p ≥ 1 The set WΔ1,p(J) is a Banach space together with the norm defined for every x ∈ WΔ1,p(J) as
x W1,p
Δ := x L p
Δ+xΔ
Moreover, the set H1
Δ(J) : = WΔ1,2(J) is a Hilbert space together with the inner product given for every (x, y) ∈ H1
Δ(J) × H1
Δ(J) by
(x, y) H1
Δ:=(x, y) L2
Δ+
xΔ,yΔ
L2
Proof Let { x n } n ∈Nbe a Cauchy sequence inWΔ1,p(J);Theorem 2.5guarantees the exis-tence ofu, g ∈ L pΔ(J o) such that{ x n } n ∈Nand{ x nΔ} n ∈Nconverge strongly inLΔp(J o) tou and
g, respectively, and so, by taking limits in the equality
J o
x n · ϕΔ (s) Δs = −
J o
xΔn · ϕ σ (s) Δs, ϕ ∈ C1
0,rd(J κ), (3.20)
we conclude thatu ∈ WΔ1,p(J) Thereby, it follows fromTheorem 3.4, that there exists
x ∈ WΔ1,p(J) such that { x n } n ∈Nconverges strongly inWΔ1,p(J) to x.
3.1 Some properties We will derive some properties of the Banach spaceWΔ1,p(J); the
first one asserts thatWΔ1,p(J) is continuously inmersed into C(J) equipped with the
supre-mum norm · C(J).
Proposition 3.6 Assume p ∈R¯ with p ≥ 1, then there exists a constant K > 0, only de-pendent on b − a, such that the inequality
x C(J) ≤ K · x W1,p
holds for all x ∈ WΔ1,p(J) and hence, the immersion WΔ1,p(J) C(J) is continuous.
Proof Fix x ∈ WΔ1,p(J) Let t, T ∈ J be such that | x(t) |:=mins ∈T | x(s) | and | x(T) |:=
maxs∈T | x(s) |; there is no harm in assumingt ≤ T The fundamental theorem of Calculus
and H¨older’s inequality lead to
x C(J) ≤ | x(t) |+
[t,T) ∩T | xΔ|(s) Δs ≤ K · x W1,p
The strong compactness criterion inC(J) andProposition 3.6allow to prove the fol-lowing compactness property inC(J).
Trang 8Proposition 3.7 Let p ∈R¯ be such that p ≥ 1 Then, the following statements are true (1) If p > 1, then the immersion WΔ1,p(J) C(J) is compact.
(2) If p = 1, then the immersion WΔ1,p(J) C(J) is compact if and only if every point of
J is isolated.
Proof Denote byᏲpthe closed unit ball inWΔ1,p(J); we know fromTheorem 3.4thatᏲp
is closed and bounded inC(J).
If p > 1, then the fundamental theorem of Calculus and H¨older’s inequality ensure
thatᏲpis equicontinuous
On the other hand, ifp =1, then it is clear thatᏲpis equicontinuous whenever every point ofJ is isolated, while if there exists t0∈ Tsuch thatt0is not isolated, then we will prove thatᏲpis not equicontinuous
LetS : =1/(b − a + 1), let δ > 0 be arbitrary and let s δ ∈(t0− δ, t0+δ) ∩ Tbe such that
s δ = t0; it is not a loss of generality assumings δ < t0
Define f δ:J → Ras
f δ:=
⎧
⎪
⎪
S
t0− s δ, ift ∈s δ, t0
∩ J ,
0, ift ∈s δ, t0
∩ J
;
(3.23)
the fundamental theorem of Calculus asserts thatF δ:J → Rgiven by
F δ( t) : =
[a,t) ∩T f δ( s) Δs, t ∈ J, (3.24) belongs toᏲp; so that, as
F δ
t0
− F δ
s δ
=
[s δ,t0 )∩T f δ(s) Δs = S, (3.25)
we conclude thatᏲpis not equicontinuous
Therefore, Arzel`a-Ascoli theorem establishes our claims
As a consequence ofProposition 3.6, we achieve the following sufficient condition for strong convergence inC(J).
Corollary 3.8 Let p ∈R¯ be such that p > 1, let { x m } m ∈N ⊂ WΔ1,p(J), and let x ∈ WΔ1,p(J).
If { x m } m ∈N converges weakly in WΔ1,p(J) to x, then { x m } m ∈N converges strongly in C(J)
to x.
Proof Suppose { x m } m ∈N converges weakly inWΔ1,p(J) to x;Proposition 3.6establishes that { x m } m ∈N converges weakly in C(J) to x and so, as { x m } m ∈N is equicontinuous,
Moreover, Proposition 3.6 allows to deduce the following equivalence between the Sobolev’s spaces onJ, WΔ1,p(J), and the usual Sobolev’s spaces on (a, b), W1,p((a, b)) Corollary 3.9 Suppose that p ∈R¯ and p ≥ 1, x : J → R and ¯ x : [a, b] → R is the exten-sion of x to [a, b] defined in ( 2.13 ) Then, x belongs to WΔ1,p(J) if and only if ¯x belongs to
W1,p((a, b)).
Trang 9Moreover, there exist two constants K1,K2> 0 which only depend on (b − a) such that the inequalities
K1· ¯x W1,p ≤ x W1,p
Δ ≤ K2· ¯x W1,p (3.26)
are satisfied for every x ∈ WΔ1,p(J) and p ∈R¯ with p ≥ 1.
Proof Let ¯ x, x Δ: [a, b] → Rbe the extensions ofx and xΔto [a, b] defined in (2.13) and (2.9), respectively; it is not difficult to deduce the following equality:
Therefore, Corollaries2.8and2.12andProposition 3.6yield to the result
As an application of the previous result, we will prove that some properties known forW1,p((a, b)) are directly transferred to WΔ1,p(J); in order to do this, we will use the
following result
Proposition 3.10 If y : [a, b] → R belongs to W1,p((a, b)) for some p ∈R¯ with p ≥ 1, then y | J belongs to WΔ1,p(J) Moreover, there exists a constant T > 0 which only depends on
(b − a) such that
y | J W1,p
Δ ≤ T · y W1,p, ∀ y ∈ W1,p
(a, b) , p ∈R,¯ p ≥1. (3.28)
Proof Let R = { t i } i ∈ I, I ⊂ N, be the set of all right-scattered points ofT, letI J o = { i ∈ I,
t i ∈ J o }and suppose y ∈ W1,p((a, b)) for some p ∈R¯ withp ≥1 The classical funda-mental theorem of Calculus allows to assert that
y | J Δ
t i
=
[t i,σ(t i)]y (s)ds
σ
t i
− t i , for every i ∈ I J o,
y | J
Δ
= y a.e on J o ∩(T\R).
(3.29)
Therefore, if p =+∞, then it is clear thaty | J ∈ WΔ1,p(J) and (3.28) holds while ifp ∈ R, then, by (2.2), we have that
y | JΔp
LΔp ≤
J o ∩(T\ R)
y p
(s)ds +
i ∈ I Jo
[t i,σ(t i)]
y p
(s)ds ≤ y W p1,p, (3.30)
moreover, as we know that
y | J
L pΔ≤(b − a)1/ p · y C([a,b]) ≤ C ·(b − a)1/ p · y W1,p, (3.31) for someC > 0, it turns out that y | J ∈ WΔ1,p(J) and (3.28) is true Next, we deduce some properties inWΔ1,p(J) from the analogous ones in W1,p((a, b)) Corollary 3.11 Let p ∈R¯ be such that p ≥ 1 Then, for every q ∈[1, +∞ ), the inmersion
WΔ1,p(J) L q
Δ(J o ) is compact.
Trang 10Proof Fix q ∈[1, +∞); as a consequence ofProposition 3.7and the fact that the inmer-sionC(J) L q
Δ(J o) is continuous, it only remains to prove thatᏲ1is compact inL qΔ(J o) wheneverJ has at least one not isolated point.
Assume the existence of a not isolated point t0∈ J and let { x n } n ∈N be a sequence
inᏲ1.Corollary 3.9ensures that{ x n } n ∈N, defined in (2.13), is a bounded sequence in
W1,1((a, b)) and hence, there exist { x n k } k ∈Nandy ∈ L q([a, b]) such that { x n k } k ∈N con-verges strongly inL q([a, b]) to y By defining x : = y | J, it is not difficult to prove that
Corollary 3.12 The Banach space WΔ1,p(J) is reflexive for every p ∈(1, +∞ ) and separable for all p ∈[1, +∞ ).
Proof Let p ∈R¯ be such thatp ≥1 We know, fromCorollary 3.9, that the operatorT p:
WΔ1,p(J) → W1,p((a, b)) given for every x ∈ WΔ1,p(J) by T p(x) : = ¯x, defined in (2.13), is lin-ear and continuous It follows fromCorollary 3.9andProposition 3.10thatT p(WΔ1,p(J))
is a closed subspace ofW1,p((a, b)) Therefore, since W1,p((a, b)) is reflexive whenever
p ∈(1, +∞) and separable wheneverp ∈[1, +∞),T p(WΔ1,p(J)) satisfies the same
Corollary 3.13 If x ∈ WΔ1,p(J) for some p ∈[1, +∞ ), then there exists a sequence of in-finitely differentiable functions with compact support inR, { y n } n ∈N such that { y n | J } n ∈N
converges strongly in WΔ1,p(J) to x.
Proof. Corollary 3.9asserts that ¯x : [a, b] → R, defined in (2.13), belongs toW1,p((a, b));
so that, there exists a sequence{ y n } n ∈Nof infinitely differentiable functions with compact support in Rsuch that { y n |[a,b] } n ∈N converges to ¯x in W1,p((a, b)) Hence, our claim
3.2 The spacesW0,Δ1,p(J) Corollary 3.13 guarantees the density of the set C rd1 (J κ) in
WΔ1,p(J) for every p ∈[1, +∞); however, for an arbitrary bounded time scale it is not true that the set of test functions defined in (3.2),C1
0,rd(J κ), is dense inWΔ1,p(J); this section is
devoted to prove some properties concerning the closure ofC1
0,rd(J κ) inWΔ1,p(J) Definition 3.14 Let p ∈ Rbe such thatp ≥1, define the setW0,Δ1,p(J) as the closure of the
setC1
0,rd(J κ) inWΔ1,p(J) Denote as H1
0, Δ(J) : = W0,1,2Δ(J).
The spacesW0,Δ1,p(J) and H1
0,Δ(J) are endowed with the norm induced by · W1,p
Δ , de-fined in (3.18), and the inner product induced by (·,·)H1
Δ, defined in (3.19), respectively SinceW0,1,Δp(J) is closed in WΔ1,p(J),Theorem 3.5andCorollary 3.12ensure thatW0,1,Δp(J) is
a separable Banach space and reflexive wheneverp > 1 and H1
0,Δ(J) is a separable Hilbert
space The spaceW0,Δ1,p(J) is characterized in the following result.
Proposition 3.15 Assume x ∈ WΔ1,p(J) Then, x ∈ W0,Δ1,p(J) if and only if x(a) =0= x(b).
... know from general theory of Sobolev’s spaces, another important class of func-tions is just the absolutely continuous funcfunc-tionsDefinition 2.9 A function f : J → Ris... First-order Sobolev’s spaces< /b>
The aim of this section is to study the first-order Sobolev’s spaces on< i>J equipped with the
LebesgueΔ-measure
Definition 3.1 Let... ⊂ N , is the set of all right-scattered points of< /i>T Then, f is absolutely continuous on J if and only if ¯f is absolutely continuous on [a, b].
Moreover,