BASIC PROPERTIES OF SOBOLEV’S SPACES ON TIME 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 Lebesgue Δ-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 Introduction Sobolev’s spaces are a fundamental tool in real analysis, for instance, in the use of variational 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 functions 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] ∩ T of an arbitrary time scale T endowed with the Lebesgue Δmeasure In Section 2, we gather together the concepts one needs to read this paper, such as the L p spaces linked to the Lebesgue Δ-measure and absolutely continuous functions on an arbitrary closed interval of T The most important part of this paper is Section p where we define the first-order Sobolev’s spaces as the space of LΔ ([a,b) ∩ T) functions p whose generalized Δ-derivative belongs to LΔ ([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 is devoted to the generalization of Sobolev’s spaces to order n ≥ 2 Preliminaries The Lebesgue Δ-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 Difference Equations Volume 2006, Article ID 38121, Pages 1–14 DOI 10.1155/ADE/2006/38121 Basic properties of Sobolev’s spaces on time scales well-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 of T; moreover, it satisfies the following equality: ⎧ ⎪ ⎪ ⎨λ + σ ti − ti · δti + μM , μ Δ = ⎪ i ∈I ⎪ σ ti − ti · δti , ⎩λ + if M ∈ T, if M ∈ T, (2.1) i ∈I where {ti }i∈I , I ⊂ N, is the set of all right-scattered points of T, M is the supremum of T, λ is the Lebesgue measure, δti is the Dirac measure concentrate at ti , and μM is a degenerate measure defined as μM (A) = if M ∈ A and μM (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 formula to calculate the Lebesgue Δ-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 where f (s)Δs = E σ ti − ti · f ti + r( f ,E), f (s)ds + (2.2) i∈IE ⎧ ⎨μM (E) · f (M), r( f ,E) = ⎩ 0, if M ∈ T, if M ∈ T, (2.3) IE := {i ∈ I : ti ∈ E} and {ti }i∈I , I ⊂ N, is the set of all right-scattered points of T Definition 2.3 Let A ⊂ T A is called Δ-null set if μΔ (A) = Say that a property P holds Δ-almost everywhere (Δ-a.e.) on A, or for Δ-almost all (Δ-a.a.) t ∈ A if there is a Δ-null set E ⊂ A such that P holds for all t ∈ A\E ¯ ≡ [−∞,+∞] be such that Definition 2.4 Let E ⊂ T be a Δ-measurable set and let p ∈ R p ¯ p ≥ and let f : E → R be a Δ-measurable function Say that f belongs to LΔ (E) provided that either p E | f | (s) Δs < ∞ if p ∈ R, (2.4) or there exists a constant C ∈ R such that |f|≤C Δ-a.e on E if p = +∞ (2.5) p Note that equality (2.2) guarantees that in order for f : T → R to belong to LΔ (T), p ∈ R, and T bounded from above, it is necessary that f (M) = We will work with the Ravi P Agarwal et al p LΔ (J o ) spaces, where J = [a,b] ∩ T, a,b ∈ T, a < b, is an arbitrary closed subinterval of T and J o = [a,b) ∩ T; we state some of their properties whose proofs can be found in [6–8] ¯ be such that p ≥ Then, the set LΔp (J o ) is a Banach space together Theorem 2.5 Let p ∈ R p with the norm defined for every f ∈ LΔ (J o ) as f ⎧ ⎪ ⎪ ⎨ p LΔ 1/ p | f | p (s)Δs , := ⎪ J o ⎪ ⎩inf C ∈ R : | f | ≤ C Δ-a.e on J o , if p ∈ R, (2.6) if p = +∞ Moreover, L2Δ (J o ) is a Hilbert space together with the inner product given for every ( f ,g) ∈ o LΔ (J ) × L2Δ (J o ) by ( f ,g)L2Δ := Jo f (s) · g(s)Δs (2.7) ¯ and p ≥ Let p ∈ R ¯ be such that 1/ p + 1/ p = Proposition 2.6 Suppose p ∈ R p o p o o Then, if f ∈ LΔ (J ) and g ∈ LΔ (J ), then f · g ∈ LΔ (J ) and f ·g L1Δ ≤ f p LΔ · g p LΔ (2.8) This expression is called Hăolders inequality and Cauchy-Schwarzs inequality whenever p = Proposition 2.7 If p ∈ R and p ≥ 1, then, the set Cc (J o ) of all continuous functions on J o p with compact support in J o is dense in LΔ (J o ) As a consequence of Proposition 2.2, one can establish the following equivalence bep tween the LΔ (J o ) spaces and the usual L p ([a,b]) spaces linked to the Lebesgue measure ¯ with p ≥ 1, let f : J → R ¯ , and let f : [a,b] → R ¯ be the extension Corollary 2.8 Let p ∈ R of f to [a,b] defined as ⎧ ⎨ f (t), f (t) := ⎩ f (ti ), if t ∈ J, if t ∈ ti ,σ ti , for some i ∈ IJ , (2.9) with IJ := {i ∈ I : ti ∈ J } and {ti }i∈I , I ⊂ N, is the set of all right-scattered points of T p Then, f ∈ LΔ (J o ) if and only if f ∈ L p ([a,b]) In this case, f p LΔ = f Lp (2.10) As we know from general theory of Sobolev’s spaces, another important class of functions is just the absolutely continuous functions Definition 2.9 A function f : J → R is said to be absolutely continuous on J, f ∈ AC(J), if for every ε > 0, there exists a δ > such that if {[ak ,bk ) ∩ T}nk=1 , with ak ,bk ∈ J, is a finite pairwise disjoint family of subintervals of J satisfying nk=1 (bk − ak ) < δ, then n k=1 | f (bk ) − f (ak )| < ε 4 Basic properties of Sobolev’s spaces on time scales These 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 , f Δ ∈ L1Δ (J o ) and f (t) = f (a) + [a,t)∩T f Δ (s)Δs, ∀t ∈ J (2.11) Absolutely continuous functions on T verify the integration by parts formula Theorem 2.11 If f ,g : J → R are absolutely continuous functions on J, then f · g is absolutely continuous on J and the following equality is valid: Jo f Δ g + f σ g Δ (s)Δs = f (b)g(b) − f (a)g(a) = Jo f g Δ + f Δ g σ (s)Δs (2.12) They are linked to the class of absolutely continuous functions on [a,b] as the following property shows Corollary 2.12 [3, Corollary 3.1] Assume that f : J → R and define f¯ : [a,b] → R as ⎧ ⎪ ⎪ ⎨ f (t), if t ∈ J, f¯(t) := ⎪ ⎪ ⎩ f ti + f σ ti − f ti σ ti − ti t − ti , if t ∈ ti ,σ ti , for some i ∈ IJ , (2.13) with IJ := {i ∈ I : ti ∈ J } and {ti }i∈I , I ⊂ N, is the set of all right-scattered points of T Then, f is absolutely continuous on J if and only if f¯ is absolutely continuous on [a,b] Moreover, for every n ∈ N, n ≥ 1, we will denote as AC n (J) := x ∈ AC(J) : xΔ ∈ AC J κ ∀ j ∈ {1, ,n} , j j (2.14) j where for every j ∈ N, j ≥ 1, J κ = [a,ρ j (b)] ∩ T First-order Sobolev’s spaces The aim of this section is to study the first-order Sobolev’s spaces on J equipped with the Lebesgue Δ-measure ¯ be such that p ≥ and u : J → R ¯ Say that u belongs to WΔ1,p (J) Definition 3.1 Let p ∈ R p ¯ such that g ∈ LΔp (J o ) and if and only if u ∈ LΔ (J o ) and there exists g : J κ → R Jo u · ϕΔ (s)Δs = − Jo g · ϕσ (s)Δs ∀ϕ ∈ C0,rd Jκ (3.1) with 1 C0,rd J κ := f : J −→ R : f ∈ Crd J κ , f (a) = = f (b) (3.2) (J κ ) is the set of all continuous functions on J such that they are Δ-differentiable and Crd κ on J and their Δ-derivatives are rd-continuous on J κ Ravi P Agarwal et al The integration by parts formula for absolutely continuous functions on J establishes that the relation p 1,p 1,p VΔ (J) := x ∈ AC(J) : xΔ ∈ LΔ J o ⊂ WΔ (J) (3.3) ¯ with p ≥ We will show that both sets are, as class of functions, is true for every p ∈ R 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: Jo ∀u ∈ Cc J o , ( f · u)(s)Δs = 0, (3.4) then f ≡ Δ-a.e on J o (3.5) Proof Fix ε > 0, the density of Cc (J o ) in L1Δ (J o ) guarantees the existence of f1 ∈ Cc (J o ) such that f − f1 L1Δ < ε, and so, by (3.4), we deduce that for every u ∈ Cc (J o ), it is true that f1 · u (s)Δs ≤ u Jo C(J o ) · f − f1 C(J o ) (3.6) A2 := s ∈ J o : f1 (s) ≤ −ε (3.7) L1Δ 0, we achieve (3.5) Lemma 3.3 Let f ∈ L1Δ (J o ) Then, a necessary and sufficient condition for the validity of the equality Jo f · ϕΔ (s)Δs = 0, for every ϕ ∈ C0,rd Jκ , (3.10) is the existence of a constant c ∈ R such that f ≡c Δ-a.e on J o (3.11) Basic properties of Sobolev’s spaces on time scales Proof The necessary condition is consequence of the fundamental theorem of Calculus Conversely, fix u ∈ Cc (J o ) arbitrary; by defining h,ϕ : J → R as ⎧ ⎪ ⎪ ⎪ ⎨u(t) − J o u(r)Δr b−a ⎪ J o u(r)Δr ⎪ ⎩− , b−a h(t) := ⎪ ϕ(t) := [a,t)∩T h(s)Δs, if t ∈ J o , , if t = b, (3.12) ∀t ∈ J, (J κ ) and so, equality (3.10) the fundamental theorem of Calculus establishes that ϕ ∈ C0,rd yields to 0= = Jo J o u(r)Δr f · u− Jo f− Jo b−a (s)Δs (3.13) f (r)Δr · u (s)Δs b−a Therefore, Lemma 3.2 allows to deduce (3.11) with c = Jo f (r)Δr/(b − a) 1,p Now, we are able to prove the characterization of functions in WΔ (J) in terms of 1,p functions in VΔ (J) 1,p ¯ with p ≥ and that (3.1) holds Theorem 3.4 Suppose that u ∈ WΔ (J) for some p ∈ R p o 1,p for g ∈ LΔ (J ) Then, there exists a unique function x ∈ VΔ (J) such that the equalities x = u, xΔ = g Δ-a.e on J o (3.14) are satisfied (J κ ) such that Moreover, if g ∈ Crd (J κ ), then there exists a unique function x ∈ Crd xΔ = g x = u Δ-a.e on J o , on J κ (3.15) Proof Define v : J → R as v(t) := [a,t)∩T g(s)Δs, ∀t ∈ J; (3.16) 1,p the fundamental theorem of Calculus guarantees that v ∈ VΔ (J) and by the integration (J κ ), by parts formula, we have that for every ϕ ∈ C0,rd Jo (v − u) · ϕΔ (s)Δs = − Jo vΔ − g · ϕσ (s)Δs = 0; (3.17) so that, Lemma 3.3 ensures the existence of a constant c ∈ R such that v − u ≡ c Δ-almost everywhere on J o As a consequence of the fundamental theorem of Calculus we conclude that function x : J → R defined as x(t) := v(t) − c for all t ∈ J is the unique function in 1,p VΔ (J) for which (3.14) is valid Ravi P Agarwal et al Furthermore, if g ∈ Crd (J κ ), then the fundamental theorem of Calculus establishes (J κ ) and xΔ = g on J κ that x ∈ Crd 1,p By identifying every function in WΔ (J) with its absolutely continuous representative 1,p 1,p in VΔ (J) for which (3.14) holds, the set WΔ (J) can be endowed with the structure of Banach space ¯ and p ≥ The set WΔ1,p (J) is a Banach space together with Theorem 3.5 Assume p ∈ R 1,p the norm defined for every x ∈ WΔ (J) as x 1,p WΔ := x p LΔ + xΔ LΔ (3.18) p Moreover, the set HΔ1 (J) := WΔ1,2 (J) is a Hilbert space together with the inner product given for every (x, y) ∈ HΔ1 (J) × HΔ1 (J) by (x, y)HΔ1 := (x, y)L2Δ + xΔ , y Δ L2Δ (3.19) 1,p Proof Let {xn }n∈N be a Cauchy sequence in WΔ (J); Theorem 2.5 guarantees the exisp p tence of u,g ∈ LΔ (J o ) such that {xn }n∈N and {xnΔ }n∈N converge strongly in LΔ (J o ) to u and g, respectively, and so, by taking limits in the equality Jo xn · ϕΔ (s)Δs = − Jo xnΔ · ϕσ (s)Δs, ϕ ∈ C0,rd (J κ ), (3.20) 1,p we conclude that u ∈ WΔ (J) Thereby, it follows from Theorem 3.4, that there exists 1,p 1,p x ∈ WΔ (J) such that {xn }n∈N converges strongly in WΔ (J) to x 1,p 3.1 Some properties We will derive some properties of the Banach space WΔ (J); the 1,p first one asserts that WΔ (J) is continuously inmersed into C(J) equipped with the supremum norm · C(J) ¯ with p ≥ 1, then there exists a constant K > 0, only deProposition 3.6 Assume p ∈ R pendent on b − a, such that the inequality x C(J) ≤K· x 1,p (3.21) 1,p WΔ 1,p holds for all x ∈ WΔ (J) and hence, the immersion WΔ (J) C(J) is continuous 1,p WΔ (J) Proof Fix x ∈ 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 assuming t ≤ T The fundamental theorem of Calculus and Hăolders inequality lead to x C(J) |x(t)| + [t,T)∩T |xΔ |(s)Δs ≤ K · x 1,p WΔ , (3.22) for some K > 0, only dependent on b − a The strong compactness criterion in C(J) and Proposition 3.6 allow to prove the following compactness property in C(J) 8 Basic properties of Sobolev’s spaces on time scales ¯ be such that p ≥ Then, the following statements are true Proposition 3.7 Let p ∈ R 1,p (1) If p > 1, then the immersion WΔ (J) C(J) is compact 1,p (2) If p = 1, then the immersion WΔ (J) C(J) is compact if and only if every point of J is isolated 1,p Proof Denote by Ᏺ p the closed unit ball in WΔ (J); we know from Theorem 3.4 that Ᏺ p is closed and bounded in C(J) If p > 1, then the fundamental theorem of Calculus and Hăolders inequality ensure that p is equicontinuous On the other hand, if p = 1, then it is clear that Ᏺ p is equicontinuous whenever every point of J is isolated, while if there exists t0 ∈ T such that t0 is not isolated, then we will prove that Ᏺ p is not equicontinuous Let S := 1/(b − a + 1), let δ > be arbitrary and let sδ ∈ (t0 − δ,t0 + δ) ∩ T be such that sδ = t0 ; it is not a loss of generality assuming sδ < t0 Define fδ : J → R as ⎧ ⎪ ⎨ S , fδ := ⎪ t0 − sδ ⎩0, if t ∈ sδ ,t0 ∩ J , if t ∈ sδ ,t0 ∩ J ; (3.23) the fundamental theorem of Calculus asserts that Fδ : J → R given by Fδ (t) := [a,t)∩T t ∈ J, (3.24) fδ (s)Δs = S, (3.25) fδ (s)Δs, belongs to Ᏺ p ; so that, as Fδ t0 − Fδ sδ = [sδ ,t0 )∩T we conclude that Ᏺ p is not equicontinuous Therefore, Arzel`a-Ascoli theorem establishes our claims As a consequence of Proposition 3.6, we achieve the following sufficient condition for strong convergence in C(J) ¯ be such that p > 1, let {xm }m∈N ⊂ WΔ1,p (J), and let x ∈ WΔ1,p (J) Corollary 3.8 Let p ∈ R 1,p If {xm }m∈N converges weakly in WΔ (J) to x, then {xm }m∈N converges strongly in C(J) to x 1,p Proof Suppose {xm }m∈N converges weakly in WΔ (J) to x; Proposition 3.6 establishes that {xm }m∈N converges weakly in C(J) to x and so, as {xm }m∈N is equicontinuous, {xm }m∈N converges strongly in C(J) to x Moreover, Proposition 3.6 allows to deduce the following equivalence between the 1,p Sobolev’s spaces on J, WΔ (J), and the usual Sobolev’s spaces on (a,b), W 1,p ((a,b)) ¯ and p ≥ 1, x : J → R and x¯ : [a,b] → R is the extenCorollary 3.9 Suppose that p ∈ R 1,p sion of x to [a,b] defined in (2.13) Then, x belongs to WΔ (J) if and only if x¯ belongs to 1,p W ((a,b)) Ravi P Agarwal et al Moreover, there exist two constants K1 ,K2 > which only depend on (b − a) such that the inequalities K1 · x¯ W 1,p ≤ x 1,p WΔ ≤ K2 · x¯ (3.26) W 1,p 1,p ¯ with p ≥ are satisfied for every x ∈ WΔ (J) and p ∈ R Proof Let x¯ , xΔ : [a,b] → R be the extensions of x and xΔ to [a,b] defined in (2.13) and (2.9), respectively; it is not difficult to deduce the following equality: xΔ = x¯ a.e on [a,b] (3.27) Therefore, Corollaries 2.8 and 2.12 and Proposition 3.6 yield to the result As an application of the previous result, we will prove that some properties known 1,p for W 1,p ((a,b)) are directly transferred to WΔ (J); in order to this, we will use the following result ¯ with p ≥ 1, Proposition 3.10 If y : [a,b] → R belongs to W 1,p ((a,b)) for some p ∈ R 1,p then y|J belongs to WΔ (J) Moreover, there exists a constant T > which only depends on (b − a) such that y|J 1,p WΔ ≤T· y W 1,p , ¯ , p ≥ ∀ y ∈ W 1,p (a,b) , p ∈ R (3.28) Proof Let R = {ti }i∈I , I ⊂ N, be the set of all right-scattered points of T, let IJ o = {i ∈ I, ¯ with p ≥ The classical fundati ∈ J o } and suppose y ∈ W 1,p ((a,b)) for some p ∈ R mental theorem of Calculus allows to assert that y|J Δ ti = y|J y (s)ds , σ ti − ti [ti ,σ(ti )] Δ =y for every i ∈ IJ o , (3.29) a.e on J o ∩ (T\R) 1,p Therefore, if p = +∞, then it is clear that y|J ∈ WΔ (J) and (3.28) holds while if p ∈ R, then, by (2.2), we have that y|J Δ p p LΔ ≤ J o ∩(T\R) y p (s)ds + i∈IJ o [ti ,σ(ti )] y p (s)ds ≤ y p W 1,p , (3.30) moreover, as we know that y|J p LΔ ≤ (b − a) 1/ p · y C([a,b]) ≤ C · (b − a) 1/ p · y W 1,p , (3.31) 1,p for some C > 0, it turns out that y|J ∈ WΔ (J) and (3.28) is true 1,p Next, we deduce some properties in WΔ (J) from the analogous ones in W 1,p ((a,b)) ¯ be such that p ≥ Then, for every q ∈ [1,+∞), the inmersion Corollary 3.11 Let p ∈ R 1,p q o WΔ (J) LΔ (J ) is compact 10 Basic properties of Sobolev’s spaces on time scales Proof Fix q ∈ [1,+∞); as a consequence of Proposition 3.7 and the fact that the inmerq q sion C(J) LΔ (J o ) is continuous, it only remains to prove that Ᏺ1 is compact in LΔ (J o ) whenever J has at least one not isolated point Assume the existence of a not isolated point t0 ∈ J and let {xn }n∈N be a sequence in Ᏺ1 Corollary 3.9 ensures that {xn }n∈N , defined in (2.13), is a bounded sequence in W 1,1 ((a,b)) and hence, there exist {xnk }k∈N and y ∈ Lq ([a,b]) such that {xnk }k∈N converges strongly in Lq ([a,b]) to y By defining x := y|J , it is not difficult to prove that q {xnk }k∈N converges strongly in LΔ (J o ) to x 1,p Corollary 3.12 The Banach space WΔ (J) is reflexive for every p ∈ (1,+∞) and separable for all p ∈ [1,+∞) ¯ be such that p ≥ We know, from Corollary 3.9, that the operator T p : Proof Let p ∈ R 1,p 1,p ¯ defined in (2.13), is linWΔ (J) → W 1,p ((a,b)) given for every x ∈ WΔ (J) by T p (x) := x, 1,p ear and continuous It follows from Corollary 3.9 and Proposition 3.10 that T p (WΔ (J)) 1,p 1,p is a closed subspace of W ((a,b)) Therefore, since W ((a,b)) is reflexive whenever 1,p p ∈ (1,+∞) and separable whenever p ∈ [1,+∞), T p (WΔ (J)) satisfies the same properties 1,p Corollary 3.13 If x ∈ WΔ (J) for some p ∈ [1,+∞), then there exists a sequence of infinitely differentiable functions with compact support in R, { yn }n∈N such that { yn |J }n∈N 1,p converges strongly in WΔ (J) to x Proof Corollary 3.9 asserts that x¯ : [a,b] → R, defined in (2.13), belongs to W 1,p ((a,b)); so that, there exists a sequence { yn }n∈N of infinitely differentiable functions with compact support in R such that { yn |[a,b] }n∈N converges to x¯ in W 1,p ((a,b)) Hence, our claim follows from equality x¯|J = x and Proposition 3.10 1,p 3.2 The spaces W0,Δ (J) Corollary 3.13 guarantees the density of the set Crd (J κ ) in 1,p WΔ (J) for every p ∈ [1,+∞); however, for an arbitrary bounded time scale it is not true 1,p that the set of test functions defined in (3.2), C0,rd (J κ ), is dense in WΔ (J); this section is 1,p (J κ ) in WΔ (J) devoted to prove some properties concerning the closure of C0,rd 1,p Definition 3.14 Let p ∈ R be such that p ≥ 1, define the set W0,Δ (J) as the closure of the 1,p 1,2 1 (J κ ) in WΔ (J) Denote as H0,Δ (J) := W0,Δ (J) set C0,rd 1,p (J) are endowed with the norm induced by · WΔ1,p , deThe spaces W0,Δ (J) and H0,Δ fined in (3.18), and the inner product induced by (·, ·)HΔ1 , defined in (3.19), respectively 1,p 1,p 1,p Since W0,Δ (J) is closed in WΔ (J), Theorem 3.5 and Corollary 3.12 ensure that W0,Δ (J) is (J) is a separable Hilbert a separable Banach space and reflexive whenever p > and H0,Δ 1,p space The space W0,Δ (J) is characterized in the following result 1,p 1,p Proposition 3.15 Assume x ∈ WΔ (J) Then, x ∈ W0,Δ (J) if and only if x(a) = = x(b) Ravi P Agarwal et al 11 1,p (J κ) Proof Firstly, suppose that x ∈ W0,Δ (J), so that there exists a sequence {xn }n∈N ⊂ C0,rd 1,p such that {xn }n∈N converges strongly in WΔ (J) to x Therefore, inequality (3.21) allows to assert that x(a) = = x(b) Conversely, assume that x(a) = = x(b) We know from Corollary 3.9 that x¯ : [a,b] → 1,p R, defined in (2.13), belongs to W0 ((a,b)) and so, there exists a sequence { yn }n∈N ⊂ ¯ By defining xn := yn |J , n ∈ N, Cc1 ((a,b)) which converges strongly in W 1,p ((a,b)) to x one can deduce that xn ∈ C0,rd (J κ ) for every n ∈ N and {xn }n∈N converges strongly in 1,p WΔ (J) to x As a straightforward consequence of the previous result, Corollary 3.9, and the char1,p 1,p acterization of W0 ((a,b)) we obtain the following criterion for belonging to W0,Δ (T) Corollary 3.16 Let p ∈ R be such that p ≥ 1, let x : J → R, and let x¯ : [a,b] → R be the 1,p 1,p extension of x to [a,b] defined in (2.13) Then, x ∈ W0,Δ (J) if and only if x¯ ∈ W0 ((a,b)) By using Proposition 3.15, we are able to prove the validity of Poincar´e’s inequality Proposition 3.17 Let p ∈ R be such that p ≥ Then, there exists a constant L > 0, only dependent on (b − a), such that x 1,p WΔ ≤ L · xΔ LΔ , p 1,p 1,p ∀x ∈ W0,Δ (J), 1,p that is, in W0,Δ (J), the norm defined for every x ∈ W0,Δ (J) as xΔ norm · WΔ1,p (3.32) p LΔ is equivalent to the 1,p Proof Choose x ∈ W0,Δ (J); the fundamental theorem of Calculus and Proposition 3.15 allow to assert that the following inequality x(t) = x(a) + [a,t)∩T xΔ (s)Δs = [a,t)∩T xΔ (s)Δs ≤ xΔ L1Δ (3.33) is valid for every t ∈ T Thus, (3.32) follows from Hăolders inequality Remark 3.18 One can check that the function defined for every x, y ∈ H0,Δ (J) as(xΔ , y Δ )L2Δ is an inner product in H0,Δ (J) and its associated norm is equivalent to the norm associated to (·, ·)HΔ1 Generalization to order n ≥ The aim of this section is to define recursively the nth-order Sobolev’s spaces on J for n,p n−1,p n ≥ 2, WΔ (J), which consist in the Δ-antiderivatives of functions in WΔ (J κ ) ¯ , p ≥ 1, and let u : J → R ¯ Say that u belongs to Definition 4.1 Let n ∈ N, n ≥ 2, let p ∈ R n,p n−1,p n−1,p κ WΔ (J) if and only if u ∈ WΔ (J) and there exists g1 : J → R such that g1 ∈ WΔ (J κ ) and Jo u · ϕΔ (s)Δs = − Jo g1 · ϕσ (s)Δs, ∀ϕ ∈ C0,rd Jκ (4.1) 12 Basic properties of Sobolev’s spaces on time scales n,p It is easy to prove the following characterization of the set WΔ (J) ¯ is such that u ∈ LΔp (J o ), then u ∈ WΔn,p (J) if and Proposition 4.2 Suppose that u : J → R j κ ¯ , j ∈ {1, ,n}, such that g j ∈ LΔp ((J κ j −1 )o ), only if there exist g j : J → R u · ϕΔ (s)Δs = − Jo Jo g1 · ϕσ (s)Δs, ∀ϕ ∈ C0,rd Jκ , (4.2) and for all j ∈ {2, ,n}, j −1 o (J κ ) g j −1 · ϕΔ (s)Δs = − j −1 (J κ ) j g j · ϕσ (s)Δs, o ∀ϕ ∈ C0,rd Jκ , (4.3) with j J κ := f : J κ C0,rd j −1 −→ R : f ∈ Crd J κ , f (a) = = f ρ j −1 (b) j (4.4) j −1 and Crd (J κ ) is the set of all continuous functions on J κ such that they are Δ-differentiable j j κ on J and their Δ-derivatives are rd-continuous on J κ j The integration by parts formula for absolutely continuous functions on closed subintervals of T establishes that the relation n,p p VΔ (J) := x ∈ AC n−1 (J) : xΔ ∈ LΔ J κ n n−1 n,p o ⊂ WΔ (J) (4.5) ¯ with p ≥ 1; moreover, both sets are, as class of functions, equivalent is true for every p ∈ R as one can check in the following result n,p ¯ with p ≥ and Theorem 4.3 Suppose that u ∈ WΔ (J) for some n ∈ N with n ≥ 2, p ∈ R p o n,p that (4.1) holds for g1 ∈ LΔ (J ) Then, there exists a unique function x ∈ VΔ (J) such that xΔ = g j j x = u Δ-a.e on J o , Δ-a.e on J κ j −1 o , ≤ j ≤ n, (4.6) j ¯ , ≤ j ≤ n, are given in Proposition 4.2 where J κ = J and g j : J κ → R n,p Inductively, one can prove that the set WΔ (J) is endowed with the structure of Banach space ¯ and p ≥ The set WΔn,p (J) is a Banach space Theorem 4.4 Assume n ∈ N, n ≥ 2, p ∈ R n,p together with the norm defined for every x ∈ WΔ (J) as n x n,p WΔ := xΔ j =0 j LΔ , p (4.7) where xΔ = x Furthermore, the set HΔn (J) := WΔn,2 (J) is a Hilbert space together with the inner product given for every (x, y) ∈ HΔn (J) × HΔn (J) by n (x, y)HΔn := j =0 xΔ , y Δ j j L2Δ (4.8) Ravi P Agarwal et al 13 n,p 1,p Properties proved for the spaces WΔ (J) can be derived for the spaces WΔ (J); for instance, we have the following n,p n−1 n−1 Proposition 4.5 The immersion WΔ (J) C n−1 (J κ ) is continuous; where C n−1 (J κ ) j is the set of all functions defined on J with n − continuous Δ-derivatives on J κ , ≤ j ≤ n − Finally, by extending, whenever it is necessary, the function xΔ n−1 xΔ n−1 ρ j (b) = xΔ n−1 ρn−1 (b) to J as ∀ j ∈ {0, ,n − 2}, (4.9) with ρ0 (b) = b, one can prove inductively the following relation between the Banach n,p spaces WΔ (J) and W n,p ((a,b)) n−1 Theorem 4.6 Let n ∈ N, n ≥ 2, let x : J → R be such that x ∈ C n−1 (J κ ) n,p Then, x ∈ WΔ (J) if and only if the function y : [a,b] → R defined for every t ∈ [a,b] as n −2 y(t) := xΔ (a) j j =0 (t − a) j + j! At xΔn−1 sn−1 dsn−1 · · · ds1 belongs to W n,p ((a,b)), where xΔn−1 : [a,b] → R is the extension of xΔ in (2.13) and n−1 s1 , ,sn−1 ∈ [a,b]n−1 : sn−1 < · · · < s1 < t At := : Jκ (4.10) n−1 → R defined (4.11) Moreover, the following equalities y n = xΔ n n Δ-a.e on J κ , y n −1 = x Δ n−1 on J κ n−1 (4.12) hold Acknowlegments This research is partially supported by D.G.I and F.E.D.E.R project MTM 2004-06652C03-01, and by Xunta of Galicia and F.E.D.E.R project PGIDIT05PXIC20702PN, Spain References [1] M Bohner and A Peterson (eds.), Advances in Dynamic Equations on Time Scales, Birkhăauser Boston, Massachusetts, 2003 [2] H Brezis, Analyse Fonctionnelle: Th`eorie et Applications, Masson, Paris, 1996 [3] A Cabada and D R Vivero, Criterions for absolute continuity on time scales, Journal of Difference Equations and Applications 11 (2005), no 11, 1013–1028 , Expression of the Lebesgue Δ−integral on time scales as a usual Lebesgue integral Appli[4] cation to the calculus of Δ−antiderivatives, Journal of Mathematical Analysis and Applications 43 (2006), 194–207 [5] G Sh Guseinov, Integration on time scales, Journal of Mathematical Analysis and Applications 285 (2003), no 1, 107–127 14 Basic properties of Sobolev’s spaces on time scales [6] E Hewitt and K Stromberg, Real and Abstract Analysis A Modern Treatment of the Theory of Functions of a Real Variable, 3rd ed., Graduate Texts in Mathematics, no 25, Springer, New York, 1975 [7] W Rudin, Real and Complex Analysis, 1st ed., McGraw-Hill, New York, 1966 , Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987 [8] Ravi P Agarwal: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA E-mail address: agarwal@fit.edu Victoria Otero–Espinar: Departamento de An´alise Matem´atica, Facultade de Matem´aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain E-mail address: vivioe@usc.es Kanishka Perera: Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA E-mail address: kperera@fit.edu Dolores R Vivero: Departamento de An´alise Matem´atica, Facultade de Matem´aticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Galicia, Spain E-mail address: lolirv@usc.es Mathematical Problems in Engineering Special Issue on Time-Dependent Billiards Call for Papers This subject has been extensively studied in the past years for one-, two-, and three-dimensional space Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles His original model was then modified and considered under different approaches and using many versions Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos) We intend to publish in this special issue papers reporting research on time-dependent billiards The topic includes both conservative and dissipative dynamics Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned Mathematical papers regarding the topics above are also welcome Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www hindawi.com/journals/mpe/ Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Guest Editors Edson Denis Leonel, Departamento de Estatística, Matemỏtica Aplicada e Computaỗóo, Instituto de Geociờncias e Ciờncias Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; edleonel@rc.unesp.br Alexander Loskutov, Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia; loskutov@chaos.phys.msu.ru Hindawi Publishing Corporation http://www.hindawi.com ... Basic properties of Sobolev? ? ?s spaces on time scales Proof Fix q ∈ [1,+∞); as a consequence of Proposition 3.7 and the fact that the inmerq q sion C(J) LΔ (J o ) is continuous, it only remains... 194–207 [5] G Sh Guseinov, Integration on time scales, Journal of Mathematical Analysis and Applications 285 (2003), no 1, 107–127 14 Basic properties of Sobolev? ? ?s spaces on time scales [6] E Hewitt... First-order Sobolev? ? ?s spaces The aim of this section is to study the first-order Sobolev? ? ?s spaces on J equipped with the Lebesgue Δ-measure ¯ be such that p ≥ and u : J → R ¯ Say that u belongs