Maximum Principles for Dynamic Equations on Time Scales and Their Applications

Abstract

We consider the second dynamic operators of elliptic type on time scales. We establish basic generalized maximum principles and apply them to obtain weak comparison principle for second dynamic elliptic operators and to obtain the uniqueness of Dirichlet boundary value problems for dynamic elliptic equations.

1. Introduction

Maximum principles play an important role in the theories for differential equations. They can be used to obtain a priori estimate and uniqueness results for differential equations and other results. The survey of classical maximum principles can be found in Protter and Weinberger [1] and references therein.

Similarly, discrete maximum principles and their relations to their continuous counterpart are very important in difference equations. They have been consequently studied; see in Cheng [2] or Kuo and Trudinger [3].

The theory of time scales was first introduced by Stefan Hilger in 1988 to unify the continuous and discrete analysis. Since then much contributions have been made to the theories of time scales; see [4–6] and references therein.

Because of the importance and the distinct behavior of maximum principles in differential and difference equations, it seems natural to study them in the time scales setting. Reference [7–9] have studied the classical maximum principles. Unfortunately, the generalized maximum principles, that is, maximum principles in 𝐻¹ setting, have not been studied yet. In this paper, we study the generalized maximum principles for dynamic operators and their applications. To our knowledge, our results are new even in difference equations.

The paper is organized as follows. In Section 2, we give some notations on time scales, introduce the Sobolev spaces 𝐻¹ on time scales, and give some basic properties of 𝐻¹. In Section 3, we establish the generalized maximum principles for dynamic operators. In Section 4, we establish the comparison principle for dynamic operators. In Section 5, we study the uniqueness results to dynamic equations.

2. Preliminaries about Time Scales

We introduce some concepts related to time scales, which can be found in [5, 6, 10–12]. A time scale 𝑇 is defined as a closed subset of 𝑅. The forward jump operator 𝜎(𝑡)∶𝑇→𝑇 and the backward jump operator 𝜌(𝑡)∶𝑇→𝑇 for 𝑡∈𝑇 are defined as 𝜎(𝑡)∶=inf{𝜏>𝑡∣𝜏∈𝑇} and 𝜌(𝑡)∶=sup{𝜏<𝑡∣𝜏∈𝑇}, respectively, with supplementation inf∅=sup𝑇, sup∅=inf𝑇. A point 𝑡∈𝑇 is called rightscattered, rightdensed, leftscattered, and leftdensed if 𝜌(𝑡)>𝑡, 𝜌(𝑡)=𝑡, and 𝜎(𝑡)<𝑡, 𝜎(𝑡)=𝑡 hold, respectively. We define 𝑇^𝑘=𝑇 if 𝑇 does not have a left-scattered maximum 𝑡_max; otherwise, 𝑇^𝑘=𝑇⧵{𝑡_max}. The graininess function 𝜇(𝑡)∶𝑇→[0,+∞) is defined by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡.

A function 𝑓∶𝑇→𝑅 is called (delta) differentiable at 𝑡∈𝑇^𝑘 with (delta) derivative 𝑓^Δ(𝑡)∈𝑅 if for any 𝜖>0, there exists a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)⋂𝑇 for some 𝛿>0) such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡𝜎(𝑡)|𝜎(𝑡)|)[−𝑠]≤𝜖−𝑠,∀𝑠∈𝑈. (1) The function 𝑓 is differentiable on 𝑇^𝑘 if 𝑓^Δ(𝑡) exists for all 𝑡∈𝑇^𝑘. The following lemma gives some basic properties of 𝑓^Δ(𝑡); for the proofs, we refer the readers to [5, 11].

Lemma 1. Let 𝑓,𝑔∶𝑇→𝑅 be two functions, and let 𝑡∈𝑇^𝑘. Then we have the following:(i)if 𝑓^Δ(𝑡) exists, then 𝑓 is continuous at 𝑡;(ii)if 𝑡 is right scattered and 𝑓 is continuous at 𝑡, then 𝑓^Δ(𝑡) exists and 𝑓^Δ(𝑡)=(𝑓(𝜎(𝑡))−𝑓(𝑡))/𝜇(𝑡);(iii)if 𝑡 is right dense and 𝑓^Δ(𝑡) exists, then 𝑓^Δ(𝑡)=lim_𝑠→𝑡((𝑓(𝑠)−𝑓(𝑡))/(𝑠−𝑡));(iv)if 𝑓^Δ(𝑡) and 𝑔^Δ(𝑡) exist, then 𝐴𝑓+𝐵𝑔 is differentiable at 𝑡 with (𝐴𝑓+𝐵𝑔)^Δ(𝑡)=𝐴𝑓^Δ(𝑡)+𝐵𝑔^Δ(𝑡) for any constants 𝐴,𝐵;(v)if 𝑓^Δ(𝑡) and 𝑔^Δ(𝑡) exist, then 𝑓𝑔 is differentiable at 𝑡 with (𝑓𝑔)^Δ(𝑡)=𝑓^𝜎(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔(𝑡)=𝑓^Δ(𝑡)𝑔^𝜎(𝑡)+𝑓(𝑡)𝑔^Δ(𝑡);(vi)let 𝑓,𝑔∶𝑇→𝑅 be such that 𝑔(𝑡)𝑔^𝜎(𝑡)≠0 and 𝑓^Δ(𝑡) and 𝑔^Δ(𝑡) exist; then 𝑓/𝑔 is differentiable at 𝑡 and 𝑓𝑔^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)−𝑓(𝑡)𝑔^Δ(𝑡)𝑔(𝑡)𝑔^𝜎(𝑡); (2)(vii)if 𝑓^Δ(𝑡) exists, then 𝑓^𝜎(𝑡)=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡).Here and in the following, we use the notation 𝑔^𝜎(𝑡)=𝑔(𝜎(𝑡)).

A function 𝑓∶𝑇→𝑅 is called rd-continuous, provided it is continuous at each right-dense point and its left-sided limit exists (finite) at each left-dense point in 𝑇, and write 𝑓∈𝐶_rd(𝑇)=𝐶_rd(𝑇,𝑅). A rd-continuous function 𝑓 with compact support is written as 𝑓∈𝐶⁰_rd(𝑇)=𝐶⁰_rd(𝑇,𝑅). We write 𝑓∈𝐶¹_rd(𝑇), provided 𝑓^Δ∈𝐶_rd(𝑇^𝑘), write 𝑓∈𝐶²_rd(𝑇), provided 𝑓^Δ is differentiable on 𝑇^𝑘2=(𝑇^𝑘)^𝑘 with 𝑓^Δ2=(𝑓^Δ)^Δ∈𝐶_rd(𝑇^𝑘2), and similarly, write 𝑓∈𝐶^1,0_rd(𝑇^𝑘), 𝐶^2,0_rd(𝑇^𝑘) if 𝑓, 𝑓^Δ, and 𝑓^Δ2 have compact support, respectively. The definition of Riemann delta integral on time scales which is similar to the classical Riemann definition of integrability is given in [6]. We present some properties of the integral in the following lemma.

Lemma 2 (see [6]). Let 𝑓,𝑔∶𝑇→𝑅 be two functions and 𝑎,𝑏∈𝑇. Then we have the following:(i)let 𝑓 and 𝑔 be Riemann delta integrable functions on [𝑎,𝑏] and 𝛼,𝛽∈𝑅. Then 𝛼𝑓+𝛽𝑔 are Riemann delta integrable and 𝛼𝑓(^𝑏_𝑎𝑡)+𝛽𝑔(𝑡)Δ𝑡=𝛼^𝑏_𝑎𝑓(𝑡)Δ𝑡+𝛽^𝑏_𝑎𝑔(𝑡)Δ𝑡; (3)(ii)∫_𝑡^𝜎(𝑡)𝑓(𝑡)Δ𝑡=𝜇(𝑡)𝑓(𝑡) for 𝑡∈𝑇^𝑘;(iii)let 𝑐∈𝑇 with 𝑎<𝑐<𝑏. If 𝑓 is Riemann delta integrable from 𝑎 to 𝑐 and from 𝑐 to 𝑏, then 𝑓 is Riemann delta integrable from 𝑎 to 𝑏 and ^𝑏_𝑎𝑓(𝑡)Δ𝑡=^𝑐_𝑎𝑓(𝑡)Δ𝑡+^𝑏_𝑐𝑔(𝑡)Δ𝑡; (4)(iv)(fundamental theorem of calculus) let 𝑓 be a continuous function on [𝑎,𝑏]⊂𝑇 such that 𝑓 is (delta) differentiable on [𝑎,𝑏). If 𝑓^Δ is Riemann delta integrable from 𝑎 to 𝑏, then ^𝑏_𝑎𝑓^Δ(𝑡)Δ(𝑡)=𝑓(𝑏)−𝑓(𝑎); (5)(v)(integration by parts) let 𝑓 and 𝑔 be continuous functions on [𝑎,𝑏] that are differentiable on [𝑎,𝑏). If 𝑓^Δ and 𝑔^Δ are Riemann delta integrable from 𝑎 to 𝑏, then ^𝑏_𝑎𝑓(𝑡)𝑔^Δ(𝑡)Δ𝑡=𝑓(𝑡)𝑔(𝑡)^𝑏_𝑎−^𝑏_𝑎𝑓^Δ(𝑡)𝑔^𝜎(𝑡)Δ𝑡; (6)(vi)if 𝑓 is Riemann delta integrable on [𝑎,𝑏], then is |𝑓| and ^𝑏_𝑎𝑓(𝑡)Δ𝑡≤^𝑏_𝑎𝑓Δ𝑡. (7)

The construction of the Δ-measure on 𝑇 and the following concepts are derived from [6]:(i)for each 𝑡₀∈𝑇⧵{max𝑇}, the single-point set 𝑡₀ is Δ-measurable, and its Δ-measure is given by 𝜇_Δ𝑡₀=𝜎𝑡₀−𝑡₀=𝜇𝑡₀; (8)(ii)if 𝑎,𝑏∈𝑇 and 𝑎≤𝑏, then 𝜇_Δ([𝑎,𝑏))=𝑏−𝑎,𝜇_Δ((𝑎,𝑏))=𝑏−𝜎(𝑎); (9)(iii)if 𝑎,𝑏∈𝑇⧵{max𝑇} and 𝑎≤𝑏, then 𝜇_Δ((𝑎,𝑏])=𝜎(𝑏)−𝜎(𝑎),𝜇_Δ([𝑎,𝑏])=𝜎(𝑏)−𝑎. (10)

The Lebesgue integral associated with the measure 𝜇_Δ on 𝑇 is called the Lebesgue delta integral. For a (measurable) set 𝐸⊂𝑇 and a measurable function 𝑓∶𝐸→𝑅, the Lebesgue delta integral of 𝑓 on 𝐸 is denoted by ∫_𝐸𝑓Δ𝑡. All the theorems of Lebesgue integral hold also for the Lebesgue delta integral on 𝑇. Comparing the Lebesgue delta integral with the Riemann delta integral on 𝑇, we have the following.

Lemma 3 (see [6]). Let [𝑎,𝑏] be a closed bounded interval in 𝑇, and let 𝑓 be a bounded real-valued function defined on [𝑎,𝑏]. If 𝑓 is Riemann delta integrable on [𝑎,𝑏], then 𝑓 is Lebsgue delta integrable on [𝑎,𝑏), and (𝑅)^𝑏_𝑎𝑓Δ𝑡=(𝐿)_[𝑎,𝑏)𝑓Δ𝑡, (11) where 𝑅 and 𝐿 indicate the Riemann delta integral and Lebesgue delta integral from 𝑎 to 𝑏, respectively.
Assume 1≤𝑝<+∞. Let 𝐿^𝑝([𝑎,𝑏]) denote the set 𝑓∶[𝑎,𝑏]⟶𝑅∣𝑓^𝑝[).𝑖𝑠𝐿𝑒𝑏𝑒𝑠𝑔𝑢𝑒𝑑𝑒𝑙𝑡𝑎𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑏𝑙𝑒𝑜𝑛𝑎,𝑏𝑖𝑛𝑡𝑖𝑚𝑒𝑠𝑐𝑎𝑙𝑒𝑠𝑒𝑛𝑠𝑒 (12) Then the space 𝐿^𝑝([𝑎,𝑏]) is a complete linear space with the norm ‖⋅‖_𝑝 defined by 𝑓_𝑝=_[𝑎,𝑏)𝑓(𝑡)^𝑝Δ𝑡^1/𝑝. (13)

Lemma 4 (Hölder inequality [4, 5]). Let 𝑓,𝑔∈𝐶_𝑟𝑑([𝑎,𝑏]), 𝑝>1, and 𝑞 be the conjugate number of 𝑝. Then ^𝑏_𝑎𝑓(𝑡)𝑔(𝑡≤)Δ𝑡_𝑏𝑎𝑓(𝑡)^𝑝Δ𝑡^1/𝑝𝑏_𝑎|𝑔(𝑡)|^𝑞Δ𝑡^1/𝑞. (14)

Lemma 5 (see [13]). For any 𝑝≥1,(a)𝐶⁰(𝑇)⊂𝐶⁰_𝑟𝑑(𝑇)⊂𝐿^𝑝(𝑇); (b)𝐶⁰(𝑇) is dense in 𝐿^𝑝(𝑇).

From Lemma 5, we see that Lemma 4 still holds for 𝑓∈𝐿^𝑝([𝑎,𝑏]), 𝑔∈𝐿^𝑞([𝑎,𝑏]).

Lemma 6 (see [13]). Suppose that (𝑢_𝑛) is a sequence in 𝐿^𝑝(𝑇), for some 𝑝≥1.(a)If ‖𝑢_𝑛−𝑢‖_𝑝→0, for some 𝑢∈𝐿^𝑝(𝑇), and if 𝑡∈𝑇 is right-scattered, then 𝑢_𝑛(𝑡)→𝑢(𝑡).(b)If (𝑢_𝑛) is a Cauchy sequence in 𝐿^𝑝(𝑇) (with respect to the norm ‖⋅‖_𝑝), then there exists a unique 𝑢∈𝐿^𝑝(𝑇) such that ‖𝑢_𝑛−𝑢‖_𝑝→0.

Following [13], we now define the generalized derivative of Lebesgue delta integrable functions.

Definition 7. Define the norm ‖⋅‖_1,𝑇 on 𝐶¹_rd(𝑇^𝑘) by ‖𝑢‖²_1,𝑇∶=‖𝑢‖²_𝑇+𝑢^Δ2_𝑇,𝑢∈𝐶¹_rd𝑇^𝑘, (15) and define the space 𝐻¹(𝑇^𝑘)⊂𝐿²(𝑇) to be the completion of 𝐶¹_rd(𝑇^𝑘) with respect to the norm ‖⋅‖_1,𝑇 and 𝐻¹₀(𝑇^𝑘) to be the completion of 𝐶^1,0_rd(𝑇^𝑘) with respect to the norm ‖⋅‖_1,𝑇.

Lemma 8 (see [13]). (a)   𝑢∈𝐻¹(𝑇^𝑘) if and only if there exists a function 𝑢^Δ𝑔∈𝐿²(𝑇) such that the following condition holds: there exists a sequence (𝑢_𝑛) in 𝐶¹_rd(𝑇^𝑘) such that 𝑢_𝑛→𝑢 and 𝑢^Δ_𝑛→𝑢^Δ𝑔 in 𝐿²(𝑇). If 𝑢∈𝐻¹(𝑇^𝑘), then the function 𝑢^Δ𝑔 is unique (in 𝐿²(𝑇) sense).
(b) If 𝑢∈𝐶¹_rd(𝑇^𝑘), then 𝑢^Δ𝑔=𝑢^Δ.

Definition 9. For any 𝑢∈𝐻¹(𝑇^𝑘), the function 𝑢^Δ𝑔 in Lemma 8 will be called the generalized derivative of 𝑢.

Remark 10. We can also define the generalized derivative of 𝑢 and the spaces 𝐻¹(𝑇^𝑘) as in [14].

The following two lemmas present basic properties of 𝐻¹(𝑇^𝑘).

Lemma 11 (see [13]). If 𝑢∈𝐻¹(𝑇^𝑘), then 𝑢∈𝐶⁰(𝑇), and there exists 𝐶>0 such that |𝑢|_0,𝑇≤𝐶‖𝑢‖_1,𝑇,𝑢∈𝐻¹𝑇^𝑘. (16) Furthermore, 𝑢(𝑡)−𝑢(𝑠)=^𝑡_𝑠𝑢^Δ𝑔Δ,𝑠,𝑡∈𝑇. (17)

Lemma 12 (see [13]). Suppose 𝑢,𝑣∈𝐻¹(𝑇^𝑘). Then(a)if the sequence (𝑢_𝑛) in 𝐶¹(𝑇^𝑘) is as in Lemma 8, then 𝑢_𝑛→𝑢 in 𝐶⁰(𝑇);(b)if 𝑢^Δ𝑔=0, then 𝑢≡ constant;(c)if 𝑡 is right scattered, then 𝑢^Δ𝑔(𝑡)=𝑢^Δ(𝑡)=(𝑢^𝜎(𝑡)−𝑢(𝑡))/(𝜎(𝑡)−𝑡);(d)if 𝛼,𝛽∈𝑅, then 𝛼𝑢+𝛽𝑣∈𝐻¹(𝑇^𝑘) and (𝛼𝑢+𝛽𝑣)^Δ𝑔=𝛼𝑢^Δ𝑔+𝛽𝑣^Δ𝑔;(e)(𝑢𝑣)^Δ𝑔=𝑢^Δ𝑔𝑣+𝑢^𝜎𝑣^Δ𝑔=𝑢𝑣^Δ𝑔+𝑢^Δ𝑔𝑣^𝜎;(f)∫^𝑡_𝑠𝑢^Δ𝑔𝑣Δ=[𝑢𝑣]^𝑡_𝑠−∫^𝑡_𝑠𝑢^𝜎𝑣^Δ𝑔Δ, 𝑠,𝑡∈𝑇^𝑘.

Remark 13. From (f), if 𝑣∈𝐻¹₀(𝑇^𝑘), then we have ∫^𝑡_𝑠𝑢^Δ𝑔𝑣Δ=−∫^𝑡_𝑠𝑢^𝜎𝑣^Δ𝑔Δ, 𝑠,𝑡∈𝑇^𝑘; hence, we can also define weak derivatives as usual Sobolev space [15].

Definition 14 (see [16]). A function 𝑓∶𝑇→𝑅 is said to be absolutely continuous on 𝑇 if for every 𝜖>0, there exists a 𝛿>0 such that if {(𝑎_𝑘,𝑏_𝑘)}^𝑚_𝑘=1 with 𝑎_𝑘,𝑏_𝑘∈𝑇 is a finite pairwise of subintervals satisfying ^𝑚_𝑘=1𝑏_𝑘−𝑎_𝑘<𝛿,then^𝑚_𝑘=1𝑓𝑏_𝑘−𝑓𝑎_𝑘<𝜖. (18)

Lemma 15 (see [17]). If 𝑔∈𝐿¹(𝑇,𝑅) and 𝑓∶𝑇→𝑅 is the function defined by 𝑓(𝑡)∶=_{[𝑎,𝑡)∩𝑇}𝑔(𝑠)Δ𝑠, (19) then 𝑓 is absolutely continuous and 𝑓^Δ(𝑡)=𝑔(𝑡)Δ-almost everywhere on 𝑇^𝑘.

Proposition 16 (see [16]). A function 𝑓∶𝑇→𝑅 is absolutely continuous on 𝑇 if and only if 𝑓 is Δ-differentiable Δ-almost everywhere on 𝑇^𝑘,𝑓^Δ∈𝐿¹(𝑇^𝑘) and _{[𝑎,𝑡)∩𝑇}𝑓^Δ(𝑠)Δ𝑠=𝑓(𝑡)−𝑓(𝑎),𝑓𝑜𝑟𝑒𝑣𝑒𝑟𝑦𝑡∈𝑇. (20)

In the following sections, we still write 𝑢^Δ𝑔 as 𝑢^Δ.

3. Generalized Maximum Principle

Let 𝑇 be a bounded time scale and set 𝑎=min𝑇,  𝑏=max𝑇; that is, 𝑇=[𝑎,𝑏], where [𝑎,𝑏] is a time scale interval. In this section, we consider the generalized maximum principle for the dynamic operators 𝐿 on 𝑇^𝑘=[𝑎,𝜌(𝑏)]: 𝐿𝑢∶=𝑎(𝑡)𝑢^ΔΔ+𝑏(𝑡)𝑢^Δ+𝑐(𝑡)𝑢. (21)

To study the generalized maximum principle, we should make clear what it means when we say a 𝐿^𝑝(𝑇) function takes some value on the boundary of 𝑇. It is well known that a usual 𝐿^𝑝(Ω) function that takes some value on the boundary 𝜕Ω is understood in the trace sense, that is, the limitation of some suitable smooth function with definite value on the boundary 𝜕Ω. The boundary value of a 𝐿^𝑝(𝑇) function is understood in the same way; that is, if 𝑢∈𝐿^𝑝([𝑎,𝑏]), 𝑢_𝑛∈𝐶_rd([𝑎,𝑏]), 𝑢_𝑛→𝑢 in 𝐿^𝑝, and 𝑢_𝑛(𝑎)=𝑐,  𝑢_𝑛(𝑏)=𝑑, then we say 𝑢(𝑎)=𝑐, 𝑢(𝑏)=𝑑. And 𝑢(𝑎)≥𝑐, 𝑢(𝑏)≤𝑑 are understood in the same way.

We define the bilinear form associated with the operator 𝐿 as follows: £[𝑢,𝑣]∶=_𝑎^𝜌(𝑏)𝑎(𝑡)𝑢^Δ𝑣^Δ−𝑏(𝑡)𝑢^Δ𝑣^𝜎−𝑐(𝑡)𝑢𝑣^𝜎Δ𝑡. (22)

We assume that 𝑎(𝑡),𝑏(𝑡),𝑐(𝑡)∈𝐿^∞([𝑎,𝑏]) satisfy the following conditions: 𝑎(𝑡)≥𝜆>0,0≤𝑏(𝑡)≤𝜆𝜇,𝑐(𝑡)<0. (23)

Theorem 17 (generalized weak maximum principle). If 𝑢∈𝐻¹(𝑇^𝑘) satisfies 𝐿𝑢≥0(≤0) in weak sense; that is, £[𝑢,𝑣]≤0(≥0) for all 0≤𝑣∈𝐻¹₀(𝑇^𝑘), then sup_𝑇𝑢≤sup_𝜕𝑇𝑢⁺inf_𝑇𝑢≥inf_𝜕𝑇𝑢⁻. (24)

Proof. If 𝑢∈𝐻¹(𝑇^𝑘), 𝑣∈𝐻¹₀(𝑇^𝑘), we have 𝑢𝑣∈𝐻¹₀(𝑇^𝑘) and (𝑢𝑣)^Δ=𝑢^Δ𝑣^𝜎+𝑢𝑣^Δ; hence we can rewrite £[𝑢,𝑣]≤0 as _𝑎^𝜌(𝑏)𝑎(𝑡)𝑢^Δ𝑣^Δ−𝑏(𝑡)𝑢^Δ𝑣^𝜎Δ𝑡≤_𝑎^𝜌(𝑏)𝑐(𝑡)𝑢𝑣^𝜎Δ𝑡≤0. (25) Equation (25) holds for all 𝑣≥0 which satisfys 𝑢𝑣≥0.
Conditions (23) and (25) imply _𝑎^𝜌(𝑏)𝑎(𝑡)𝑢^Δ𝑣^ΔΔ𝑡≤𝜆𝜇_𝑎^𝜌(𝑏)𝑢^Δ𝑣^𝜎Δ𝑡. (26) In the special case that 𝑏(𝑡)≡0, we can easily obtain the result by choosing 𝑣=max{𝑢−𝑙,0}, where 𝑙=sup_𝜕𝑇𝑢⁺. In general case, we deduce by contradiction. Suppose that 𝑙<sup_𝑇𝑢∶=𝑀; we can then choose 𝑘 satisfying 𝑙<𝑘<𝑀 and set 𝑣=(𝑢−𝑘)⁺. Then we have 𝑣∈𝐻¹₀(𝑇^𝑘) and 𝑣^Δ=𝑢^Δ,𝑢>𝑘,0,𝑢≤𝑘, (27) and hence, we obtain by (26) that _𝑎^𝜌(𝑏)𝑎(𝑡)𝑣^Δ2Δ𝑡≤𝜆𝜇_𝑎^𝜌(𝑏)𝑣^Δ𝑣^𝜎Δ𝑡. (28) The condition on 𝑎(𝑡) and Hölder’s inequality imply _𝑎^𝜌(𝑏)𝑣^Δ2Δ𝑡≤𝜇_𝑎^𝜌(𝑏)𝑣^Δ𝑣^𝜎Δ𝑡≤𝜇𝑣^𝜎₂𝑣^Δ₂, (29) from which, we have 𝑣^Δ₂≤𝜇𝑣^𝜎₂. (30) Applying embedding theorem and Hölder’s inequality, we get 𝑣^𝜎₂≤𝐶𝑣^𝜎_𝑞≤𝐶𝑣^Δ₂≤𝐶𝑣^𝜎₂≤𝐶supp𝑣^{𝜎(1/2)−(1/𝑞)}𝑣^𝜎_𝑞, (31) where 𝐶=𝐶(𝑇,𝜇), |𝐸| denotes the 𝜇_Δ measure of set 𝐸. Hence, supp𝑣^𝜎≥𝐶^{2𝑞/(𝑞−2)}. (32) Since the above inequality does not depend on 𝑘, it still holds as 𝑘 tends to 𝑀, that is, 𝑢^𝜎, that is, 𝑢 must attains its superemum on a set of positive 𝜇_Δ measure. Hence, the set of the points on which 𝑢 attains its superemum must contain either an interval [𝑐,𝑑]⊂[𝑎,𝜌(𝑏)] (in time scale sense) or at least one right-scattered point 𝑡₀.
In the first case where 𝑢(𝑡)≡𝑀 in [𝑐,𝑑], then 𝑢^Δ(𝑡)≡0 and 𝑢>0 in [𝑐,𝑑]. Choosing 0≤𝑣∈𝐻¹₀([𝑐,𝑑]), we have 0<−^𝑑_𝑐𝑐(𝑡)𝑢𝑣^𝜎=Δ𝑡_𝑑𝑐𝑎(𝑡)𝑢^Δ𝑣^Δ−𝑏(𝑡)𝑢^Δ𝑣^𝜎−𝑐(𝑡)𝑢𝑣^𝜎Δ𝑡≤0, (33) which is a contradiction.
In the second case, we have 𝑢(𝑡₀)=𝑀; if 𝑢^𝜎(𝑡₀)=𝑢(𝑡₀), then proceeding as before in the interval [𝑐,𝑑]=[𝑡₀,𝜎(𝑡₀)], we then get a contradiction. If 𝑢^𝜎(𝑡₀)<𝑢(𝑡₀), then 𝑢^Δ(𝑡₀)<0. Set 𝑣=𝑋_{[𝑡0,𝜎(𝑡0))}(𝑢−𝑙)⁺, where 𝑋(𝐸) is the characteristic function of the set 𝐸; then we have 0≤𝑣∈𝐻¹₀(𝑇^𝑘); hence, we deduce from £[𝑢,𝑣]≤0 that ^𝜎(𝑡0)_𝑡0𝑎(𝑡)𝑢^Δ2−𝑏(𝑡)𝑢^Δ(𝑢−𝑙)^+𝜎−𝑐(𝑡)𝑢(𝑢−𝑙)^+𝜎Δ𝑡≤0, (34) while ^𝜎(𝑡0)_𝑡0𝑎(𝑡)𝑢^Δ2−𝑏(𝑡)𝑢^Δ(𝑢−𝑙)^+𝜎−𝑐(𝑡)𝑢(𝑢−𝑙)^+𝜎𝑡Δ𝑡=𝑎₀𝑢^Δ𝑡₀²−𝑏𝑡₀𝑢^Δ𝑡₀𝑢^𝜎𝑡₀−𝑙⁺−𝑐𝑡₀𝑢𝑡₀𝑢^𝜎𝑡₀−𝑙⁺𝜎𝑡₀−𝑡₀>0; (35) the above two inequalities contradict each other.

Theorem 18 (strong maximum principle). If 𝑢∈𝐻¹(𝑇^𝑘) satisfies 𝐿𝑢≥0(≤0) in weak sense; that is, £[𝑢,𝑣]≤0(≥0) for all 0≤𝑣∈𝐻¹₀(𝑇^𝑘), suppose further that 𝑢 attains its nonnegative maximum (nonpositive minimum) at inner point of (𝑎,𝑏) and at 𝑎 (at 𝑏), and then 𝑢 is a constant.

Proof. Suppose that 𝑢 is not a constant and 𝑢 attains its maximum 𝑀 at 𝑡₀∈[𝑎,𝑏). Then there exists at least one point 𝑡₁∈(𝑎,𝑏) such that 𝑢(𝑡₁)<𝑀=𝑢(𝑡₀). Then we may assume 𝑡₀<𝑡₁<𝑏. In the first case, we may assume 𝑢(𝑡)<𝑀 in (𝑡₀,𝑡₁). If 𝑡₀ is right scattered, then 𝑢^Δ(𝑡₀)<0. Choosing 𝑣=𝑋_{[𝑡0,𝜎(𝑡0))}𝑢⁺, we deduce from £[𝑢,𝑣]≤0 that ^𝜎(𝑡0)_𝑡0𝑎(𝑡)𝑢^Δ2−𝑏(𝑡)𝑢^Δ𝑢^+𝜎−𝑐(𝑡)𝑢𝑢^+𝜎Δ𝑡≤0, (36) while ^𝜎(𝑡0)_𝑡0𝑎(𝑡)𝑢^Δ2−𝑏(𝑡)𝑢^Δ𝑢^+𝜎−𝑐(𝑡)𝑢𝑢^+𝜎=𝑎𝑡Δ𝑡₀𝑢^Δ𝑡₀²−𝑏𝑡₀𝑢^Δ𝑡₀𝑢^𝜎𝑡₀⁺−𝑐𝑡₀𝑢𝑡₀𝑢^𝜎𝑡₀⁺𝜎𝑡₀−𝑡₀>0; (37) the above two inequalities contradict each other. If 𝑡₀ is right dense, Lemma 11 implies that 𝑢 is continuous on [𝑎,𝜌(𝑏)), especially at 𝑡₀, and there exists a neighbourhood [𝑡₀,𝑡₀+ℎ)(ℎ>0) of 𝑡₀ such that 𝑢>0 and is decreasing on [𝑡₀,𝑡₀+ℎ]; hence 𝑢 is uniformly continuous on [𝑡₀,𝑡₀+ℎ] and so absolutely continuous on [𝑡₀,𝑡₀+ℎ]; therefore, Proposition 16 implies 𝑢^Δ≤0𝜇_Δ a.e. on [𝑡₀,𝑡₀+ℎ). Choosing 𝑣=𝑋_{[𝑡0,𝑡0+ℎ)}𝑢⁺, we can also obtain a similar contradiction.

Remark 19. From the proof of Theorem 18, we see that the result is also true if only that 𝑢 attains its nonnegative maximum at 𝑎 (nonpositive minimum at 𝑏).

4. Weak Comparison Principle

It is well known that the comparison principle plays essential role in the theory of partial differential equations. In this section we study the counterpart for dynamic equations on 𝑇=[𝑎,𝑏] by applying the weak maximum principle.

Theorem 20. If 𝑢∈𝐻¹(𝑇^𝑘) satisfies 𝐿𝑢≥0(≤0) in weak sense, that is, £[𝑢,𝑣]≤0(≥0), for all 0<𝑣∈𝐻¹₀(𝑇^𝑘), 𝑢(𝑎)≤0(≥0), and 𝑢(𝑏)≤0(≥0), then 𝑢(𝑡)≤0(≥0) for all 𝑡∈𝑇.

Proof. We assume that 𝑢∈𝐻¹(𝑇^𝑘) satisfies 𝐿𝑢≥0 in weak sense; then by Theorem 17, we have 𝑢(𝑡)≤sup_𝑇𝑢≤sup_𝜕𝑇𝑢⁺=0. (38)

We can easily deduce from Theorem 20 the following.

Corollary 21. If 𝑢,𝑣∈𝐻¹(𝑇^𝑘) satisfies 𝐿𝑢≥0, 𝐿𝑣≤0 in weak sense, that is, £[𝑢,ℎ]≤0, £[𝑣,ℎ]≥0, for all 0<ℎ∈𝐻¹₀(𝑇^𝑘), 𝑢(𝑎)≤𝑣(𝑎), and 𝑢(𝑏)≤𝑣(𝑏), then 𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈𝑇.

Corollary 22. If 𝑢,𝑣∈𝐻¹₀(𝑇^𝑘) satisfies 𝐿𝑢≥0, 𝐿𝑣≤0 in weak sense, that is, £[𝑢,ℎ]≤0, £[𝑣,ℎ]≥0, for all 0<ℎ∈𝐻¹₀(𝑇^𝑘), then 𝑢(𝑡)≤0, 𝑣(𝑡)≥0 for all 𝑡∈𝑇.

Corollary 23. If 𝑢,𝑣∈𝐻¹(𝑇^𝑘) satisfies 𝐿𝑢≥𝑓(𝑡), 𝐿𝑣≤𝑓(𝑡), 𝑓(𝑡)∈𝐿²(𝑇^𝑘) in weak sense, that is, £[𝑢,ℎ]≤∫_𝑎^𝜌(𝑏)𝑓(𝜏)ℎ(𝜏)Δ𝜏, £[𝑣,ℎ]≥∫_𝑎^𝜌(𝑏)𝑓(𝜏)ℎ(𝜏)Δ𝜏, for all 0<ℎ∈𝐻¹₀(𝑇^𝑘), 𝑢(𝑎)≤𝑣(𝑎), and 𝑢(𝑏)≤𝑣(𝑏)), then 𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈𝑇.

Definition 24. If 𝑢∈𝐻¹(𝑇^𝑘) satisfies 𝐿𝑢≥𝑓(𝑡)(≤𝑓(𝑡)), 𝑓(𝑡)∈𝐿²(𝑇^𝑘) in weak sense, we say that 𝑢 is a weak subsolution (supper solution) to dynamic equation 𝐿𝑢=𝑓(𝑡).

Corollary 23 asserts that if the supper-solution and subsolution to dynamic equation 𝐿𝑢=𝑓(𝑡) attain the same value on the boundary of 𝑇, then the supper solution is not less than the subsolution.

5. Uniqueness Results

We now consider the following dynamic equation: 𝐿𝑢=𝑓(𝑡),𝑡∈𝑇^𝑘,𝑢(𝑎)=𝑢𝜌(𝑏)=0, (39) where 𝑓(𝑡)∈𝐿²(𝑇^𝑘).

Theorem 25. There exists at most one solution to dynamic equation (39) in 𝐻¹₀(𝑇^𝑘).

Proof. Suppose that there exist two solutions 𝑢,𝑣∈𝐻¹₀(𝑇^𝑘) to (39). Set 𝑤=𝑢−𝑣; then 𝑤 satisfies 𝐿𝑤=0,𝑡∈𝑇^𝑘,𝑤(𝑎)=𝑤𝜌(𝑏)=0. (40) Then Theorem 17 implies 0=inf_𝜕𝑇𝑤⁻≤inf_𝑇𝑤≤𝑤(𝑡)≤sup_𝑇𝑤≤sup_𝜕𝑇𝑤⁺=0, from which we deduce that 𝑢≡𝑣.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This project is supported by Natural Science Foundation of China (no. 10971061), Hunan Provincial Natural Science Foundation of China (no. 11JJ6005), and the program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.