Maximum Principles for Dynamic Equations on Time Scales and Their Applications

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 H setting, have not been studied yet. In this paper, we study the generalizedmaximum 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 H on time scales, and give some basic properties of H. 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.


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][5][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][8][9] have studied the classical maximum principles.Unfortunately, the generalized maximum principles, that is, maximum principles in  1 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  1 on time scales, and give some basic properties of  1 .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.

Preliminaries about Time Scales
We introduce some concepts related to time scales, which can be found in [5,6,[10][11][12] ( The function  is differentiable on   if  Δ () exists for all  ∈   .The following lemma gives some basic properties of  Δ (); for the proofs, we refer the readers to [5,11].
Here and in the following, we use the notation   () = (()).
A function  :  →  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 , and write  ∈  rd () =  rd (, ).A rd-continuous function  with compact support is written as  ∈  0 rd () =  0 rd (, ).We write  ∈  1 rd (), provided ), and similarly, write  ∈  1,0 rd (  ),  2,0 rd (  ) if ,  Δ , and  Δ 2 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 ,  :  →  be two functions and ,  ∈ .Then we have the following: (iv) (fundamental theorem of calculus) let  be a continuous function on The construction of the Δ-measure on  and the following concepts are derived from [6]: (i) for each  0 ∈  \ {max }, the single-point set  0 is Δ-measurable, and its Δ-measure is given by (ii) if ,  ∈  and  ≤ , then (iii) if ,  ∈  \ {max } and  ≤ , then The Lebesgue integral associated with the measure  Δ on  is called the Lebesgue delta integral.For a (measurable) set  ⊂  and a measurable function  :  → , the Lebesgue delta integral of  on  is denoted by ∫  Δ.All the theorems of Lebesgue integral hold also for the Lebesgue delta integral on .Comparing the Lebesgue delta integral with the Riemann delta integral on , we have the following.Lemma 3 (see [6]).Let [, ] be a closed bounded interval in , and let  be a bounded real-valued function defined on Lemma 5 (see [13]).For any  ≥ 1, From Lemma 5, we see that Lemma 4 still holds for  ∈   ([, ]),  ∈   ([, ]).Lemma 6 (see [13]).Suppose that (  ) is a sequence in   (), for some  ≥ 1.
Following [13], we now define the generalized derivative of Lebesgue delta integrable functions.
Remark 10.We can also define the generalized derivative of  and the spaces  1 (  ) as in [14].
The following two lemmas present basic properties of  1 (  ).

Generalized Maximum Principle
Let  be a bounded time scale and set  = min ,  = max ; that is,  = [, ], where [, ] is a time scale interval.In this section, we consider the generalized maximum principle for the dynamic operators  on   = [, ()]: To study the generalized maximum principle, we should make clear what it means when we say a   () function takes some value on the boundary of .It is well known that a usual   (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   () function is understood in the same way; that is, if  ∈   ([, ]),   ∈  rd ([, ]),   →  in   , and   () = ,   () = , then we say () = , () = .And () ≥ , () ≤  are understood in the same way.
Remark 19.From the proof of Theorem 18, we see that the result is also true if only that  attains its nonnegative maximum at  (nonpositive minimum at ).

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  = [, ] by applying the weak maximum principle.
Proof.We assume that  ∈  1 (  ) satisfies  ≥ 0 in weak sense; then by Theorem 17, we have We can easily deduce from Theorem 20 the following.