Maximum Principles for Dynamic Equations on Time Scales and Their Applications
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.
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  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  or Kuo and Trudinger .
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 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 on time scales, and give some basic properties of . 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 is defined as a closed subset of . The forward jump operator and the backward jump operator for are defined as and , respectively, with supplementation , . A point is called rightscattered, rightdensed, leftscattered, and leftdensed if , , and , hold, respectively. We define if does not have a left-scattered maximum ; otherwise, . The graininess function is defined by .
A function is called (delta) differentiable at with (delta) derivative if for any , there exists a neighborhood of (i.e., for some ) such that The function is differentiable on if exists for all . The following lemma gives some basic properties of ; for the proofs, we refer the readers to [5, 11].
Lemma 1. Let be two functions, and let . Then we have the following:(i)if exists, then is continuous at ;(ii)if is right scattered and is continuous at , then exists and ;(iii)if is right dense and exists, then ;(iv)if and exist, then is differentiable at with for any constants ;(v)if and exist, then is differentiable at with ;(vi)let be such that and and exist; then is differentiable at and (vii)if exists, then .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 . A rd-continuous function with compact support is written as . We write , provided , write , provided is differentiable on with , and similarly, write , if , , and 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 . We present some properties of the integral in the following lemma.
Lemma 2 (see ). Let be two functions and . Then we have the following:(i)let and be Riemann delta integrable functions on and . Then are Riemann delta integrable and (ii) for ;(iii)let with . If is Riemann delta integrable from to and from to , then is Riemann delta integrable from to and (iv)(fundamental theorem of calculus) let be a continuous function on such that is (delta) differentiable on . If is Riemann delta integrable from to , then (v)(integration by parts) let and be continuous functions on that are differentiable on . If and are Riemann delta integrable from to , then (vi)if is Riemann delta integrable on , then is and
The construction of the -measure on and the following concepts are derived from :(i)for each , the single-point set is -measurable, and its -measure is given by (ii)if and , then (iii)if and , then
The Lebesgue integral associated with the measure on is called the Lebesgue delta integral. For a (measurable) set and a measurable function , the Lebesgue delta integral of on is denoted by . 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 ). Let be a closed bounded interval in , and let be a bounded real-valued function defined on . If is Riemann delta integrable on , then is Lebsgue delta integrable on , and
where and indicate the Riemann delta integral and Lebesgue delta integral from to , respectively.
Assume . Let denote the set Then the space is a complete linear space with the norm defined by
Lemma 4 (Hölder inequality [4, 5]). Let , , and be the conjugate number of . Then
Lemma 5 (see ). For any ,(a); (b) is dense in .
From Lemma 5, we see that Lemma 4 still holds for , .
Lemma 6 (see ). Suppose that is a sequence in , for some .(a)If , for some , and if is right-scattered, then .(b)If is a Cauchy sequence in (with respect to the norm ), then there exists a unique such that .
Following , we now define the generalized derivative of Lebesgue delta integrable functions.
Definition 7. Define the norm on by and define the space to be the completion of with respect to the norm and to be the completion of with respect to the norm .
Lemma 8 (see ). (a) if and only if there exists a function such that the following condition holds: there exists a sequence in such that and in . If , then the function is unique (in sense).
(b) If , then .
Definition 9. For any , the function in Lemma 8 will be called the generalized derivative of .
Remark 10. We can also define the generalized derivative of and the spaces as in .
The following two lemmas present basic properties of .
Lemma 11 (see ). If , then , and there exists such that Furthermore,
Lemma 12 (see ). Suppose . Then(a)if the sequence in is as in Lemma 8, then in ;(b)if , then constant;(c)if is right scattered, then ;(d)if , then and ;(e);(f), .
Remark 13. From (f), if , then we have , ; hence, we can also define weak derivatives as usual Sobolev space .
Definition 14 (see ). A function is said to be absolutely continuous on if for every , there exists a such that if with is a finite pairwise of subintervals satisfying
Lemma 15 (see ). If and is the function defined by then is absolutely continuous and -almost everywhere on .
Proposition 16 (see ). A function is absolutely continuous on if and only if is -differentiable -almost everywhere on and
In the following sections, we still write as .
3. Generalized Maximum Principle
Let be a bounded time scale and set , ; 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 function that takes some value on the boundary is understood in the trace sense, that is, the limitation of some suitable smooth function with definite value on the boundary . The boundary value of a function is understood in the same way; that is, if , , in , and , , then we say , . And , are understood in the same way.
We define the bilinear form associated with the operator as follows:
We assume that satisfy the following conditions:
Theorem 17 (generalized weak maximum principle). If satisfies in weak sense; that is, for all , then
Proof. If , , we have and ; hence we can rewrite as
Equation (25) holds for all which satisfys .
Conditions (23) and (25) imply In the special case that , we can easily obtain the result by choosing , where . In general case, we deduce by contradiction. Suppose that ; we can then choose satisfying and set . Then we have and and hence, we obtain by (26) that The condition on and Hölder’s inequality imply from which, we have Applying embedding theorem and Hölder’s inequality, we get where , denotes the measure of set . Hence, Since the above inequality does not depend on , it still holds as tends to , that is, , that is, must attains its superemum on a set of positive measure. Hence, the set of the points on which attains its superemum must contain either an interval (in time scale sense) or at least one right-scattered point .
In the first case where in , then and in . Choosing , we have which is a contradiction.
In the second case, we have ; if , then proceeding as before in the interval , we then get a contradiction. If , then . Set , where is the characteristic function of the set ; then we have ; hence, we deduce from that while the above two inequalities contradict each other.
Theorem 18 (strong maximum principle). If satisfies in weak sense; that is, for all , suppose further that attains its nonnegative maximum (nonpositive minimum) at inner point of and at (at ), and then is a constant.
Proof. Suppose that is not a constant and attains its maximum at . Then there exists at least one point such that . Then we may assume . In the first case, we may assume in . If is right scattered, then . Choosing , we deduce from that while the above two inequalities contradict each other. If is right dense, Lemma 11 implies that is continuous on , especially at , and there exists a neighbourhood of such that and is decreasing on ; hence is uniformly continuous on and so absolutely continuous on ; therefore, Proposition 16 implies a.e. on . Choosing , 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 attains its nonnegative maximum at (nonpositive minimum at ).
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 by applying the weak maximum principle.
Theorem 20. If satisfies in weak sense, that is, , for all , , and , then for all .
Proof. We assume that satisfies in weak sense; then by Theorem 17, we have
We can easily deduce from Theorem 20 the following.
Corollary 21. If satisfies , in weak sense, that is, , , for all , , and , then for all .
Corollary 22. If satisfies , in weak sense, that is, , , for all , then , for all .
Corollary 23. If satisfies , , in weak sense, that is, , , for all , , and , then for all .
Definition 24. If satisfies , in weak sense, we say that is a weak subsolution (supper solution) to dynamic equation .
Corollary 23 asserts that if the supper-solution and subsolution to dynamic equation attain the same value on the boundary of , then the supper solution is not less than the subsolution.
5. Uniqueness Results
We now consider the following dynamic equation: where .
Theorem 25. There exists at most one solution to dynamic equation (39) in .
Proof. Suppose that there exist two solutions to (39). Set ; then satisfies Then Theorem 17 implies , from which we deduce that .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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.
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