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 [

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 [

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 [

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 [

The paper is organized as follows. In Section

We introduce some concepts related to time scales, which can be found in [

A function

Let

if

if

if

if

if

let

if

A function

Let

let

let

(fundamental theorem of calculus) let

(integration by parts) let

if

The construction of the

for each

if

if

The Lebesgue integral associated with the measure

Let

Assume

Let

For any

From Lemma

Suppose that

If

If

Following [

Define the norm

(a)

(b) If

For any

We can also define the generalized derivative of

The following two lemmas present basic properties of

If

Suppose

if the sequence

if

if

if

From (f), if

A function

If

A function

In the following sections, we still write

Let

To study the generalized maximum principle, we should make clear what it means when we say a

We define the bilinear form associated with the operator

We assume that

If

If

Conditions (

In the first case where

In the second case, we have

If

Suppose that

From the proof of Theorem

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

If

We assume that

We can easily deduce from Theorem

If

If

If

If

Corollary

We now consider the following dynamic equation:

There exists at most one solution to dynamic equation (

Suppose that there exist two solutions

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.