^{1}

^{2}

^{1}

^{2}

This work is concerned with the monotone iterative technique for partial dynamic equations of first order on time scales and for this purpose, the existence, uniqueness, and comparison results are also established.

The study of dynamic equations on time scales is an area
of research that has received a lot of attention. Many authors have been
devoted to the qualitative research of ordinary differential equations on time
scales. We can refer to monographs by Bohner and Peterson [

Lakshmikantham et al. [

For the detail of basic notions concerned with time scales, we refer to [

A time scale is a nonempty closed subset of the real
number

If

Let

In this section, we generalize existing ideas of the time scales calculus to the multivariate case.

Firstly, consider the product

the forward jump operator

the back jump operator

the graininess function

Secondly, we introduce the definitions of time-scale
derivatives for the function

Because we will need notation for partial derivatives
with respect to time-scale variables

The following definitions of these partial derivatives are now given.

Let

Finally, we define the partial derivative of

Let

In this section, we consider the initial value problem (IVP) for partial dynamic
equation of first order on time scales

We define the set

Now, we begin to prove the following comparison result.

Assume that

Then

Let us prove the theorem for strict inequalities firstly. For example,
we suppose that _{1}), we obtain _{0}) is not strict, and on one
hand, if _{0}) and (A_{2}), we have
_{0}), (A_{2}), and Lemma

Assume that (A_{1}) and (A_{2}) hold. Suppose further that (A_{3}) for each _{4}) for each

Then there exists a unique solution

By (A_{3}) and (A_{4}),

To show uniqueness of solution of (

we are now in a position to describe the monotone iterative technique which yields monotone sequences on time scales. We prove the following result specifically.

Assume that (A_{0}), (A_{1}), and (A_{3}) hold with _{5}) for some _{6}) for each

Then there exist monotone sequences

Consider the linear IVP

By (A_{5}), we have
_{0}) holds. Also, if _{2}) is satisfied for _{6}), (A_{4}) is satisfied relative to

Defining a mapping

Let

To prove (ii), let

Consider the sequence

Finally, we show that

We note that if (A_{4}) holds, then

This work is supported by the Key Project of Chinese Ministry of Education (207014) and the Natural Science Foundation of Hebei Province of China (A2006000941).