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This paper deals with the existence and uniqueness of periodic solutions for the first-order
functional differential equation

In this paper, we are concerned with the existence and uniqueness of periodic solutions for the first-order functional differential equation (cf., e.g., [

Functional differential equations with periodic delays such as those stated above appear in a number of ecological, economical, control and physiological, and other models. One important question is whether these equations can support periodic solutions, and whether they are unique. The existence question has been studied extensively by many authors (see, e.g., [

We will tackle the existence and uniqueness question by fixed point theorems for mixed monotone operators. We choose this approach because such fixed point methods, besides providing the usual existence and uniqueness results, sometimes may also provide additional numerical schemes for the computation of solutions.

We first recall some useful terminologies (see [

Every cone

Assume that

Let

A function

Assume

Let

Assume that

The proof of the following theorem can be found in [

Let

there exists

there exists

Condition (A3) in Theorem

A real

It is well known (see, e.g., [

Now let

The functions

We remark that the term quasi is used in the above definition to remind us that they are different from the traditional concept of lower and upper solutions (cf. (

Let

We need two basic assumptions in the main results:

for any

there exist

Suppose that conditions (

for any

there exist

The mapping

We will prove that

Suppose that conditions (

there exist

for any

for any

We assert that

Suppose that conditions (

for any

there exists

for any

Set

Suppose that conditions (

there exists

for any

We may easily prove that

Suppose that conditions (

if

for any

It is easily seen that

Suppose that conditions (

there exists

Then (

Set

Next we will prove that

Suppose that conditions (

for any

there exist

Indeed, it is easily seen that

If

Suppose that

for any

there exist

Indeed, from

As an example, consider the equation

Indeed, let

Other examples can be constructed to illustrate the other results in the previous section.

The first author is supported by Natural Science Foundation of Shanxi Province (2008011002-1) and Shanxi Datong University, by Development Foundation of Higher Education Department of Shanxi Province, and by Science and Technology Bureau of Datong City. The second author is supported by the National Science Council of R. O. China and also by the Natural Science Foundation of Guang Dong of P. R. China under Grant number (951063301000008).