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Resonance and nonresonance periodic value problems of first-order differential systems are studied. Several new existence and uniqueness of solutions for the above problems are obtained. To establish such results sufficient conditions of limit forms are given. A necessary and sufficient condition for existence of nontrivial solution is also proved.

This paper is concerned with the existence and uniqueness of solutions of the nonresonance periodic boundary value problems (BVP) for the first-order differential system:

The paper is also concerned with the existence and uniqueness of the resonance periodic boundary value problem of first-order:

Recently, by using fixed point methods and Leray-Schauder degree theory, for (

Let

Let

However, we find that Lemma

The purpose of this paper is to establish several new existence and unique theorem for BVP (

In the proof of the existence theorem below, we will use the following fixed point theorem.

Let

Let

Assume that (

Lemma

Obviously, from (

Assume that (

There exist function

From Lemma

Define the map

Now we apply Shaeffer's theorem to prove that BVP (

From Lemma

In particular, choose, respectively,

In Theorem

We only prove the corollary for (

Let

Next, we give a necessary and sufficient condition for BVP (

Assume that (

Suppose that (

Suppose that BVP (

For the resonance BVP

Since Lemma

Assume that (

There exist functions

BVP (

Assume that (

The result can be obtained from Lemma

It is possible that in Theorems

Assume that all the conditions of Theorem

By Theorem

Similarly, we have the following result.

Assume that all the conditions of Theorem

In this section, we will establish uniqueness results of the solutions for BVP (

Consider the Banach space

Assume that (

First, to show that (

Let

Next, if (

Now, consider BVP

Similarly, the following result may be obtained.

Assume that (

To provide an exact estimations of

In particular, for

Applying Theorems

Assume that (

Assume that (

Since when there is at least a

In this section, we will present examples which highlight the theory of this paper. We also compare our results with known ones.

Consider BVP (

Consider BVP of scalar differential equation:

The results obtained in the previous sections are new even for functions

From the results of this paper, one easily sees that for both cases of nonresonance (

The results of this paper can be generalized impulsive periodic boundary problems of first-order; see [

This work is supported by the Fund of the Doctoral Program Research of the Education Ministry of China (20040108003), the Sciences Foundation of Shanxi (2009011005-3), and the Major Subject Foundation of Shanxi.