^{1}

^{2}

^{2}

^{1}

^{2}

By introducing subdifferentiability of lower semicontinuous
convex function

The existence of periodic solutions for differential system has received a great deal of attention in the last few decades. Different from ordinary differential equations and partial differential equations that do not contain delay variate, it is very difficult to study the existence of periodic solutions for functional differential equations. For this reason, many mathematicians developed different approaches such as the averaging method [

However, the critical point theory was rarely used in the literature, and most of the existing results were established for autonomous functional differential equations while little was done for nonautonomous equations via critical point theory.

In this paper, by using critical point and operator equation theories, we study the existence of the following second-order nonlinear and nonautonomous mixed-type functional differential equation:

Our basic assumptions are the following

there exists a continuously differentiable function

Fix

It is obvious that

Moreover,

Let us consider the functional

For all

It is then easy to see that

It is not difficult to see that (

Since

In this paper, our main tool is the following.

Let

there exist constants

The rest of this paper is organized as follows. Subdifferentiability of lower semicontinuous convex function

Let

Let

By the definition of Subdifferentiability of function

If

One has

One has

Let

Since

Let

We divide our proof into two parts. First we show that

(i) The case when

From the definition of

Next, to prove

Consider the two convex sets

(ii) For all

Let

One has

Define an operator

For

Our objective is to define the conjugate function of

for

The conjugate function of function

Then

Let

The above expression holds since

Let

Let

In fact, for all

If

In this section, we discuss the existence of solutions to operator equation (

Under assumptions

To prove this theorem, we state and prove the following lemmas first.

Let

It is not difficult to verify that

We may obtain solutions to (

So if

The following two conditions are equivalent:

For all

Let

Let

By the convexity of function

Let

By Corollary

Now we show that (

Similar arguments to the proof of Lemma

Next we show that (

Finally, we show that (

There exist constants

By (

We next show the validity of the second part of the inequality.

For all

Let

(I) First, we show that the terms in

Since

Now suppose to the contrary that

(II) For all

For the chosen constants

By repeating the above argument on

From (

We next use the Maintain Pass Theorem to prove Theorem

(i) We show that

First, we show that

On the other hand, by Lemmas

So by (

By (

Let

By (

(iii) It is obvious that

Let

From (i), (ii), (iii), and the Maintain Pass Theorem, we conclude that problem (

In this section, we present a remark and an example to illustrate our main result.

For assumptions

Finally, as an application, we consider the following example.

Consider the equation

Moreover,

for

Thus, (

The project is supported by NNSF of China (10771055), Doctoral Fund of Ministry of Education of China (200805321017).