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We are concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems

As is known to all, the search for periodic as well as homoclinic and heteroclinic solutions of Hamiltonian systems has a long and rich history. In present paper, we particularly focus our attention on the existence of homoclinic solutions of second order nonautonomous singular Hamiltonian systems. For the results on the literature of periodic solutions for such singular systems, we refer the reader to the book [

Second order Hamiltonian systems are systems of the following form:

For singular systems, one assumes that

The present paper is concerned with the existence of homoclinic solutions for the following second order nonautonomous singular Hamiltonian systems:

there is a constant

for all

there is a neighbourhood

Now we are in the position to state our main result.

Under the conditions of (A) and (

The assumption (

In the case of autonomous singular Hamiltonian systems, the first result on existence of a homoclinic orbit using variational methods was obtained by Tanaka [

In the case of planar autonomous systems, more extensive existence and multiplicity results were obtained. Indeed, under essentially the same conditions as above with

On the other hand, in the case of

there exists

Here we must point out that all the results mentioned above are obtained for the case that

The remaining part of this paper is organized as follows. In Section

In this section, we investigate the approximating problem

In order to obtain a critical point of

Let

Since

Now, one introduces a minimax procedure for

Choose

For any given

In view of Propositions

In this section, we give some estimates on the solutions

There is a constant

In what follows, we denote by

Suppose that

Define the function

In view of (A) and (

It only remains to show that

The following proposition gives us an

Using (

By Proposition

Consider

Let

In this section, we construct a homoclinic solution of

In what follows, we focus our attentions to show that

In view of (

As the last step of the proof of Theorem

Here, we just check the case that

Since

Up to now, we are in the position to give the proof of our main result.

In view of Propositions

The authors declare that there is no conflict of interests regarding the publication of this paper.

The project is supported by the National Natural Science Foundation of China (Grant no. 11101304).