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The efficient conditions guaranteeing the existence of homoclinic solutions to second-order singular differential equation with

In the last years, homoclinic solutions for Hamiltonian systems and differential and difference systems have been studied by several authors. Based on variational methods and critical points theory, Rabinowitz [

In recent years, homoclinic solution problems of second-order singular differential equation have raised concerns. Bonheure and Torres [

Motivated by the above work, this paper is devoted to the study of the existence of homoclinic solutions to second-order singular differential equation with

The distinctive contributions of this paper are outlined as follows:

The problem (

Due to singularity, it is very difficult for estimating priori bound. In order to overcome this difficulty, we develop a new technique introduced in [

A unified framework is established to handle second-order equations with singularity term and

The following sections are organized as follows: In Section

In this section, we give some notations and lemmas which will be used in this paper. The set of all positive integers is denoted by N. Let

For each

The equation

The Brouwer degree

Assume that

Then, problem

For investigating the existence of homoclinic solutions to (

In the present paper, we list the following assumptions:

(H_{1}).

(H_{2}).

(H_{3}).

Let

Obviously, the existence of

Here, we give the main results of the present paper in the following theorem.

Assume that the assumptions (H_{1})–(H_{3})

Let

Let

There exist

This implies that

By (

In view of monotonicity of

On the other hand,

We claim that

In fact, if (

By (

Thus, we have

Now, we estimate the bound of

It follows from (

Integrating (

Let

Thus

Since

In view of Lemma

From (

Now, we will show

Multiplying (

From (_{1}) and (H_{2}), we have

In view of (H_{3}) and (

In view of (

It follows by (

In view of (

From (

Thus,

Next, we prove

Furthermore, by (

This section presents two examples that demonstrate the validity of our theoretical results.

Obviously, _{1})-(H_{3}) hold. Based on Theorem

Obviously,

In this paper, we study a class of second-order singular equation with

No data were used to support this study.

The authors declare that they have no competing interests.

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

The work is supported by the Natural Science Foundation of Jiangsu High Education Institutions of China (Grant No. 17KJB110001).