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We investigate a class of variable coefficients singular third-order differential equation with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. By applications of Green’s function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established.

Generally speaking, differential equations with singularities have been considered from the very beginning of the discipline. The main reason is that singular forces are ubiquitous in applications, the most obvious examples being gravitational and electromagnetic forces. In 1965, Ding [

Ding’s work has attracted the attention of many specialists in differential equations. More recently, the method of lower and upper solutions, Poincaré-Birkhoff twist theorem, Mawhin’s topological degree theorem, Schauder’s fixed point theorem, and Krasnoselskii fixed point theorem in a cone have been employed to investigate the existence of positive periodic solutions of singular second order differential equations (see, e.g., [

At the beginning, most of work concentrated on second-order singular differential equations, as in the references we mentioned above. Recently, there have been published some results on third-order singular differential equation (see [

In the above papers, the authors investigated singular third-order equations with constant coefficients. However, the study on the singular third-order equation with variable coefficients is relatively infrequent. Motivated by Torres et al. [

As far as we know, studies on third-order differential equation with variable coefficients are rather infrequent, especially those focused on the research of singular third-order differential equations with variable coefficients. The main difficulty lies in the calculation of Green’s function of the third-order differential equation with variable coefficients, being more complicated than in the constant-coefficient case. Therefore, in Section

Let

Firstly, we consider

There exist differentiable

Next, we will consider

Suppose that

Suppose (

Let

Therefore, we know that the solution of (

Assume that (

From (

There exist an

Let us define

Similarly to (

Assume

From Lemma

Firstly, we establish the existence of positive periodic solutions for third-order differential equation (

Let

Then

For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:

(

(

(

Under Lemmas

Let

Define the cone

Define the operator

Now, if

Assume that (

If

If

Assume that (

If

If

We split

Noting that

Thus, it is clear that

On the other hand, if

Now let

Assume that (

From the definition of

If

Assume that (

If

If

We split

Let

Assume

Assume that (

From the definition of

If

Assume that (

If we choose

The conclusions of Lemmas

Assume that (

If

Now we consider the cases that

Assume that (

If

If

By selecting

Assume that (

If

Again, if

Assume that (

If

If

By selecting

In this section, we present out main results for the existence and multiplicity of positive periodic solutions of singular third-order equation of repulsive type (

Let (

If

If

There exists a

(a) Since

On the other hand, by the condition

(b) Again, since

On the other hand, in view of the assumptions

(c) First we note that

On the other hand, in view of the assumption

When

Let (

If

If

There exists a

(a) Since

On the other hand, since

(b) First, since

On the other hand, since

(c) Since

On the other hand, in view of the assumption

In this case, replacing assumptions (

We illustrate our results with some examples.

Consider the following singular equation:

Comparing (

Consider the following singular equation:

Comparing (

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

Research is supported by NSFC Project (nos. 11326124 and 11271339).