This paper is concerned with the existence and nonexistence of positive solutions to the singular third-order

Singular boundary value problems for nonlinear ordinary differential equations arise in a variety of areas of applied mathematics, physics, chemistry, and so on. For earlier works, see [

Very recently, motivated by Ma [

where

By using Guo-Krasnosel’skii fixed point theorem, the author established the following results.

Suppose that (

Suppose that (

Suppose that (

Being directly inspired by the previously mentioned works, we will consider the existence and nonexistence of positive solutions to the following third-order

We assume that (

From

Our main results are the following.

Let (

Let (

The proof of previous theorems is based on the Schauder fixed-point theorem.

BVP

We do not assume any monotonicity condition on the nonlinearity as in [

It is obvious that Theorem

In this section, we present some notation and preliminary lemmas.

Let

Suppose that

Then

Let

for every

for every

Suppose that (

has a unique nonnegative solution

The proof of the uniqueness is standard and hence is omitted here. Now we prove the existence of the solution.

From (

In fact, from (

Thus

Similarity,

By Lemma

Again applying (

This together with (

By Lemma

It is easy to see that

The proof is complete.

For any

Suppose that (

The proof is similar to Lemma 2.4 in [

Suppose that (

has a unique solution

From (

Now set

This together with the fact that

In this section, we will prove our main results.

We divide the proof into three steps.

Let

Then

Let

Define a closed convex subset in

and an operator

Suppose that

In fact, from Lemma

The claim is proved. Using the Schauder fixed point theorem, we conclude that

Suppose to the contrary that BVP of (

On the other hand, since

This implies that

Thus

which is a contradiction.

Now we consider the following third-order

Since

By Lemma

Let

Suppose to the contrary that

If

If

In fact, from the fact that

Let

We easily verify that

This contradicts with the fact that

We easily obtain

If

By Lemma

Summarizing the previous discussion, we assert that

Since

That is,

Consider the boundary value problem

But we cannot apply Theorem B [

The authors are grateful to the anonymous referee for his or her constructive comments and suggestions which led to improvement of the original paper. X. Han is supported by the NNSF of China (no.11101335).