We are concerned with the following third-order three-point boundary value problem:

Third-order differential equations arise from a variety of different areas of applied mathematics and physics, for example, in the deflection of a curved beam having a constant or varying cross-section, a three-layer beam, electromagnetic waves or gravity driven flows, and so on [

Recently, the existence of single or multiple positive solutions to some third-order three-point boundary value problems (BVPs for short) has received much attention from many authors; see [

In 2008, Palamides and Smyrlis [

Very recently, for the third-order three-point BVP with sign-changing Green’s function

In 2013, Li et al. [

It is worth mentioning that there are other types of works on sign-changing Green’s functions which prove the existence of sign-changing solutions, positive in some cases; see Infante and Webb’s papers [

Motivated greatly by the above-mentioned works, in this paper, we consider the following third-order three-point BVP:

To end this section, we state some fundamental definitions and the two-fixed-point theorem due to Avery and Henderson [

Let

A functional

Let

Let

For the BVP

The BVP (

It is simple to check.

Now, for any

After a direct computation, one may obtain the expression of Green’s function

It is not difficult to verify that the

Let

Let

For

For

Obviously,

Throughout this paper, for any

Let

By Lemma

In the remainder of this paper, we always assume that

for each

for each

Let

Now, we define an operator

For convenience, we denote

Suppose that there exist numbers

First, we define the increasing, nonnegative, and continuous functionals

Next, for any

Now, we assert that

To prove this, let

Then, we assert that

To see this, suppose that

Finally, we assert that

In fact, the constant function

To sum up, all the hypotheses of Theorem

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