Positive Solutions for Third-Order Boundary Value Problems with the Integral Boundary Conditions and Dependence on the First-Order Derivatives

Abstract

In this paper, by the use of a new fixed point theorem and the Green function of BVPs, the existence of at least one positive solution for the third-order boundary value problem with the integral boundary conditions is considered,where there is a nonnegative continuous function. Finally, an example which to illustrate the main conclusions of this paper is given.

1. Introduction

Third-order boundary value problems are originated in a variety of different fields of applied mathematics and physics, for example, deflection of a buckling beam with a fixed or varying cross-section, three-layer beams, electromagnetic waves, and flood tides from gravitational blowing. In recent year, researches on third-order nonlinear boundary value problems have received widespread attention, and many excellent results have been obtained, in references [1–10].

As we all know, boundary value problems with integral boundary conditions can describe many valuable phenomena more accurately. The study of many problems in the fields of heat conduction, chemical engineering, groundwater flow, thermoelasticity, plasma physics, etc. can be reduced to the study of boundary value problems with integral boundary conditions. However, in recent years, although the third-order boundary value problems with integral boundary conditions have received widespread attention, there are relatively few researches on the third-order boundary value problems with integral boundary conditions, in references [11–13].

By using the Guo-Krasnoselskii fixed point theorem, Zhao and Cunchen [11] investigated the existence and nonexistence of at least one or two monotone positive solutions for the following third-order boundary value problem with integral boundary conditions:

By using the mixed monotone operator method, He and Xiaoling [13] proved the existence and uniqueness of positive solutions for the following third-order ordinary differential equations with integral boundary conditions:

All the above works were done under the assumption that derivative is not involved explicitly in the nonlinear term . In this paper, we are concerned with the existence of positive solutions for the third-order boundary value problem with the integral boundary conditions:

Throughout, we assume are continuous;

2. Preliminary

Let be the Banach space equipped with the norm

Lemma 1 (see [12]). Let . Then for any the problem has a unique solution where, By (5), we get
Let

Lemma 2 (see [11]). For any , have (i)(ii)

Lemma 3. If then the unique solution of problem (1) satisfies

Proof. By Lemma 1, we get By Lemma 2, we get So, For we have The proof is completed.

Let be a Banach space and a cone. Suppose are two continuous convex functionals satisfying for and for and for where is a constant.

Theorem 4 (see [14]). Let be constants and (13) two bounded open sets in . Set

Assume is a completely continuous operator satisfying the following:

_(A1)

_(A2)

_(A3) There is a such that and for all and

Then, has at least one fixed point in

3. The Main Results

Let be the Banach space equipped with the norm and is a cone in

Define two continuous convex functionals and for each , and then and for for

In the following, we denote

We will suppose that there are such that satisfies the following growth conditions: for ; for ; and for .

Let

Let

We denote

Lemma 5. Suppose hold. Then, is completely continuous.

Proof. For by Lemma 3, we have
So, we can get
In the following, we will show that is completely continuous.
At first we show that is continuous.
Let ; it satisfies ; then, there is a constant such that then If is uniformly continuous on we get Next, we show that is compact.
Let be bounded, and then, there is such that For , we have where It is clear that is a bounded set in ; because is uniformly continuous on for , there exists such that for
For we have Therefore is equicontinuous. Using the Arzela-Ascoli theorem, a standard proof yields is completely continuous.

Theorem 6. Suppose - hold. Then, BVP (1) has at least one positive solution satisfying

Proof. Take two bounded open sets in and
By Lemma 5, is completely continuous, and there is a such that , for all and
By Lemma 2 and , for we get By Lemma 2, for there is
So, by , we get By , for we have Theorem 4 implies there is such that So, is a positive solution for BVP (1) satisfying Thus, Theorem 6 is completed.

4. Example

Example 1. Consider the following boundary value problem where, In this problem, we know that ; then, we can get Choose then

Furthermore, we obtain

If we take then we get .

Then, all the conditions of Theorem 6 are satisfied. Therefore, by Theorem 6 we know that boundary value problem (1) has at least one positive solution satisfying

Data Availability

This paper is a basic theoretical study without data support.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors declare that the work was realized in collaboration with same responsibility. All authors read and approved the final manuscript.

Acknowledgments

The project is supported by the Department of Education Science and Technology Research Project of Jiangxi Province in 2021 (item nos. GJJ212513 and GJJ219004).