Abstract

In this paper, we consider a nonlinear third-order three-point boundary value problem and give the existence and uniqueness of solutions by constructing Green’s function and using its properties. The methods used here are based on Darbo’s fixed point theorem combined with the technique of measure of noncompactness. Finally, as applications, two examples are given to illustrate our main results.

1. Introduction

Multipoint boundary value problem (BVP) for the third- and higher-order differential equations plays an important part in various fields, such as fluid mechanics, physics, engineering, and many other branches of applied mathematics (see [114]). In the past decades, third-order three-point boundary value problems have been widely investigated. In particular, Anderson in [1] considered a right focal problemwhere , , are real numbers and , and the Krasnoselskii, Leggett, and Williams fixed point theorems were used to prove the existence of at least three solutions of (1).

In [2], the authors solved the existence of the solutions of the BVPwhere is a continuous function, , , , , . The coincidence degree theory of Mawhin is the fundamental tool to deduce the existence results.

In 2019, the authors in [3] studied the existence and uniqueness of solutions of the BVPwhere , , , , and its primary tools are contracting mapping theorem and variation of parameters formula; namely, it first deals with

And then, the solution of BVP (3) can be expressed aswhere are constants determined by its boundary value conditions.

Motivated greatly by above-mentioned works, in this paper, we consider the existence and uniqueness of solutions to the following BVPwhere , , is continuous and . The methods used are a measure of noncompactness and Darbo’s fixed point theorem, which prove the existence of solutions, and at the same time, its uniqueness also holds by Banach contraction principle, which is different from [3]. A new direction of research with the presentation of the notion of a measure of noncompactness was opened up in 1930 by Kuratowski, which can be applied to prove the existence results related to various integral, differential equations, integro-differential equations as well as their systems. As an important application of this measure, Darbo’s fixed point theorem generalizes the Schauder-fixed point theorem and Banach contraction principle, especially in proving the existence of solutions for classes of nonlinear equations (see[1530]). Comparing with other papers, our advantage is that the solution we get is in a small sphere rather than in the whole space.

The rest of the paper is organized as follows. In Section 2, we give the noncompactness, Darbo’s theorem, and related notations. In Section 3, the corresponding Green’s function and its properties are given. In Section 4, we prove our main results on the existence and uniqueness for the solutions and give some examples to verify our results’ effectiveness and applicability.

2. Preliminaries

We start this section by introducing some necessary definitions, notations, and basic results required for further development.

Let be subset of a metric space , we note and are closure and convex of , respectively; and , respectively, denote the family of all nonempty and bounded subsets, and nonempty and relatively compact subsets of the metric space . Then,(1)finite set is a finite net of if and only if for any , any ; there exists , such that(2) is totally bounded if and only if it has a finite net.(3) is relatively compact if and only if any sequence of has a convergent subsequence.(4) is compact if and only if any sequence of has a convergent subsequence, and its limit is in . is compact if is relatively compact and closed.

Definition 1. (see [31]). The Kuratowski measure of noncompactness defined on bounded set of Banach space iswhere denotes the diameter of the set , that is,

Definition 2. (see [32]). The Hausdorff measure of noncompactness defined on bounded set of Banach space iswhereThe definition of the Hausdorff measure of noncompactness for the set can equivalently be stated as follows:

Definition 3. (see [32]). The measure of noncompactness defined on bounded set of Banach space isThe definition of the measure of noncompactness for the set can also equivalently be stated as follows:Almost all known measures of noncompactness possess the property that they are equal to zero on the family of all relatively compact sets in a given space.

Lemma 1. (see [32]). Let be a Banach space, and be bounded subsets of , . Then,(1) if and only if is a relatively compact(2)the family ker is nonempty and ker (3)(4)(5), where (6), where (7)(8), for any , where (9)where denotes the distance of Hausdorff between and ; i.e.,(10), , for , if ; then, In this article, we work on infinite space , which denotes the space of all continuous functions defined on the interval . For , we define the usual norm as follows:It is well known that is complete.
The measure of noncompactness in can be formulated as follows ([33]).
Let be a nonempty and bounded subset of the space and for , , , letandwhere

Lemma 2 (Agarwal and O’Regan [34]). Let be a closed, convex subset of a Banach space . Then, every compact, continuous map has at least one fixed point.
Using the measure of noncompactness, Darbo generalized Lemma 2, namely, Darbo’s fixed point theorem.

Lemma 3 (Darbo [35]). Let be a nonempty, bounded, closed, and convex subset of a Banach space and be a continuous mapping. Assume that there is a constant such thatThen, has a fixed point.

Lemma 4 (Ahmad et al. [36]). Let be a closed subset of Banach space and be a strictly contractive operator; i.e., there exists a constant such thatThen, has a unique fixed point.

3. The Construction of Green’s Function

First of all, let us construct the Green’s function for the BVPwhere , , .

Lemma 5. The above BVP (22) has the solutionwhereand

Proof. We easily know that , where , , are constants. As a result,by using the boundary condition , which yields that , and thus,Substituting the values of and into the above equation, one hasApplying the Gaussian formula, we haveandTherefore,This completes the proof.

Lemma 6. The Green’s function from Lemma 5 satisfies the following properties:(1) is continuous(2) for

Proof. The continuity of is obvious, and next, we consider (2). First, we note that(i)When ,(ii)When ,Therefore, is decreasing with respect to for fixed , soandThis completes the proof.
Now, we shall present our main results concerning the existence and uniqueness of solutions for problem (6). Let us introduce the following conditions:
There exists a nonnegative constant such that.
There exists a positive number satisfying the following inequality:where .

4. Main Results

We now give the existence and uniqueness of solutions for BVP (6).

Theorem 1. Under the conditions , BVP (6) has at least a solution.

Proof. First of all, define an operator on byThen, is a solution of BVP (6) if and only if it is a fixed point of . Next, we are going to divide the progress into three steps.

Step 1. We need to show the operator is continuous.
According to Lemma 6 and condition , for any , there exists such that, for any and , we haveIn fact,So,

Step 2. We prove , where denotes a closed sphere of which center is 0 and radius is .
For any , , from and Lemma 6 (2), we haveThis impliesCombining condition with (44), we know that there exists such that maps into itself.

Step 3. Now, we prove .
First, for all , , based on the proof process of inequality (40), we can obtainand it follows thatBesides, for all , , by the continunity of , we know thatcombining (46) with conditions and , for any , we havewhich implies thatConsequently,Let , we obtainIn view of (45) and (50), we easily obtainThus, by Lemma 3, we conclude that has at least a solution in ; that is, BVP (6) has at least a solution in . This completes the proof.

Corollary 1. In fact, Theorem 1 not only proves the existence of solutions of BVP (6) but also illustrates its uniqueness.

Proof. According to the proof of inequality (40) and condition , we know, namely, operator is a contraction mapping, so has a unique solution in by construction mapping theorem in Lemma 4; that is, BVP (6) has a unique solution in the whole space . This completes the proof.
To be honest, if we only need to know the BVP (6) has an unique solution in its domain, the Lipschitz conditions and are enough. But, it is better to find the more accurate range of the solution in practical matters and concrete situations, so we could choose Theorem 1 to identify the scope of the solution.
In the following, we give two concrete examples to illustrate our main results.

Example 1. Consider the following problem:Here, we haveBy easy calculation, we getandchoose , with the help of Theorem 1, the problem (51) has at least a solution in .

Example 2. Consider the following problemHere, we haveBy easy calculation, we getandchoose , with the help of Theorem 1, the problem (55) has a solution in .

5. Conclusion

In this paper, we study the nonlinear third-order three-point boundary value problem (6). First, we construct Green’s function for the third-order three-point boundary value problem (22) and discuss its properties. Second, based on Green’s function and its properties, we define a solution operator . Then, we prove that has at least a fixed point by using Darbo’s fixed point theorem combined with the technique of measure of noncompactness. The fixed point is unique by Banach fixed point theorem. Therefore, the existence and uniqueness of solutions for (6) have been established. Finally, as applications, two examples are given to illustrate our main results. It should be pointed out that the method used here can be applied to impulsive differential equation boundary value problems, such as [37].

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

The researchers would like to thank Shanxi Vocational University of Engineering Science and Technology for funding the publication of this project.