Existence and Uniqueness of Solutions for a Third-Order Three-Point Boundary Value Problem via Measure of Noncompactness

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.

In 2019, the authors in [3] studied the existence and uniqueness of solutions of the BVP where η ∈ (a, b), k ∈ R, f: C [a, b] ⟶ R, f(t, 0) ≠ 0, and its primary tools are contracting mapping theorem and variation of parameters formula; namely, it rst deals with And then, the solution of BVP (3) can be expressed as where λ 0 , λ 1 , λ 2 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 BVP is continuous and f(t, 0) ≠ 0. e 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 [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]). Comparing with other papers, our advantage is that the solution we get is in a small sphere rather than in the whole space. e 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.

Preliminaries
We start this section by introducing some necessary definitions, notations, and basic results required for further development.
Let X be subset of a metric space (E, d), we note X and co(X) are closure and convex of X, respectively; M E and N E , respectively, denote the family of all nonempty and bounded subsets, and nonempty and relatively compact subsets of the metric space (E, d). en, (1) finite set N ⊂ X is a finite ϵ− net of X if and only if for any ϵ > 0, any x ∈ X; there exists y ∈ N, such that (2) X is totally bounded if and only if it has a finite ϵ− net. (3) X is relatively compact if and only if any sequence of X has a convergent subsequence. (4) X is compact if and only if any sequence of X has a convergent subsequence, and its limit is in M. M is compact if M is relatively compact and closed.
Definition 1 (see [31]). e Kuratowski measure of noncompactness α(·) defined on bounded set S of Banach space E is where diam(S i ) denotes the diameter of the set S i , that is, Definition 2 (see [32]). e Hausdorff measure of noncompactness χ(·) defined on bounded set S of Banach space E is where e definition of the Hausdorff measure of noncompactness for the set S can equivalently be stated as follows: Definition 3 (see [32]). e Istr� atescu measure of non- e definition of the Istr� atescu measure of noncompactness for the set S can also equivalently be stated as follows: Journal of Mathematics 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 E be a Banach space, S and T be bounded subsets of E, a ∈ R. en, where d h (S, T) denotes the distance of Hausdorff between S and T; i.e., It is well known that (C[a, b], ‖ · ‖) is complete. The measure of noncompactness in C[a, b] can be formulated as follows ( [33]).
Let X be a nonempty and bounded subset of the space and where diam Then, T has a fixed point.
Then, T has a unique fixed point.

The Construction of Green's Function
First of all, let us construct the Green's function for the BVP Lemma 5. e above BVP (22) has the solution where and Proof. We easily know that where c 1 , c 2 , c 3 are constants. As a result, by using the boundary condition x ″ (a) � 0, which yields that c 3 � 0, and thus, Journal of Mathematics 3 Substituting the values of a, b and η into the above equation, one has Applying the Gaussian formula, we have and erefore, is completes the proof. □ Lemma 6. e Green's function G(t, s) from Lemma 5 satisfies the following properties: Proof. e continuity of G(t, s) is obvious, and next, we consider (2). First, we note that (ii) When a ≤ s ≤ t ≤ b, erefore, G(t, s) is decreasing with respect to t for fixed s ∈ [a, b], so and is 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: (C 1 ) ere exists a nonnegative constant L such that ere exists a positive number d 0 satisfying the following inequality: where c � max a≤t≤b |f(t, 0)|.

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

G(t, s)f(s, x(s))ds, t ∈ [a, b].
(39) en, x(t) is a solution of BVP (6) if and only if it is a fixed point of H. Next, we are going to divide the progress into three steps.
Step 1. We need to show the operator H is continuous.
For any x ∈ C[a, b], t ∈ [a, b], from (C 1 ), (C 2 ) and Lemma 6 (2), we have Journal of Mathematics 5 Combining condition (C 3 ) with (44), we know that there exists d 0 > 0 such that H maps B d 0 into itself.
Step 3. Now, we prove α( First, for all x, y ∈ B d 0 , t ∈ [a, b], based on the proof process of inequality (40), we can obtain and it follows that Besides, for all |t 1 − t 2 | < ε, t 1 , t 2 ∈ [a, b], by the continunity of G(t, s), we know that combining (46) with conditions (C 1 ) and (C 2 ), for any x, y ∈ B d 0 , we have which implies that Consequently, Let T ⟶ ∞, we obtain In view of (45) and (50), we easily obtain us, by Lemma 3, we conclude that H has at least a solution in B d 0 ⊂ C [a, b]; that is, BVP (6) has at least a solution in B d 0 ⊂ C [a, b]. is completes the proof.

Corollary 1. In fact, eorem 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 (C 2 ), we know, namely, operator H: C [a, b] ⟶ C[a, b] is a contraction mapping, so H has a unique solution in C[a, b] by construction mapping theorem in Lemma 4; that is, BVP (6) has a unique solution in the whole space C [a, b]. is 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 (C 1 ) and (C 2 ) 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 eorem 1 to identify the scope of the solution.
In the following, we give two concrete examples to illustrate our main results.
Journal of Mathematics Example 1. Consider the following problem: Here, we have By easy calculation, we get and choose d 0 � 2.8, with the help of eorem 1, the problem (51) has at least a solution x(t) in B d 0 .
By easy calculation, we get and c � max 0≤t≤1 e t cost � ecos1 ≈ 2.718, ζ � 25 32 choose d 0 � 9.8, with the help of eorem 1, the problem (55) has a solution x(t) in B d 0 .

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 H. en, we prove that H has at least a fixed point by using Darbo's fixed point theorem combined with the technique of measure of noncompactness. e fixed point is unique by Banach fixed point theorem. erefore, 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
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e author declares no conflicts of interest.