Abstract

In this paper, we study the existence of positive solutions for the following nonlinear second-order third-point semi-positive BVP. We derive an explicit interval of positive parameters, which for any in this interval, the existence of positive solutions to the boundary value problem is guaranteed under the condition that are all superlinear (sublinear), or one is superlinear, the other is sublinear.

1. Introduction

In the applied mathematical field, three-point BVP can describe many phenomena. Moshinsky [1] introduced the vibrations of a guy wire with a uniform cross-section and composed of parts of different densities using a multipoint BVP. Timoshenko [2] also revealed that the theory of elastic stability can be used by the method of a three-point BVP. Il’in and Moviseev [3] were the first to study this aspect. Since then, more general nonlinear BVP have been studied by several authors [4–25].

In their paper [7], Ma and Wang obtained the existence of positive solutions for a three-point BVP by Krasnoselskii’s fixed theorem:where is a positive constant, , , , and there exists such that

In our paper, we study the existence of positive solutions of second-order third-point semipositive BVP:where , are positive parameters, , , and . And our paper also allows that are both semipositive and lower unbounded.

Our main tool is the following fixed point index theory.

Theorem 1 [4]. We suppose that is a cone in , in which is a real Banach space, the open bounded set is in , , , and . Suppose operator can be completely continuous and satisfies one of the following conditions:(i), ; , (ii), ; , Then, operator has at least one fixed point in .

Theorem 2 [4]. We suppose that is a cone in , in which is a real Banach space, the open bounded set is in , , , , and . Suppose operator is completely continuous and satisfies the following conditions:

Then, operator has at least two fixed points and in , and and .

2. Preliminaries and Lemmas

We set a Banach space with norm . We know of the following lemmas from Ref. [6].

Lemma 3. Settingas the positive solution of the equation, we have:

Then, is strictly increasing on , and .

Lemma 4. Settingas the positive solution of the equation, we have:Then, is strictly decreasing on .

From Lemma 3 and Lemma 4, we know that , . In the rest of our paper, the following condition is used:(C1), whereis given by Lemma 3

Throughout this paper, we shall use the following notation:where .

Obviously, from Ref. [6], we can be assured that when (C1) holds, the BVPis equivalent to the following integral equation:where .

Set . From (6), for , we know that

We present some other lemmas that are important to our main results.

Lemma 5 [7]. Assume that for any , is the solution of the following BVP:Then, we have

Lemma 6. Assume that is a solution of the following BVP:where . Then, there exists constant and satisfies

Proof. For , we can haveObviously, for , we havewhere .
By the same method, we can know thatwhere .
So, by choosing constant , we have☐

Lemma 7 [7]. Let , . Define the following function:Then(i) is a nondecreasing function for (ii)For g assumptions:(C2)From (C2), there exists a function, which satisfieswhere . is given by Lemma 6.(C3),(C4),(C5),whereLet , andwhere , .
For any , we set

From Lemma 6, letting , then is the positive solution of problem (2) if and only if is the solution of the following problem:and , ; here, , is given by (21).

Defining the cone in , we have

Obviously, problem (18) is equivalent to

Defining the operator , we have

Obviously and is completely continuous.

3. Our Main Three Results

Theorem 8. Suppose condition (C1), condition (C2), and condition (C3) hold. Then, for the small number , problem (2) has at least one positive solution.

Proof. Firstly, we choose sufficiently small which satisfies the following:Letting , for any , , by the definition of operator , we haveThus, we haveSecondly, by (C3), we know that there exists constant which satisfiesLetting , then . Set , for any , , we haveTherefore, we have .
Thus, by the definition of and (30), we can haveWe haveThen, by Lemma 5, we haveTherefore, by the definition of , we haveThen, by (29), (35) and Theorem 1, operator has at last one fixed point , i.e., is the solution of problem (2), and it is easy to know .
Finally, by (C2) and Lemma 3, we haveThus, is the positive solution of problem (2).☐

Theorem 9. We suppose that condition (C1), (C2), and (C4) hold, and the following condition also holds:(C6)There exist constant , and we have

Then, for the small number , problem (2) has at least one positive solution.

Proof. Firstly, let , andSet , for any , we haveThus. Therefore, by (C6) and the definition of operator , we haveFor , we haveWe can know that by the above discussion, we haveSecondly, by (C4), we can haveThen, there exists constant which satisfiesLetting , then . Set , for any , we haveThus, we haveSo, we haveThen, we can have by the definition of ☐

Then, similar to the proof of heorem 8, we have that result of heorem 9 by Theorem 1.

Theorem 10. Suppose condition (C1), condition (C2), and condition (C5) hold. Then, for sufficiently small , problem (2) has at least two positive solutions.

Proof. Firstly, by Lemma 7, there exists constant which satisfiesTherefore, setting , for any , by the above discussion, for the quite small , we haveWe haveThen, we haveSecondly, by (C5), there exists a constant , which satisfiesLetting , and , for any , we haveThen, .
Therefore, by the definitions of and the above discussion, we haveThus, we haveFinally, lettingthen, . Set , for any , by the definition of operator , we haveThus, we haveThen, similar to the proof of heorem 8, we have the result of heorem 10 by Theorem 2.☐

Remark 11. The results of these three theorems in our paper also hold under the condition in which nonlinear are both lower semicontinuous.

Remark 12. We can obtain the results of Theorem 10 if we replace condition (C5) with (C6), .

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

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

Authors’ Contributions

All authors contributed equally to the manuscript, and all authors typed, read, and approved the final manuscript.

Acknowledgments

The authors were supported by the Project of Shandong Province Higher Educational Science and Technology Program (J16LI01).