Abstract
In our paper, we consider the positive solutions of the nonlinear -order -point semipositive BVP. In this BVP equation, we allow that can change the symbol for ; by using the fixed point index theory, the existence of positive solutions and many positive solutions are obtained under the condition that is superlinear or sublinear.
1. Introduction
In our paper, we study the nonlinear -order -point semipositive problems: with the following boundary value conditions: where , , ; are constants, .
For the differential equations, I offer wonderful tools for describing various natural phenomena arising from natural sciences, for example, [1–16]. Very few authors discussed cases (1) and (2). In this paper, under the condition that is superlinear or sublinear, we focus on the existence of positive solutions for the nonlinear -order -point semipositive BVP (1) and (2) under the conditions that is continuous.
2. Preliminaries and Lemmas
Let is a Banach space and where . And with norm .
Throughout this paper, we shall use the following notation: where .
It is well known from papers [6, 7] that is a nonnegative continuous function, and is the Green’s function of the BVP:
Let
It is obvious that for and by the properties of Euler integral, we have and . Surely, for , and for , we have
Suppose the following conditions hold:
(): .
For , let
is the Green’s function of the BVP:
By direct computation, we know that.
Lemma 1 (see [6]). defined as above have the following properties: where By Lemma 1 and (7) and (8), it is obvious that where ,
Lemma 2. Suppose satisfies the following problem: where . Then,
Proof. By , we know that so, Therefore,
Lemma 3. Suppose satisfies the following problem: where . Then, for any , there exists constant such that
Proof. Let , and then by Lemma 2, we can obtain the results.
Lemma 4. Suppose satisfies the following problem: where . Then, there exists constant such that
Proof. For , we can have Obviously, for , By the same method, we can get that So, we can choose the constant And we have This completes the proof of Lemma 4.
In the rest of the paper, we also make the following assumptions:
() , and where ,, is constant in Lemma 4, and is the function in Lemma 1.
We denote a cone : where .
Set
By Lemma 4, we set , and for ,
Then, is solution of BVP (1) if and only if is the positive solution of the following BVP()
Clearly, BVP() is equivalent to the equation
i.e.,the fixed point problem with operator given by
3. The Existence of Single Positive Solution
In this section, we present our main results by fixed point index theory.
Theorem 5. If conditions () and () hold. For , and also satisfies
() For , there has
() For , there has
where
Then, the higher-order nonlinear -point semipositive BVP (1) and (2) has at least one solution such that lies between and .
Theorem 6. If conditions () and () hold. And also satisfies
()
() where
Then, the higher-order nonlinear -point semipositive boundary value problem (1) and (2) have a solution such that lies between and .
Proof of Theorem 5. Firstly, let and of :
Then, for ,
so, for ,
And then by (), for ,
Therefore, we know that
Another, for , we know that
and then by (),
Therefore,
Then, we know that
Therefore, by (40) and (44), ,
Then, operator has a fixed point and .
Finally, using Lemmas 3 and 4, we know
i.e., is the solution of BVP (1) and (2). This completes the proof of Theorem 5.
Proof of Theorem 6. Copying Theorem 5. First, by , for , there exists the number , as , we know that
So, for , thus by (47), we know that
So, condition () holds.
Next, using (), , then for , there exists a number , as , we know that
Let , thus, by (49), () holds. Then, we have that the results of Theorem 6 holds.
4. The Existence of Many Positive Solutions
Next, we will discuss the existence of many positive solutions.
Theorem 7. If (), (), and hold. And also satisfies the following conditions:
()
() where
Then, the semipositive boundary value problems (1) and (2) have at least two solutions such that
Theorem 8. If (), (), and hold. And also satisfies the following conditions:
()
() where
Then, the semipositive boundary value problems (1) and (2) have at least two solutions such that .
Proof of Theorem 7. Copying Theorem 5. First, for , From (), there exists a constant which satisfy
Let . Then, for and , we know that
so, for , we know that
Therefore, from (52), we could obtain the following:
Then,
Next, using condition (), for any , there exists a constant which satisfies
We can choose a constant which satisfies .
Set . Then,
where and . So, for , we have
And then for , we can easily obtain that
Therefore,
Finally, letting , for any , by , Lemma 2, we can also easy to obtain that
Then,
Therefore, by (57)–(65) and , we could obtain the following results:
Then, have fixed point , and fixed point and .
Finally, using Lemmas 3 and 4, for , we have
i.e., are the positive solutions of (1) and (2).
Proof of Theorem 8. Using the proof of Theorem 5. We only need to discuss the operator which is given as (33).
First, by , for , there exists a constant which satisfy
Set , then for , and
and then by (), we have
So, we have
Then, by ((1)), we have
Next, letting
It is easy to know that is monotone increasing for .
Thus, by , and Therefore, for any , there exists such
Set , then for any ,
i.e., . Then, by ((1)), we have
Next, similar to Theorem 5, we set , and for any , by Lemma 2, condition , we can also know that
Then,
Therefore,
Then, have fixed point , and fixed point and .
Finally, using Lemmas 3 and 4,
here . Then, are the solutions of BVP (1) and (2). The proof of Theorem 8 is complete.
5. Application
Example 1. Let , we consider the following semipositive BVP for : with the following boundary value conditions:
Clearly,
By direct calculating, we have
Therefore, using Lemma 4, let such that . Then, holds.
By directly calculating, we can be easy to know that . So, conditions and hold. Then, let such that . Then by Theorem 6, we have Example 1 has at least one positive solution and .
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
Disclosure
The preprint of this manuscript can be found in the following: https://assets.researchsquare.com/files/rs-2387096/v1_covered.pdf?c=1671766530.
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 typed, read, and approved the final manuscript.
Acknowledgments
The author was supported by the Natural Foundation of Shandong Province (ZR2021MA038).