1. Introduction and Main Results
In this paper, we consider the three-point boundary value problem
(1.1)
where , , , and is a positive parameter.
The m-point boundary value problem for linear second-order ordinary differential equations was initiated by Ilin and Moiseev [1] [2]. Since then, there are many results on the existence of general nonlinear multi-point boundary value problems, see [3] [4] [5] [6] and their references. For examples, in [6], Rynne studied the $m$-point boundary value problem
where , , with and presented the existence of the sign changing solutions by Rabinowitz bifurcation theorem. Especially, Rynne ([7]) discussed the three-point boundary value problem
and showed the solvability and non-solvability results from either the half-eigenvalue or the Fucik spectrum approach. As we known, the method of upper and lower solutions is very important for the study of the boundary value problems, see [8]-[18]. Therefore, establishing the method of upper and lower solutions for three-point boundary value problems is necessary and important.
In [19], when f is nondecreasing on x, Du and Zhao got the methods of upper and lower solutions of
and used iterative techniques to study the existence of positive solutions. And in [3] when f is decreasing on u, Du and Zhao considered the existence and uniqueness of positive solutions of the problem
by constructing lower and upper solutions. Wei ([15]) constructed the method of upper and lower solutions for three-point boundary value problems and gave the sufficient and necessary conditions for the existence of positive solutions of the problem
On the other hand, singular boundary problems arise in the contexts of chemical heterogeneous catalysts, non-Newtonian fluids and also the theory of heat conduction in electrically conducting materials, see [20]-[25] for a detailed discussion. An interesting result comes from [25], in which, using method of upper and lower solutions, Shi and Yao discussed the following problem
where , and is a positive parameter. Under various appropriate assumptions on , Shi and Yao obtained the existence and uniqueness of classical solutions.
Motivated by above works, under various appropriate assumptions on p, q and , we will obtain the existence and uniqueness of positive solution of problem (1.1) for in different circumstances. In our proof, the upper and lower solutions theorem (see [16]) plays an important role in the paper.
Define
The main results of this paper are stated in the following theorems.
Theorem 1.1. When ,
1) If , there exists such that the problem (1.1) has at least one C[0,1] positive solution for .
2) For , (1.1) has a maximal solution and is increasing with respect to .
Theorem 1.2. When ,
1) If , (1.1) has at least one C[0,1] positive solution for all .
2) If , (1.1) has an unique positive solution for all .
3) in (2) is increasing with respect to .
Theorem 1.3. When ,
1) If , there exists a such that the problem (1.1) has at least one C[0,1] posit- ive solution for .
2) For , in (1) is increasing with respect to .
Remark 1.1: Note in Theorem (1.1). This is different from the conditions in [3] [15] [19] because in these references.
Remark 1.2: The unique result in Theorem 1.2 is different from that in [3] because we remove the monotonicity of nonlinearity f in x.
Remark 1.3: Note is sigh-changing in Theorem 1.3. This is different from the conditions in [3] [15] [19] because in these references and is different from conditions in [1] [2] [4] [5] [6] [7] [26] because f is continuous at in these references.
This paper is organised as follows. Some preliminary lemmas are stated and proved in Section 2. And Section 3 is devoted to prove the results.
2. Preliminaries
In this section, we first consider the following problem
(2.1)
where , and .
Let is differential continuous on with norm
,
where . Obviously, is a Banach space. Now we give the definitions of lower and upper solutions for problem (2.1).
Definition 2.1. A function is called a lower solution to the problem (2.1), if and satisfies
(2.2)
Upper solution is defined by reversing the above inequality signs in problem (2.2).
If there exists a lower solution and an upper solution to problem (2.1) such that , then is called a couple of upper and lower solutions of problem (2.1).
Set
We list a lemma for the eigenvalues and eigenfunctions for the following linear problem
(2.3)
Lemma 2.1. (see [6]) The spectrum of problem (2.3) consists of a strictly increasing sequence ofeigenvalues , , with eigenfuctions . In addition,
1) ;
2) has exact simple zeros in , and is strictly positive on .
Lemma 2.2. Suppose that . Then, for each , the problem
(2.4)
has an unique solution in C[0,1].
Proof. Assume that and satisfies that
and
respectively. Define
and
Then
and
Hence, is a C[0,1] solution to problem(2.4). Since , Lemma 2.1 guarantees that problem (2.4) has an unique C[0,1] solution. The proof is complete.
Theorem 2.1. Let and be lower and upper solutions of (2.1) such that . Let and be a continuous function that satisfies
(2.5)
Suppose is an -Carathéodory-function such that
(2.6)
Then the problem (2.1) has at least one solution such that for all ,
Proof. The proof proceeds in five steps.
Step 1. We consider a new modified problem. From (2.5), there is an be large enough so that
(2.7)
And (2.6) guarantees that there is an with such that
(2.8)
Define then
(2.9)
and
(2.10)
Choose a and consider the new boundary value problem
(2.11)
where , .
Step 2. We discuss the existence of a solution of (2.11).
Now Lemma 2.2 guarantees that for each , the linear problem
has an unique C[0,1] solution
For , define
and
From (2.9) and (2.10), we have
which implies that the functions belonging to and are bounded and equicontinuous. The Arzela-Ascoli Theorem guarantees that is relatively compact. The proof of the continuity of T is standard. Using the Schauder’s fixed point theorem, we assert that T has at least one fixed point .
Step 3. The solution x of (2.11) is such that .
We prove that for only. In fact, suppose that there exist a such that . Since , . Let , . Then and .
Let ,
It is obvious that for all , and . If , then there exists a such that . If , obviously and . Since
, there exists
such that also. Hence, (i.e., ) and . On the other hand, since
This is a contradiction.
A similar argument holds to prove for all .
Hence, from (2.10), one know that x satisfies that
(2.12)
Step 4. The solution x of (2.11) is such that .
On the contrary, suppose that there is a such that . Without loss of generality, we assume that . Since and with , there is a such that . Without loss of generality, we assume that for all . Observe that, for all , ,
Then, from (2.12), one has
This contradicts to (2.7).
Hence , which together with guarantees that
Step 5. We claim that satisfies (2.1).
Since and , by (2.8), (2.10) and (2.12), we have
that is, is a solution of (2.1). The proof is complete.
Now we consider the following problem
(2.13)
where , and .
Now we give the definitions of lower and upper solutions for problem (2.13).
Definition 2.2. (see [16]) A function is called a lower solution to the problem (2.13), if and satisfies
(2.14)
Upper solution is defined by reversing the above inequality signs in problem (2.14).
By Theorem 2.1, we have following result.
Corollary 2.1. Suppose that there exists a lower solution and an upper solution of problem (2.1) such that , and there exists such that for all . Then the problem (2.13) has at least one C[0,1] solution satisfies , .
Remark 2.1: This result can be found in [15]. So our theorem improves the works in the previous literature.
Lemma 2.3. Suppose that is a continuous functions such that is strictly decreasing for at each . Let satisfies:
1) , ;
2) , and , , ;
3) .
Then , .
Proof. By , we know that and exist and then .
Suppose conversely on [0,1]. We may assume without loss of generality that there exists such that . Let
It’s obvious that and , , where denote Dini derivatives.
For , there are three cases.
1) . Then , , for all .
2) and , , for all , where denotes Dini derivatives.
3) and , for all . Since , then there is such that
Combining above (1), (2) and (3), there is a such that
and
Let , . Then we have
(2.15)
On the other hand,
for and on . This implies . This contradicts (2.15), so . The proof is complete.
By analogous methods in [19], we establish the following maximal theorem, which can be used in the proof of the uniqueness of positive solutions.
Lemma 2.4. (maximal theorem) Suppose that , and , if such that for , then for .
3. Proofs of Main Theorems
In this section, we’ll always assume that .
(A) The proof of Theorem 1.1.
Proof.
1) We consider the problem
(3.1.1)
where , , , , and is a positive parameter.
In [19], when is increasing in x, the problem
has an unique positive solution. From that, suppose that is an unique positive solution of the problem
(3.1.2)
where , .
Set . Then
Thus Combining it with (3.1.2) we obtain
Consequently, is a upper solution of (3.1.1).
Set , where M is a positive constant and is the first eigenfunction. Then
By Lemma 2.1 we have , . Thus there exists and such that
a) On , choosing , then we have
.
b) On , choosing , then we have
Fixing , then
and
Set . Then we have
Hence, , .
It follows from Lemma (2.1) that
and
Set . Then for all . Thus we choose and , then is a couple of upper and lower solutions of (3.1.1).
We choose , then for all . It’s easy to see that . From Corollary 2.1, the problem (3.1.1) has at least one C[0,1] positive solution satisfying for .
2) (Existence of the maximal solution) We observe the problem
(3.1.3)
From [19], we note the unique solution of (3.1.3) is for any . In (1) we obtained the solution of (3.1.1) then we have
and is decreasing in x. Noting that by (1). From Lemma 2.3, we have .
Let , and be the solution of
(3.1.4)
for , with defined in (3.1.3). Let be a solution of (3.1.1).
In (3.1.4), letting we have
(3.1.5)
Combining (3.1.5) with (3.1.3) we have for . By maximum principle, we have . Similarly, we can obtain that .
Furthermore, we observe problem (3.1.1)
Combining it with (3.1.5) we have
thus for . It’s easy to verify that for by maximum principle. By similar method we can obtain for .
Furthermore, we have is bounded from below by .
Because is a solution to (3.1.3),
Suppose that
,
, then
and
is increasing on
. By integration of
from t to
, we have
.
So . Similarly, by integration of from to t, we can obtain . For giving , we have
We can find K large such that . Then
, (3.1.4)
We define an operator , then . It follows from (3.1.4) that is a uniformly bounded and equicontinuous functions in [0,1]. Obviously, is uniformly continuous in a bounded and closed domain , i.e., for all , there exists a such that when , , , we have . Since , there exists a such that . From (3.1.4), for the above , there exists such that when , we have .
Therefore, for all , there exists such that when , we have
.
Thus is equicontinuous. Using Arzela-Ascoli theorem, there exists a subsequence such that . Without loss of generality, we assume that
(3.1.5)
In the following, we shall show that is a C[0,1] positive solution of (3.1.1).
Fixing , then can be stated
(3.1.6)
Fixing , by Lagrange mean value theorem, there exists such that .
So there exists such that . Since is bounded in [0,1], we may assume that , ,
Thus
i.e.,
Thus both and are bounded. Then they all have a convergence subsequence. Without loss of generality, we note the subsequences are and . And fixing , we assume .
In equation (3.1.6), letting we have
for , i.e., . Therefore is a C[0,1] positive solution of (3.1.1). Therefore is the maximal solution of (3.1.1).
Next we shall verify the dependence on of maximal solution .
Let H = { : (3.1.1) has a C[0,1] positive solution with }.
Obviously, by (1), . Let . and be the corresponding maximal solution of (3.1.1) for . Then for any , . By Lemma (2.3), in [0,1]. Just replacing by in above proof. We can easily find that
Combining it with boundary conditions, we can obtain that is a couple of lower and upper solutions of (3.1.1) for . One can be prove that there is a solution of (3.1.1) with such that
Therefore . Moreover, by (ii), for any , .
This completes the proof of Theorem 1.1.
(B) The proof of Theorem 1.2.
Proof. 1) We consider the problem
(3.2.1)
where , , , , and is a positive parameter.
Now we consider an approximate problem of (3.2.1) as follows
(3.2.2)
where , , .
Let very small. We’ll verify that is a lower solution of (3.2.2). Indeed, when n is big enough, we can obtain that is close to 0. Since (see [6]), we can deduce
and , which imply that is a lower solutions of (3.2.2).
In the following, we’ll construct an upper solution of (3.2.2). Let
,
where M is big enough for . We can obtain
and . It’s easy to see that is na upper solution of (3.2.2).
Choosing , then , for all . It’s easy to verify that . Because that is small and n is big enough, . From Corollary 2.1, is a couple of upper and lower solutions of (3.2.2). And for all , (3.2.2) has at least one C[0,1] positive solution such that .
In the following, we shall obtain a result as follows, there exists a subsequence and such that .
Since , is bounded. Therefore is a uniformly bounded sequence of functions in [0,1]. Because is a C[0,1] positive solution of (3.2.2), satisfies
Suppose that , , then and is increasing on . By integration of from t to , we have
So . We can find a such that . And by integration of from to t, we have
So . For above K, we have , i.e., .
For giving , we have
Then . The above inequality can be rewritten as
(3.2.3)
We now define an operator , then . It follows from (3.2.3) that is a uniformly bounded and equicontinuous functions in [0,1]. Obviously, is uniformly continuous in a bounded and closed domain , i.e., for all , there exists a such that for , . Since , there exists a such that . From (3.2.3), for the above , there exists such that for .
Therefore, for all , there exists such that
for . Consequently, is equicontinuous. Using Arzela-Ascoli theorem, there exists a subsequence such that . Without loss of generality, we assume that
(3.2.4)
In the following, we shall show that is a C[0,1] positive solution of (3.2.1). Fixing , can be stated
(3.2.5)
Fixing , by Lagrange mean value theorem, there exists such that .
So there exists such that . Since is bounded in [0,1], we may assume that , .
We can obtain
and .
Therefore both and are bounded. They all have a convergence subsequence. Without loss of generality, we note the subsequences are and . And fixing , we assume .
From (3.2.5), letting , we obtain
By derivation twice of , we have
Combining it with (3.2.4), we can obtain that is a C[0,1] positive solution of (3.2.1).
2) We study the uniqueness of positive solution of problem (3.2.1).
Let . Obviously, when , is integrable over (0,1). Since , is absolutely integrable over (0,1). Then both and exist, i.e., .
Suppose conversely that , are two positive solutions of the problem (3.2.1), on [0,1]. We may assume without loss of generality that there exists such that . Let
It’s obvious that and
Let . Then we have
(3.2.6)
On the other hand,
for and on . This implies , contradicts (3.2.6), so . Thus the positive solution of (3.2.6) is unique.
3) We assume that and , are the corresponding unique positive solutions to (3.2.1). Obviously, . In (3.2.1), is continuous.
Since , , it’s easy to see that is decreasing for at each .
for , , and . Therefore, by Lemma 2.3,
So is increasing with respect to .
This completes the proof of Theorem 1.2.
(C) The proof of Theorem 1.3.
Proof.
1) We consider the problem
(3.3.1)
where , , , , and is a positive parameter.
Since , then by Theorem 1.1, there exists a , such that for , the problem
has a maximal solution . Let . We observe that
and
Consequently, is a lower solution of :
On the other hand, the problem
has a solution for any . Then
So we have
Therefore, is an upper solution of . Since , , , , and is decreasing in x, by Lemma 2.3,
Obviously, there exists a minimal solution of , satisfying . Similarly, taking and as a couple of lower and upper solutions for , we conclude that there exists a minimal solution of such that
.
Repeating the above arguments, we obtain a sequence which is decreasing in k. Therefore, similar to the proof of Theorem 1.2 (1), we obtain a solution , and .
2) (Dependence on ) Let , and be the corresponding solutions of (3.3.1) for and which we obtained in (1). We observe that
is an upper solution of , and
Therefore , since is a minimal solution of which satisfies . Therefore we must have .
Thus Theorem 1.3 is true.
Funding
This work is supported by the National Natural Science Foundation of China (61603226) and the Fund of Natural Science of Shandong Province (ZR2018MA022).
Availability of Data and Materials
Not applicable.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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