Existence of solutions for second-order three-point integral boundary value problems at resonance
© Liu and Ouyang; licensee Springer 2013
Received: 30 May 2013
Accepted: 13 August 2013
Published: 3 September 2013
A class of second-order three-point integral boundary value problems at resonance is investigated in this paper. Using intermediate value theorems, we obtain a sufficient condition for the existence of the solution for the equation. An example is given to demonstrate our main results.
MSC:34B10, 34B16, 34B18.
Keywordsintegral boundary value problem resonance fixed point theorem intermediate value theorem
where , and .
In the last few decades, many authors have studied the multi-point boundary value problems for linear and nonlinear ordinary differential equations by using various methods, such as Leray-Schauder fixed point theorem, coincidence degree theory, Krasnosel’skii fixed point theorem, the shooting method and Leggett-Williams fixed point theorem. We refer the readers to [1–10] and references therein. Also, there are a lot of papers dealing with the resonant case for multi-point boundary value problems, see [11–17].
where and . Using the Leggett-Williams norm-type theorem, they obtained the existence of a positive solution for problem (1.3)-(1.4).
Problem (1.1)-(1.2) with and was studied by Tariboon and Sitthiwirattham in . They obtained the existence of at least one positive solution. In this paper, we are interested in the existence of the solution for problem (1.1)-(1.2) under the condition , which is a resonant case.
In this paper, using some properties of the Green function and intermediate value theorems, we establish a sufficient condition for the existence of positive solutions of problem (1.1)-(1.2).
The rest of the paper is organized as follows. The main results for problem (1.1)-(1.2) under the condition are given in Section 2. In Section 3, we give some lemmas for our results. We prove our main result in Section 4, and finally an example is given to illustrate our result.
2 Some lemmas and main results
In this section, we first introduce some lemmas which will be useful in the proof of our main results.
then Ω is a Banach space.
Lemma 2.1 
T has a fixed point in , or
there exist and with .
According to (2.5) it is easy to see that (2.1) holds.
On the other hand, if is a solution of equation (2.1), deriving both sides of (2.1) two order, it is easy to show that is also a solution of problem (1.1)-(1.2).
Therefore, problem (1.1)-(1.2) is equivalent to the integral equation (2.1) with the function defined in (2.2). The proof is completed. □
Lemma 2.3 For any , is continuous, and for any .
for . The proof is completed. □
By a simple computation, the new Green function has the following properties.
- (H)and there exist two positive continuous functions such that(2.17)
for any .
Our results are the following theorems.
Lemma 2.6 Assume that and (2.19) hold. Then the operator T is completely continuous in Ω.
Proof It is not difficult to check that T maps Ω into itself. Next, we divide the proof into three steps.
Step 1. is continuous with respect to .
Thus the operator T is continuous in Ω.
Step 2. T maps a bounded set in Ω into a bounded set.
This implies that the operator T maps a bounded set into a bounded set in Ω.
Step 3. T is equicontinuous in Ω.
It suffices to show that for any and any , as . There are the following three possible cases:
Case (i) ;
Case (ii) ;
Case (iii) .
Because of Step 1 to Step 3, it follows that the operator T is completely continuous in Ω. The proof is completed. □
Lemma 2.7 Assume that and (2.17) and (2.19) hold. Then the integral equation (2.16) has at least one solution for any real number μ.
To use Lemma 2.1 to prove the existence of a fixed point of the operator T, we need to show that the second possibility of Lemma 2.1 should not happen.
Obviously, (2.25) contradicts our assumption that . Therefore, by Lemma 2.1, it follows that T has a fixed point . Hence, the integral equation (2.21) has at least a solution . The proof is completed. □
3 The proof of Theorem 2.1
In this section, we prove Theorem 2.1 by using Lemmas 2.5-2.7 and the intermediate value theorem.
Proof of Theorem 2.1 From the right-hand side of (2.21), we know that (2.21) is continuously dependent on the parameter μ. So, we just need to find μ such that , which implies that .
Obviously, is continuously dependent on the parameter μ. Our aim here is to prove that there exists such that , we only need to prove that and .
which contradicts our assumption.
Then is not empty.
Obviously, we get that , . So, we have from (H) that is not empty.
This contradicts (3.9). Thus, we have proved that . By a similar method, we can also prove that .
Notice that is continuous with respect to . It follows from the intermediate value theorem  that there exists such that , that is, , which satisfies the second boundary value condition of (1.2). The proof is completed. □
In this section, we give an example to illustrate our main result.
Thus the conditions of Theorem 2.1 are satisfied. Therefore problem (4.1)-(4.2) has at least a nontrivial solution.
The work was partially supported by the Natural Science Foundation of Hunan Province (No. 13JJ3074), the Foundation of Science and Technology of Hengyang city (No. J1) and the Scientific Research Foundation for Returned Scholars of University of South China (No. 2012XQD43).
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