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.
Keywords:integral boundary value problem; resonance; fixed point theorem; intermediate value theorem
We are interested in the existence of the solutions for the following second-order three-point integral boundary value problems at resonance:
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].
In , Infante and Zima studied the existence of solutions for the following n-point boundary value problem with resonance:
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.
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 2.2Problem (1.1)-(1.2) is equivalent to the following integral equation:
Combining (2.3) with (2.4), we have
According to (2.5) it is easy to see that (2.1) holds.
Thus, problem (1.1)-(1.2) is equivalent to the following integral equation:
Now we let
Our results are the following theorems.
Theorem 2.1Assume that (H) holds. If
then problem (1.1)-(1.2) has at least one solution, where
We define an operator T on the set Ω as follows:
Proof It is not difficult to check that T maps Ω into itself. Next, we divide the proof into three steps.
Suppose that is a sequence in Ω, and converges to . Because of being continuous with respect to and from Lemma 2.4, it is obvious that is uniformly continuous with respect to . Then, for any positive number ε, there exists an integer N. When , we have
It follows from (2.21) and (2.22) that
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.
Because of Step 1 to Step 3, it follows that the operator T is completely continuous in Ω. The proof is completed. □
Proof We only need to present that the operator T is a priori bounded. Set
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.
Here we use the inequality
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.
We rewrite (2.16) for any given real number μ as follows:
From (3.1), it suffices to show that there exists μ such that
Firstly, we prove that . On the contrary, we suppose that . Then there exists a sequence with such that , which implies that the sequence is bounded. Notice that the function is continuous with respect to and . So, it is impossible to have
Thus we get that
Since we have from (H) that
by (3.2), (3.5) and (3.6), we have
which contradicts our assumption.
from which it follows that
which implies 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.
Example Consider the boundary value problem
So, we have
Now we take
It is easy to check that
Thus the conditions of Theorem 2.1 are satisfied. Therefore problem (4.1)-(4.2) has at least a nontrivial solution.
The authors declare that they have no competing interest.
Each of the authors HL and ZO contributes to each part of this study equally and read and approved the final vision of the manuscript.
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).
Anderson, D: Multiple positive solutions for a three-point boundary value problem. Math. Comput. Model.. 27, 49–57 (1998). Publisher Full Text
Webb, JRL, Lan, KQ: Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal.. 27(1), 91–115 (2006)
Sun, JP, Li, WT, Zhao, YH: Three positive solutions of a nonlinear three-point boundary value problem. J. Math. Anal. Appl.. 288, 708–716 (2003). Publisher Full Text
Kwong, MK, Wong, JSW: Solvability of second-order nonlinear three-point boundary value problems. Nonlinear Anal.. 73, 2343–2352 (2010). Publisher Full Text
Ma, R: Positive solutions for second-order three-point boundary value problems. Appl. Math. Lett.. 14, 1–5 (2001). Publisher Full Text
Han, XL: Positive solutions of a nonlinear three-point boundary value problem at resonance. J. Math. Anal. Appl.. 336, 556–568 (2007). Publisher Full Text
Sun, YP: Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem. Nonlinear Anal.. 66, 1051–1063 (2007). Publisher Full Text
Liu, B: Solvability of multi-point boundary value problems at resonance - part IV. Appl. Math. Comput.. 143, 275–299 (2003). Publisher Full Text
Przeradzki, B, Stauczy, R: Solvability of a multi-point boundary value problem at resonance. J. Math. Anal. Appl.. 264(2), 253–261 (2001). Publisher Full Text
Kosmatov, N: Multi-point boundary value problems on an unbounded domain at resonance. Nonlinear Anal.. 68(8), 2158–2171 (2008). Publisher Full Text
Ma, R: Multiplicity results for a three-point boundary value problem at resonance. Nonlinear Anal.. 53(6), 777–789 (2003). Publisher Full Text
Han, X: Positive solutions for a three point boundary value problem at resonance. J. Math. Anal. Appl.. 336, 556–568 (2007). Publisher Full Text
Infante, G, Zima, M: Positive solutions of multi-point boundary value problems at resonance. Nonlinear Anal.. 69, 2458–2465 (2008). Publisher Full Text
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