Abstract
In this paper, we focus on the following secondorder multipoint integral boundaryvalue problem:
where
MSC: 34B10, 34B15, 34B18.
Keywords:
shooting method; integral boundaryvalue problem; positive solution1 Introduction
For the study of nonlinear secondorder multipoint boundaryvalue problem, many results have been obtained by using all kinds of fixed point theorems related to a completely continuous map defined in a Banach space. We refer the reader to [19] and the references therein. Some of the results are so classical that little work can exceed; however, most of these papers are concerned with problems with boundary conditions of restrictions either on the slope of solutions and the solutions themselves, or on the number of boundary points [2,58,10].
In [8], Ma investigated the existence of positive solutions of the nonlinear secondorder mpoint boundary value problem
where
Set
The author obtained the existence of a positive solution to (1.1)(1.2) under the
case
Recently, Tariboon [9] considered threepoint boundaryvalue problem (1.1) with the integral boundary condition
where
Such a boundary condition might be more realistic in the mathematical models of thermal
conductivity, groundwater flow, thermoelectric flexibility and plasma physics, because
it describes the fluid properties in a certain continuous medium. Under the assumption
that
However, the method used in the previous two papers is Krasnoselskii’s fixed point theorem in a cone, which relates to constructing a completely continuous cone map in a Banach space, and the proof is somewhat procedural.
Constructively, Agarwal [11] explored the solution of multipoint boundary value problems by converting BVPs to equivalent IVPs, which is called shooting method. After that Man Kam Kwong [4,12] used the shooting method to consider secondorder multipoint boundary value problems. In [12], Kwong studied the existence of a positive solution to the following threepoint boundary value problem:
The principle of the shooting method used in [12] is converting BVP (1.4)(1.5) into finding suitable initial slopes
vanishes for the first time after
If we can find two solutions
and
where
In this paper, we try to employ the shooting method to establish the existence results of positive solutions for (1.1) with the more generalized multipoint integral boundary condition
where
The purpose of this article lies in two aspects. One is to explore the application
of the shooting method in a more complicated multipoint integral boundary value problem,
which demonstrates another way in studying BVPs. The other one is to establish new
criteria for the existence of positive solutions to (1.1)(1.7) under the case
For the sake of convenience, we denote
Let
In this paper, we always assume:
(H_{1})
Under the assumption, it is not difficult to prove that the initial problem (1.1)(1.6)
has at least one solution defined on
Further, we introduce the comparison results derived from [4,12], which evolved from the Sturm comparison theorem.
Theorem 1.1Let
and suppose thatF, G, gare nonnegative continuous functions on a certain intervalIfor
If
The paper is arranged as follows. In the next section, we put forward the basic principle
of the shooting method used in this paper, and show that BVP (1.1)(1.7) has no positive
solution when
2 Preliminaries
Lemma 2.1If there exist two initial slopes
(i) the solution
(ii) the solution
Proof Since the solutions of (1.1)(1.6) depend on the initial value continuously, then
from (1.8), it implies that
Therefore,
Lemma 2.2Let
Proof Assume that (1.1)(1.7) has a positive solution u.
If
which contradicts with the convexity of u.
If
In the rest of this paper, we always assume:
(H_{2})
□
3 Main results
Theorem 3.1Assume that (H_{1})(H_{2}) holds. Suppose
(i)
(ii)
Then problem (1.1)(1.7) has a positive solution.
Proof (i) Since
We claim that there exists a positive number
Let
then
From (1.8), (3.1) and combining the integral mean value theorem with Theorem 1.1, we have
where
The second inequality in (i) means that there exists a number M large enough such that
For this M, there exist two numbers δ and
and there exists another number
In view of (H_{2}) and (3.3), it is not difficult to verify that
which implies from (3.4) that
By Lemma 2.1 and (3.2)(3.5), there exists a number
Now, we prove for (ii).
In view of
For this N, there exist a number ϵ small enough and a number
Obviously,
Let
where
Since
By the convexity of
which yields
Let
and
From (3.6) and (3.7), we have
By Lemma 2.1, the proof for (ii) is complete. □
Theorem 3.2Assume that (H_{1})(H_{2}) holds. Suppose
Then problem (1.1)(1.7) has a positive solution under the case
(i)
(ii)
Proof Note the computation of
then
Now, let us consider the special superlinear case or the sublinear case. It is not difficult to verify the following corollaries.
Corollary 3.1Assume that
(i)
(ii)
Then problem (1.1)(1.7) has a positive solution.
Corollary 3.2If
Then, problem (1.1)(1.7) has a positive solution under the case
(i)
(ii)
4 Conclusion and examples
The tool which we used for the analysis in this article is the shooting method derived
from [4,12]; however, we considered a more general problem which involves integral boundaryvalue
and multiplicity of boundarypoint. The meaningful work that we have done lies in
the following three aspects. The first one is that we transform the integral problem
into a singlepoint value one by using the integral mean value theorem. The other
one is that we construct a quadratic function and a sine function as the comparison
functions because it does not take effect to construct two sine functions as in [12]. Finally, we established the new criteria for the existence of positive solutions
to (1.1)(1.7) under the case
Example 4.1 Consider the BVP
where
It is not difficult to see that
In view of
Therefore, the condition (ii) of Theorem 3.2 is satisfied. A numerical simulation
(Figure 1) for Example 4.1 demonstrates that BVP (4.1)(4.2) has a positive solution
Figure 1. Numerical simulation for Example 4.1.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The work was carried out in collaboration between all authors. HL practised the methods and organized this paper. ZG found the topic of this paper and suggested the methods. LG finished the Matlab program of numerical simulation. All authors have contributed to, seen and approved the manuscript.
Acknowledgements
The authors would like to thank the editors and the anonymous referees for their valuable suggestions on the improvement of this paper. First author was partially supported by the Scientific Research Fund of Hunan Provincial Educational Department (1200361), Project of Science and Technology Bureau of Hengyang, Hunan Province (2012KJ2). Second author was partially supported by the Doctor Foundation of University of South China ( No. 5XQD20069), the Foundation of Science and Technology Department of Hunan Province (No. 2009RS3019), the Natural Science Foundation of Hunan Province (No. 13JJ3074) and the Subject Lead Foundation of University of South China (No. 2007XQD13).
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