In this paper, we focus on the following second-order multi-point integral boundary-value problem:
where , for and are given constants. The proof is based on the shooting method. By constructing a quadratic function and a sine function as the shooting objects and combining the integral mean value theorem with the comparison principle, we consider the existence of positive solutions to the BVP respectively under the case and the case . The method is concise and some new criteria are established.
MSC: 34B10, 34B15, 34B18.
Keywords:shooting method; integral boundary-value problem; positive solution
For the study of nonlinear second-order multi-point boundary-value 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 [1-9] 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 [2,5-8,10].
In , Ma investigated the existence of positive solutions of the nonlinear second-order m-point boundary value problem
where , for , , , , and there exists a such that .
The author obtained the existence of a positive solution to (1.1)-(1.2) under the case and (super-linear case) or the case and (sub-linear case) when .
Recently, Tariboon  considered three-point boundary-value 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 , Tariboon and the author proved that problem (1.1)-(1.3) has at least one positive solution in the super-linear case or in the sub-linear one.
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  explored the solution of multi-point 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 second-order multi-point boundary value problems. In , Kwong studied the existence of a positive solution to the following three-point boundary value problem:
The principle of the shooting method used in  is converting BVP (1.4)-(1.5) into finding suitable initial slopes such that the solution of equation (1.4) with the initial value condition
vanishes for the first time after . Denote by the solution of (1.4)-(1.6) provided it exists. Then solving the boundary value problem is equivalent to finding m such that
If we can find two solutions and of (1.4) such that
where , for , then there must exist a number m between and such that is the solution of (1.4)-(1.5). By constructing two sine functions as the shooting objects and combining with the comparison principle, the author obtained some better results than those via fixed point techniques for the existence of positive solutions to (1.4)-(1.5).
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 multi-point integral boundary condition
where , for and are given constants. Following the principle of the shooting method, there are two obstacles we encounter. The first one is that the boundary condition involves integral from 0 to ( ), so we transform the integral problem into a single-point problem by using the integral mean value theorem. The other difficulty is that we cannot obtain the existence results by constructing two sine functions as in  because of the particularity of in . Therefore, we construct a quadratic function and a sine function as the objective ones.
The purpose of this article lies in two aspects. One is to explore the application of the shooting method in a more complicated multi-point 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 and the case .
For the sake of convenience, we denote
Let be the solution of (1.1)-(1.6) and define
In this paper, we always assume:
(H1) , , .
Under the assumption, it is not difficult to prove that the initial problem (1.1)-(1.6) has at least one solution defined on . In fact, after translating second-order differential equation (1.1) into one-order equations, one can draw the conclusion .
Theorem 1.1Let , , be the solution of the initial value problems, respectively,
and suppose thatF, G, gare nonnegative continuous functions on a certain intervalIfor and such that
If does not vanish in , then for , it yields
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 . In Section 3, the general criteria are established for the existence of positive solutions to (1.1)-(1.7) under the case . Moreover, we present the special results in the form of corollaries corresponding to the super-linear case or the sub-linear case. Finally, we come to the conclusion and an example is presented to illustrate our results.
Lemma 2.1If there exist two initial slopes and such that
(i) the solution of (1.1)-(1.6) remains positive in and ;
(ii) the solution of (1.1)-(1.6) satisfies for and ; then multi-point boundary value problem (1.1)-(1.7) has a positive solution with the slope between and .
Proof Since the solutions of (1.1)-(1.6) depend on the initial value continuously, then from (1.8), it implies that is continuous on m. In view of the intermediate value theorem of continuous functions, there exists a number between and such that , that is,
Therefore, is the solution of (1.1)-(1.7). □
Lemma 2.2Let , then (1.1)-(1.7) has no positive solution.
Proof Assume that (1.1)-(1.7) has a positive solution u.
If , then , the convexity of u implies that ( ) and
which contradicts with the convexity of u.
If , then , that is, for . If there exists such that , then and , which contradicts with the convexity of u. Therefore for .
In the rest of this paper, we always assume:
3 Main results
Theorem 3.1Assume that (H1)-(H2) holds. Suppose and there exists a constant such that
(i) ; or
Then problem (1.1)-(1.7) has a positive solution.
Proof (i) Since , we can choose a positive number such that
We claim that there exists a positive number small enough such that for . The claim is based on the convexity of the function and the Sturm comparison theorem (see ). Hence,
From (1.8), (3.1) and combining the integral mean value theorem with Theorem 1.1, we have
where and such that .
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 such that
and there exists another number such that for . Set
In view of (H2) and (3.3), it is not difficult to verify that
which implies from (3.4) that for . Thus, by the convexity of and Theorem 1.1, we have
By Lemma 2.1 and (3.2)-(3.5), there exists a number between and such that is the positive solution of (1.1)-(1.7). The proof for (i) is complete.
Now, we prove for (ii).
In view of , we can choose a number N large enough such that
For this N, there exist a number ϵ small enough and a number large enough such that and for . Therefore
Obviously, as . Thus approximately for as .
Let , . Similar to (3.2), we obtain
where and such that .
Since , then there exist two positive numbers and σ small enough such that
By the convexity of , for these σ and , there exists a positive number τ small enough such that
From (3.6) and (3.7), we have and for . Thus
By Lemma 2.1, the proof for (ii) is complete. □
Theorem 3.2Assume that (H1)-(H2) holds. Suppose and there exists a constant such that
Then problem (1.1)-(1.7) has a positive solution under the case
(i) ; or
Proof Note the computation of in Theorem 3.1. In (3.2), if we substitute with
then , and all the steps in the following are the same as in Theorem 3.1. □
Now, let us consider the special super-linear case or the sub-linear case. It is not difficult to verify the following corollaries.
Corollary 3.1Assume that and
(i) , ; or
(ii) , .
Then problem (1.1)-(1.7) has a positive solution.
Corollary 3.2If and there exists a constant such that
Then, problem (1.1)-(1.7) has a positive solution under the case
(i) , ; or
(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 boundary-value and multiplicity of boundary-point. 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 single-point 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 . Finally, we established the new criteria for the existence of positive solutions to (1.1)-(1.7) under the case and the case . Obviously, (1.7) vanishes to (1.3) when and the sup-linear case or the sub-linear case is sufficient for the conditions in Theorem 3.1 and Theorem 3.2, so some of our results are more general or better than those via fixed point techniques. However, in Theorem 3.2, whether the transcendental equation has a solution is somewhat difficult to verify. It can be seen that each method has its pros and cons.
Example 4.1 Consider the BVP
It is not difficult to see that
In view of , Matlab software gives and . Hence
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 such that .
Figure 1. Numerical simulation for Example 4.1.
The authors declare that they have no competing interests.
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.
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. 5-XQD-2006-9), 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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