Existence results of a kind of Sturm-Liouville type singular boundary value problem with non-linear boundary conditions
© Zhao and Ge; licensee Springer. 2012
Received: 27 January 2012
Accepted: 21 August 2012
Published: 5 September 2012
By using the Leggett-Williams fixed-point theorem, a series of Sturm-Liouville type boundary value problems are studied. We impose sufficient conditions on f and get the existence of at least three positive solutions. The interesting point is that the boundary conditions are non-linear.
where , , , , , and , satisfy
(i) , and ;
(ii) for ;
(iii) is increasing on x.
(i) and is odd;
(ii) for ;
(iii) is increasing on x.
We will assume throughout:
(C1) , which implies that may be singular at or/and . , and is not identically zero on any subinterval of .
(C2) , .
Boundary value problems (BVPs) for ordinary differential equations play a very important role in both theory and application. They are used to describe a large number of physical, biological and chemical phenomena. And in analyzing these non-linear phenomena, many mathematical models give rise to the problems for which only nonnegative solutions make sense. Therefore, since the publication of the monograph Positive Solutions of Operator Equations in 1964 by the academician Krasnosel’skii, many research articles on the theory of positive solutions of nonlinear problems have appeared [1–5]. In this vast field of research, particular attention has been focused on the BVPs with linear boundary conditions. However, BVPs with non-linear boundary conditions have not received much attention. Obviously, studying such kind of BVPs is more practical but more challenging. The main challenge is that it is not easy to construct an operator and, in turn, to get its solutions.
where , , is a nonnegative measurable function, is a nonnegative continuous function on , and , are all continuous functions defined on . By using the theory of fixed point index, the authors get the existence of at least two positive solutions. For more references about the classical Sturm-Liouville BVP, we refer the readers to [7–17].
BC ‘, (∗)’ which is the special case of BC (1.2) and (1.3) can be called Sturm-Liouville type BC, the reason is that when , , BC (∗) become the classic Sturm-Liouville BC. To the best knowledge of the authors, few papers have dealt with equation (1.1) subject to the non-linear BC (1.2) and (1.3). So to fill this gap, motivated by the results mentioned above, in this paper we consider BVP (1.1), (1.2) and (1.1), (1.3). At the end of this paper, we also give its generalization which is more universal.
On the other hand, the main tool used in this paper is the Leggett-Williams fixed point theorem, which is different from . In order to apply this fixed point theorem to our paper, we need to make use of some techniques such as the definition of the cone and the zooming of inequalities etc. The interesting point in this paper is that the boundary condition is the non-linear Sturm-Liouville type boundary condition.
2 Background material and the fixed point theorem
In this section, for convenience, we present the main definitions and the theorem that will be used in this paper.
Definition 2.1 Let E be a real Banach space. A nonempty closed convex set is called a cone if it satisfies the following two conditions:
(i) for all ,
(ii) implies .
Every cone induces an ordering in E given by if and only if .
for all and .
Let a, b, r be constants, , .
Theorem 2.1 (Leggett-Williams )
Let be a completely continuous map and α be a nonnegative continuous concave functional on P such that for , there holds . Suppose there exist a, b, d with such that
(i) and for all ;
(ii) for all ;
(iii) for all with .
In what follows, by we denote the inverse to , where .
3 Existence results for BVP (1.1), (1.2)
3.1 Some useful lemmas
Obviously, is strictly increasing on . At the same time, we can notice that , and . Further, is continuous, thus there must exist one point such that , which means that , must intersect at a unique point σ.
Hence, for , the solution of (3.1) can be expressed as (3.2), which completes our proof. □
Remark 3.1 In fact, for , the solution of (3.1) can be expressed both by (♢) and (♡), but just for convenience, we write it into two parts.
Lemma 3.2 Let satisfy all the conditions in Lemma 3.1, then the solution of BVP (3.1) is concave on . What is more, .
Proof means , so is concave on .
Further, when , , similarly, we can get . Thus for all . The proof is complete. □
3.2 Existence of positive solutions to BVP (1.1), (1.2)
In this subsection, we try to impose some growth conditions on the non-linear term f which allows us to apply Theorem 2.1 to establish the existence of triple positive solutions to BVP (1.1), (1.2).
Let , and let it be endowed with the maximum norm .
In our main results, we will make use of the following lemmas.
Lemma 3.3 For , then .
Similarly, when . Thus , which completes our proof. □
Remark 3.2 From Lemma 3.3, we know that when , there holds .
Lemma 3.4 Suppose that (C1) and (C2) hold, then is completely continuous.
Obviously, , this implies that T is concave on . Further, on , and on , . Thus is well defined.
It is easy to prove that is continuous with respect to x. For more details, see [, p.6].
For any bounded subset Ω of P, it can be easily verified that is bounded and equicontinuous. Thus by the Arzela-Ascoli theorem, is relatively compact, therefore, is completely continuous. □
Theorem 3.5 Assume that (H1) holds, there exist constants such that . Further suppose that
(S1) , for ;
(S2) , for ;
(S3) , for .
Proof Problem (1.1), (1.2) has a solution if and only if it solves the operator equation . Thus we set out to verify that the operator satisfies the Leggett-Williams fixed point Theorem 2.1, which will prove the existence of at least three fixed points of .
Now, we will prove the main theorem step by step.
Similarly, (S2) implies that .
Step 2. We try to prove Theorem 2.1(i) is satisfied.
Choose . Obviously, , thus .
Thus, Theorem 2.1(i) holds.
Step 3. We try to prove that Theorem 2.1(iii) holds.
Hence, Theorem 2.1(iii) also holds. An application of Theorem 2.1 means that BVP (1.1), (1.2) has at least three positive solutions satisfied (3.12). The proof is complete. □
4 Existence results for BVP(1.1), (1.3)
4.1 Some useful lemmas
Proof Essentially, we can prove it using the same discussion as in Lemma 3.1. Here, we omit it. □
Lemma 4.2 Let satisfy all the conditions in Lemma 3.1, then the solution of BVP (4.1) is concave on . What is more, .
4.2 Existence of positive solutions to BVP (1.1), (1.3)
In this subsection, we try to impose some growth conditions on the non-linear term f which allow us to apply Theorem 2.1 to establish the existence of triple positive solutions to problem (1.1).
Let E, cone P and the norm be defined as in Section 3.
Lemma 4.3 Suppose that (C1) and (C2) hold, then is completely continuous.
Theorem 4.4 Assume that (H2) holds, there exist constants such that . Further suppose that
(S4) , for ;
(S5) , for ;
(S6) , for .
Proof By making use of Theorem 2.1, we can prove the result by essentially the same method. Here we omit it. □
5 A generalization result
In addition to (H1), (H2), we assume at least one of the following conditions holds:
(H3) There exists a positive constant such that ;
(H4) There exists a positive constant such that .
Without loss of generality, we suppose (H4) holds. Essentially by the same method, we can get the multiple existence results for BVP (1.1), (5.1). In what follows, we just list the main results but the proofs are all omitted.
Lemma 5.2 Let satisfy all the conditions in Lemma 3.1, then the solution of BVP (5.2) is concave on . What is more, .
Lemma 5.3 Suppose that (C1) and (C2) hold, then is completely continuous.
Theorem 5.4 Assume that (H1), (H4) hold, there exist constants such that . Further suppose that
(S7) , for ;
(S8) , for ;
(S9) , for .
In this section, we give examples to illustrate our main results.
And we can see that , , , . Choose , , , . By direct calculation, we can get , . Also the following conditions hold.
(i) , for ;
(ii) , for ;
(iii) , for .
Choose , , , . Obviously, , , , , By direct calculation, we can get that , , . Also the following conditions hold.
(i) , for ;
(ii) , for ;
(iii) , for .
The authors were very grateful to the anonymous referee whose careful reading of the manuscript and valuable comments enhanced presentation of the manuscript. The study was supported by Pre-research project and Excellent Teachers project of the Fundamental Research Funds for the Central Universities 2-9-2012-141.
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