On positive solutions for nonhomogeneous m-point boundary value problems with two parameters

This paper is concerned with the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point boundary value problems with two parameters. The proof is based on the fixed-point theorem, the upper-lower solutions method, and the fixed-point index.

MSC: 34B10, 34B18.

Many authors have studied the existence, nonexistence, and multiplicity of positive solutions for multipoint boundary value problems by using the fixed-point theorem, the fixed point index theory, and the lower and upper solutions method. We refer the readers to the references [1-4]. Recently, Hao, Liu and Wu [5] studied the existence, nonexistence, and multiplicity of positive solutions for the following nonhomogeneous boundary value problems:

where , (), , , may be singular at and/or . They showed that there exists a positive number such that the problem has at least two positive solutions for , at least one positive solution for and no solution for by using the Krasnosel’skii-Guo fixed-point theorem, the upper-lower solutions method, and the topological degree theory.

Inspired by the above references, the purpose of this paper is to study the following more general nonhomogeneous boundary value problems:

where λ, μ are positive parameters, , . The main result of the present paper is summarized as follows.

Theorem 1.1Assume the following conditions hold:

(H1) are nonnegative parameters;

(H2) is continuous, does not vanish identically on any subinterval ofand, whereis given in Sect. 2;

(H3) is nondecreasing with respect tou, respectively, that is,

And eitheror;

(H4) There exist constantssuch thatand, respectively, for all;

(H5) , .

If, then there exists a bounded and continuous curve Γ separatinginto two disjoint subsetsandsuch that (1) has at least two positive solutions for, one positive solution for, and no solution for. Moreover, letbe the parametric representation of Γ, where

Then on, the functionis continuous and nonincreasing, that is, if, we have.

For the proof of Theorem 1.1, we also need the following lemmas.

Lemma 1.2[6]

LetEbe a Banach space, Ka cone inEand Ω bounded open inE. Letandbe condensing. Suppose thatfor alland all. Then

Lemma 1.3[6]

LetEbe a Banach space andKa cone inE. For, define. Assume thatis a compact map such thatfor. Iffor all, then

Lemma 2.1[5]

Assume that. Ifwith, then the Green function for the homogeneous BVP

is given by

Moreover, the Green function satisfies the following properties:

(i) for, andis continuous on;

(ii) for all.

Lemma 2.2Assume that (H1)-(H5) hold. If, thenis a solution of (1) if and only ifsatisfies the following nonlinear integral equation:

Proof Integrating both sides of (1) from 0 to t twice and applying the boundary conditions, then we can obtain

Furthermore, by Lemma 2.1, we can obtain

 □

Let E denote the Banach space with the norm . A function is said to be a solution of (1) if satisfies (1). Moreover, from Lemma 2.2, it is clear to see that is a solution of (1) is equivalent to the fixed point of the operator T defined as

In addition, define a cone as

where . Then we have

Lemma 2.3If (H1)-(H3) hold, thenis completely continuous.

The proof procedure of Lemma 2.3 is standard, so we omit it.

Now, we will establish the classical lower and upper solutions method for our problem. As usual, we say that is a lower solution for (1) if

Similarly, we define the upper solution of the problem (1):

Lemma 2.4Let, be lower and upper solutions, respectively, of (1) such that. Then (1) has a nonnegative solutionsatisfyingfor.

Proof Define

It is clear to see that is a bounded, convex and closed subset in Banach space E. Now we can prove that .

For any , from (H3), we have

On the other hand, we also have

From above inequalities, we obtain that .

Therefore, by Schauder’s fixed theorem, the operator T has a fixed point , which is the solution of (1). □

Lemma 3.1Assume (H1)-(H5) hold and Σ be a compact subset of. Then there exists a constantsuch that for alland all possible positive solutionsof (1) at, one has.

Proof Suppose on the contrary that there exists a sequence of positive solutions of Eq. (1) at such that for all and

Then , and thus

Since Σ is compact, the sequence has a convergent subsequence which we denote without loss of generality still by such that

and at least or .

Case (I). If , we have for n sufficient large. Then by (H5), there exists a such that

where L satisfies

Since , for n sufficient large, we

This is a contradiction.

Case (II). If , then we have for n sufficient large. Since , there exists a such that

where M satisfies

Since , then for n sufficient large, we have

This is a contradiction. □

Lemma 3.2Assume (H1)-(H4) hold. If (1) has a positive solution at, then Eq. (1) has a positive solution atfor all.

Proof Let be the solution of Eq. (1) at , then be the upper solution of (1) at with . Since or , is not a solution of (1), but it is the lower solution of (1) at . Therefore, by Lemma 2.4, we obtain the result. □

Lemma 3.3Assume (H1)-(H5) hold. Then there existssuch that Eq. (1) has a positive solution for all.

Proof Let be the unique solution of

It is clear to see that is a positive solution of (3). Let , , then by (H4), we know that and . Set , we have

which implies that is an upper solution of (3) at . On the other hand, 0 is a lower solution of (1) and . By (H3), 0 is not a solution of (1). Hence, (1) has a positive solution at , Lemma 3.2 now implies the conclusion of Lemma 3.3. □

Define a set S by

Then it follows from Lemma 3.3 that and is a partially ordered set.

Lemma 3.4Assume (H1)-(H5) hold. Thenis bounded above.

Proof Let and be a positive solution of (1) at , then we have

by (H4). Furthermore, we can obtain that

 □

Lemma 3.5Assume (H1)-(H5) hold. Then every chain inShas a unique supremum inS.

Lemma 3.6Assume (H1)-(H5) hold. Then there exists asuch (1) has a positive solution atfor all, no solution atfor all. Similarly, there exists asuch that (1) has a positive solution atfor all, and no solution atfor all.

Lemma 3.7Assume (H1)-(H5) hold. Then there exists a continuous curve Γ separatinginto two disjoint subsetsandsuch thatis bounded andis unbounded, Eq. (1) has at least one solution for, and no solution for. The functionis nonincreasing, that is, if

then

Lemma 3.8Let. Then there existssuch thatis an upper solution of (1) atfor all, whereis the positive solution of Eq. (1) corresponding to somesatisfying

Proof From (H4), there exists constant such that

Then by the uniform continuity of f and g on a compact set, there exist such that

for all and .

Let , then we have

and

From above inequalities, it is clear to see that , is an upper solution of (1) at for all . □

Proof of Theorem 1.1 From above lemmas, we need only to show the existence of the second positive solution of (1) for . Let , then there exists such that

Let be the positive solution of (1) at . Then for given by Lemma 3.8 and for all , denote

Define the set

Then D is bounded open set in E and . The map T satisfies and is condensing, since it is completely continuous. Now let , then there exists such that either . Then by (H) and Lemma 3.8, we obtain

for all . Thus, for all and , Lemma 1.2 now implies that

Now for some fixed λ and μ, it follows from assumption (H4) that there exists a such that

where L satisfies

Let where is given by Lemma 3.1 with Σ a compact set in containing . Let

Then it follows from Lemma 3.1,

Moreover, for , we have

Furthermore, we have

Thus, and it follows from Lemma 1.3 that

By the additivity of the fixed-point index,

which yields

Hence, T has at least one fixed point in and another one in ; this shows that in , (1) has at least two positive solution. □

Example Consider the following boundary value problem:

where , , and .

The authors declare that they have no competing interests.

In this paper, the author studies the existence, multiplicity, and nonexistence of positive solutions for nonhomogeneous m-point boundary value problems with two parameters. The proof is based on the upper-lower solutions method and fixed-point index. All authors typed, read, and approved the final manuscript.

The authors would like to thank the referees for valuable comments and suggestions for improving this paper. The first author is supported financially by the Fundamental Research Funds for the Central Universities.

1. Kong, L, Kong, Q: Higher order boundary value problems with nonhomogeneous boundary conditions. Nonlinear Anal.. 72, 240–261 (2010). Publisher Full Text

2. Kong, L, Wong, JSW: Positive solutions for higher order multi-point boundary value problems with nonhomogeneous boundary conditions. J. Math. Anal. Appl.. 367, 588–611 (2010). Publisher Full Text

3. Ma, R: Positive solutions for nonhomogeneous m-point boundary value problems. Comput. Math. Appl.. 47, 689–698 (2004). Publisher Full Text

4. Yang, X: Existence of positive solutions for 2m-order nonlinear differential systems. Nonlinear Anal.. 61, 77–95 (2005). Publisher Full Text

5. Hao, X, Liu, L, Wu, Y: On positive solutions of an m-point nonhomogeneous singular boundary value problem. Nonlinear Anal.. 73, 2532–2540 (2010). Publisher Full Text

6. Guo, D, Lakshmikantham, V: Nonlinear Problems in Abstract Cones, Academic Press, New York (1988)