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On positive solutions for nonhomogeneous m-point boundary value problems with two parameters
Boundary Value Problems volume 2012, Article number: 87 (2012)
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.
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  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 1.1 Assume the following conditions hold:
(H1) are nonnegative parameters;
(H2) is continuous, does not vanish identically on any subinterval of and , where is given in Sect. 2;
(H3) is nondecreasing with respect to u, respectively, that is,
And either or ;
(H4) There exist constants such that and , respectively, for all ;
(H5) , .
If , then there exists a bounded and continuous curve Γ separating into two disjoint subsets and such that (1) has at least two positive solutions for , one positive solution for , and no solution for . Moreover, let be the parametric representation of Γ, where
Then on , the function is continuous and nonincreasing, that is, if , we have .
For the proof of Theorem 1.1, we also need the following lemmas.
Lemma 1.2 
Let E be a Banach space, K a cone in E and Ω bounded open in E. Let and be condensing. Suppose that for all and all . Then
Lemma 1.3 
Let E be a Banach space and K a cone in E. For , define . Assume that is a compact map such that for . If for all , then
Lemma 2.1 
Assume that . If with , then the Green function for the homogeneous BVP
is given by
Moreover, the Green function satisfies the following properties:
for , and is continuous on ;
for all .
Lemma 2.2 Assume that (H 1)-(H 5) hold. If , then is a solution of (1) if and only if satisfies 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 2.3 If (H 1)-(H 3) hold, then is 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 2.4 Let , be lower and upper solutions, respectively, of (1) such that . Then (1) has a nonnegative solution satisfying for .
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). □
3 Proof of Theorem 1.1
Lemma 3.1 Assume (H 1)-(H 5) hold and Σ be a compact subset of . Then there exists a constant such that for all and all possible positive solutions of (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 3.2 Assume (H 1)-(H 4) hold. If (1) has a positive solution at , then Eq. (1) has a positive solution at for 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 3.3 Assume (H 1)-(H 5) hold. Then there exists such 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 3.4 Assume (H 1)-(H 5) hold. Then is bounded above.
Proof Let and be a positive solution of (1) at , then we have
by (H4). Furthermore, we can obtain that
Lemma 3.5 Assume (H 1)-(H 5) hold. Then every chain in S has a unique supremum in S.
Lemma 3.6 Assume (H 1)-(H 5) hold. Then there exists a such (1) has a positive solution at for all , no solution at for all . Similarly, there exists a such that (1) has a positive solution at for all , and no solution at for all .
Lemma 3.7 Assume (H 1)-(H 5) hold. Then there exists a continuous curve Γ separating into two disjoint subsets and such that is bounded and is unbounded, Eq. (1) has at least one solution for , and no solution for . The function is nonincreasing, that is, if
Lemma 3.8 Let . Then there exists such that is an upper solution of (1) at for all , where is the positive solution of Eq. (1) corresponding to some satisfying
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
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,
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 .
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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.
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.