Existence of Positive Solutions for Nonlinear Eigenvalue Problems
© Sheng-PingWang et al. 2010
Received: 2 June 2009
Accepted: 2 February 2010
Published: 8 March 2010
We use a fixed point theorem in a cone to obtain the existence of positive solutions of the differential equation, , , with some suitable boundary conditions, where is a parameter.
where and are nonnegative constants, and .
In the last thirty years, there are many mathematician considered the boundary value problem (BVP λ ) with , see, for example, Chu et al. , Chu et al. , Chu and Zhau , Chu and Jiang , Coffman and Marcus , Cohen and Keller , Erbe , Erbe et al. , Erbe and Wang , Guo and Lakshmikantham , Iffland , Njoku and Zanolin , Santanilla .
In 1993, Wong  showed the following excellent result.
Theorem 1 A (see ).
where for , then, there exists such that the boundary value problem (BVP λ ) with has a positive solution in for , while there is no such solution for in which
Seeing such facts, we cannot but ask "whether or not we can obtain a similar conclusion for the boundary value problem (BVP λ )." We give a confirm answer to the question.
First, We observe the following statements.
on , then is the Green's function of the differential equation in with respect to the boundary value condition .
(2) , is a cone in the Banach space with .
Lemma 1 B (see ).
Suppose that be defined as in . Then, we have the following results.
for and )
for and )
Lemma 1 C (see [10, Lemmas and ]).
Let be a real Banach space, and let be a cone. Assume that and is completely continuous. Then
where is the fixed point index of a compact map , such that for , with respect to .
2. Main Results
Now, we can state and prove our main result.
where and . We now separate the rest proof into the following three steps.
It follows from Steps (1) and (2) and the property of the fixed point index (see, for example, [10, Theorem ]) that the proof is complete.
There are many functions and positive constants satisfying (2.27). For example, Suppose that and . Let on , then on and
Then, we have the following results.
It follows from Remark 2.2 that the hypothesis (2.2) of Theorem 2.1 is satisfied if .
which satisfies the hypothesis (2.1) of Theorem 2.1.
which satisfies the hypothesis (2.1) of Theorem 2.1.
Hence, we have the following two cases.
By Cases (i), (ii) and Remark 2.2, we see that the hypothesis (2.2) of Theorem 2.1 is satisfied if .
We immediately conclude the following corollaries.
(BVP λ ) has at least one positive solution for if one of the following conditions holds:
It follows from Remark 2.3 and Theorem 2.1 that the desired result holds, immediately.
on for some and .
Thus, we complete the proof.
on , for some .
Thus, we completed the proof.
To illustrate the usage of our results, we present the following examples.
If we take , then it follows from of Corollary 2.4 that (BVP.1) has a solution if .
If we take , then it follows from of Corollary 2.4 that (BVP.2) has a solution if .
Hence, it follows from Corollary 2.5 that (BVP.3) has two solutions if .
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