- Research Article
- Open Access
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
- Differential Equation
- Real Number
- Partial Differential Equation
- Ordinary Differential Equation
- Positive Constant
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 ).
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.
Lemma 1 B (see ).
Lemma 1 C (see [10, Lemmas and ]).
Now, we can state and prove our main result.
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.
Then, we have the following results.
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.
We immediately conclude the following corollaries.
It follows from Remark 2.3 and Theorem 2.1 that the desired result holds, immediately.
Thus, we complete the proof.
Thus, we completed the proof.
To illustrate the usage of our results, we present the following examples.
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