© Yanqiu Zhou et al. 2011
Received: 16 November 2010
Accepted: 12 January 2011
Published: 20 January 2011
We define the concept of the generalized -concave operators, which generalize the definition of the -concave operators. By using the iterative method and the partial ordering method, we prove the existence and uniqueness of fixed points of this class of the operators. As an example of the application of our results, we show the existence and uniqueness of solutions to a class of the Hammerstein integral equations.
1. Introduction and Preliminary
Let be a cone in real Banach space and the partial ordering ≤ defined by , that is, if and only if . The concept and properties of the cone can be found in [3–5]. People studied how to solve (1.1) by using the iterative method and the partial ordering method (see [1–11]).
In , Krasnosel'skiĭ gave the concept of concave operators and studied the existence and uniqueness of the fixed point for the operator by the iterative method. The concept of concave operators was defined by Krasnosel'skiĭ as follows.
In many papers, the authors studied concave operators and obtained some results (see [3–5, 8–15]). In this paper, we generalize the concept of concave operators, give a concept of the generalized concave operators, and study the existence and uniqueness of fixed points for this class of operators by the iterative method. Our results generalize the results in [3, 4, 7, 15].
2. Main Result
To prove the following Theorem 2.5, we will use the definition of the normal cone as follows.
Let be a normal cone of . If operator is increasing and generalized -concave, and is irrelevant to in (2.2), then has the only one fixed point . Moreover, constructing successively the sequence for any initial , we have .
Let be a real numbers set, and let , , then is a real Banach space and is a normal and solid cone in . Let . Considering the equation: . Obviously, is a generalized -concave operator and satisfies all the conditions of Theorem 2.5. Hence has the only one fixed point . Moreover, we know by computing.
In Example 3.2, we know that operator doesn't satisfy the definition of -concave operators. Therefore, we can't obtain the fixed point of by the fixed point theorem of -concave operators. The -concave operators' fixed points are all positive, but here 's fixed point is negative.
The project is supported by the National Science Foundation of China (10971179), the College Graduate Research and Innovation Plan Project of Jiangsu (CX10S−037Z), the Graduate Research and Innovation Programs of Xuzhou Normal University Innovation Plan (2010YLA001).
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