- Research Article
- Open Access
© 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.
- Banach Space
- Real Number
- Integral Equation
- Natural Number
- Iterative Method
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].
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).
- Collatz L: The theoretical basis of numerical mathematics. Mathematics Asian Studies 1966, 4: 1–17.Google Scholar
- Collatz L: Function analysis as the assistant tool for Numerical Mathematics. Mathematics Asian Studies 1966, 4: 53–60.Google Scholar
- Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.MATHGoogle Scholar
- Guo D: Nonlinear Functional Analysis. Science and Technology Press, Shandong, China; 1985.Google Scholar
- Sun Jingxian: Nonlinear Functional Analysis and Applications. Science Press, Beijing, China; 2007.Google Scholar
- Krasnosel'skii MA, et al.: Approximate Solution of Operator Equations. Wolters-Noordhoff; 1972.View ArticleGoogle Scholar
- Krasnosel'skiĭ MA, Zabreĭko PP: Geometrical Methods of Nonlinear Analysis, Fundamental Principles of Mathematical Sciences. Volume 263. Springer, Berlin, Germany; 1984:xix+409.View ArticleGoogle Scholar
- Guo D: The partial order in non-linear analysis. Shandong Science and Technology Press, Ji'nan, China; 2000.Google Scholar
- Zhao Z: Uniqueness and existence of fixed point on some mixed monotone mappings in ordered linear spaces. Journal of Systems Science and Complexity 1999,19(4):217–224.MATHGoogle Scholar
- Sun JX, Liu LS: An iterative solution method for nonlinear operator equations and its applications. Acta Mathematica Scientia. Series A 1993,13(2):141–145.MathSciNetGoogle Scholar
- Sun JX: Some new fixed point theorems of increasing operators and applications. Applicable Analysis 1991,42(3–4):263–273.MathSciNetMATHGoogle Scholar
- Wang WX, Liang ZD: Fixed point theorems for a class of nonlinear operators and their applications. Acta Mathematica Sinica. Chinese Series 2005,48(4):789–800.MathSciNetMATHGoogle Scholar
- Liang ZD, Wang WX, Li SJ: On concave operators. Acta Mathematica Sinica (English Series) 2006,22(2):577–582. 10.1007/s10114-005-0687-1MathSciNetView ArticleMATHGoogle Scholar
- Krasnosel'skii MA: Positive Solution of Operators Equations. Noordoff, Groningen, The Netherlands; 1964.Google Scholar
- Zhai CB, Li YJ: Fixed point theorems for u 0 -concave operators and their applications. Acta Mathematica Scientia. Series A 2008,28(6):1023–1028.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.