The Iterative Method of Generalized -Concave Operators
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Research Article The Iterative Method of Generalized u0 -Concave Operators Yanqiu Zhou, Jingxian Sun, and Jie Sun Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China Correspondence should be addressed to Jingxian Sun, [email protected] Received 16 November 2010; Accepted 12 January 2011 Academic Editor: N. J. Huang Copyright q 2011 Yanqiu Zhou et al. 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. We define the concept of the generalized u0 -concave operators, which generalize the definition of the u0 -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 In 1, 2, Collatz divided the typical problems in computation mathematics into five classes, and the first class is how to solve the operator equation Ax x
1.1
by the iterative method, that is, construct successively the sequence xn1 Axn
1.2
for some initial x0 to solve 1.1. Let P be a cone in real Banach space E and the partial ordering ≤ defined by P , that is, x ≤ y if and only if y − x ∈ P . 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.
2
Fixed Point Theory and Applications
In 7, Krasnosel’ski˘ı gave the concept of u0 -concave operators and studied the existence and uniqueness of the fixed point for the operator by the iterative method. The concept of u0 -concave operators was defined by Krasnosel’ski˘ı as follows. Let operator A : P → P and u0 > θ. Suppose that i for any x > θ, there exist α αx > 0 and β βx > 0, such that αu0 ≤ Ax ≤ βu0 ;
1.3
ii for any x ∈ P satisfying α1 u0 ≤ x ≤ β1 u0 α1 α1 x > 0, β1 β1 x > 0 and any 0 < t < 1, there exists η ηx, t > 0, such that Atx ≥ 1 η tAx.
1.4
Then A is called an u0 -concave operator. In many papers, the authors studied u0 -concave operators and obtained some results see 3–5, 8–15. In this paper, we generalize the concept of u0 -concave operators, give a concept of the generalized u0 -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 In this paper, we always let P be a cone in real Banach space E and the partial ordering ≤ defined by P . Given w0 ∈ E, let P w0 {x ∈ E | x ≥ w0 }. Definition 2.1. Let operator A : P w0 → P w0 and u0 > θ. Suppose that i for any x > w0 , there exist α αx > 0 and β βx > 0, such that αu0 w0 ≤ Ax ≤ βu0 w0 ;
2.1
ii for any x ∈ P w0 satisfying α1 u
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