Fixed Point Theorems for Nonlinear Operators with and without Monotonicity in Partially Ordered Banach Spaces
© Hui-Sheng Ding et al. 2010
Received: 30 September 2009
Accepted: 6 December 2009
Published: 12 January 2010
We establish two fixed point theorems for nonlinear operators on Banach spaces partially ordered by a cone. The first fixed point theorem is concerned with a class of mixed monotone operators. In the second fixed point theorem, the nonlinear operators are neither monotone nor mixed monotone. We also provide an illustrative example for our second result.
Fixed point theorems for nonlinear operators on partially ordered Banach spaces have many applications in nonlinear equations and many other subjects (cf., e.g., [1–7] and references therein); in particular, various kinds of fixed point theorems for mixed monotone operators are proved and applied (see, e.g., [1, 3, 5, 7] and references therein).
In Section 2, a fixed point theorem for a class of mixed monotone operators is established. In Section 3, without any monotonicity assumption for a class of nonlinear operators, we obtain a fixed point theorem by using Hilbert's projection metric.
Let us recall some basic notations about cone (for more details, we refer the reader to ). Let be a real Banach space. A closed convex set in is called a convex cone if the following conditions are satisfied:
2. Monotonic Operators
The proof is divided into 4 steps.
Next, by making some needed modifications in the proof of [3, Theorem ], one can show that has a fixed point . Suppose that is a fixed point of . It follows from the definition of and that for all . Then, by the normality of , we get . So is the unique fixed point of in .
Compared with [7,Remark ], the nonlinear operator in Theorem 2.1 is more general, and so Theorem 2.1 may have a wider range of applications.
3. Nonmonotonic Case
First, let us recall some definitions and basic results about Hilbert's projection metric (for more details, see ).
Then, the following holds.
We will also need the following result.
Theorem 3.3 is a generalization of the classical Banach's contraction mapping principle. There are many generalizations of the classical Banach's contraction mapping principle (see, e.g., [10, 11] and references therein), and these generalizations play an important role in research work about fixed points of nonlinear operators in partially ordered Banach spaces; see, for example,  and the proof of the following theorem.
Now, we are ready to present our fixed point theorem, in which no monotone condition is assumed on the nonlinear operator.
We divided the proof into 2 steps.
Corollary 3.5 is an improvement of [1,Corollary ] in the sense that there is lower semicontinuous on , and the corresponding conditions need to hold on the whole interval .
4. An Example
In this section, we give an example to illustrate Theorem 3.4. Let us consider the following nonlinear delay integral equation:
Next, let us investigate the existence of positive almost periodic solution to (4.1). For the reader's convenience, we recall some definitions and basic results about almost periodic functions (for more details, see ).
By Lemma 4.2 and [3, Corollary ], it is not difficult to verify that is an operator from to . In addition, in view of (4.2), one can verify that
The authors are very grateful to the referees for valuable suggestions and comments. In addition, Hui-Sheng Ding acknowledges support from the NSF of China (10826066), the NSF of Jiangxi Province of China (2008GQS0057), and the Youth Foundation of Jiangxi Provincial Education Department(GJJ09456); Jin Liang and Ti-Jun Xiao acknowledge support from the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805).
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