Fixed Point Theorems for Nonlinear Operators with and without Monotonicity in Partially Ordered Banach Spaces

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).

Stimulated by [7, 8], we investigate further, in this paper, the existence of fixed points of nonlinear operators with and without monotonicity in partially ordered Banach spaces.

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 [2]). Let be a real Banach space. A closed convex set in is called a convex cone if the following conditions are satisfied:

(i)if , then for any ,

(ii)if and , then .

A cone induces a partial ordering in by

For any given ,

A cone is called normal if there exists a constant such that

where is the norm on .

Throughout this paper, we denote by the set of nonnegative integers, the set of real numbers, a real Banach space, a convex cone in , an element in ( is the zero element of ), and the following set:

Theorem 2.1.

Suppose that the operator satisfies the following.

(S1) is increasing, is decreasing, and is decreasing.

(S2) There exist a constant and a function such that for each and , and

(S3)There exist such that , , and

(S4)There exists a constant such that, for all with ,

Then has a unique fixed point in , that is, .

Proof.

The proof is divided into 4 steps.

Step 1.

Let and

For each , there exists a nonnegative integer such that , that is, . Now, by (S2), we deduce, for all ,

Moreover, by (S3), we get

Hence, in the following proof, one can assume that in (S2) and (S3) without loss.

Step 2.

Fix . Then, there exists such that . Let

Then is an operator from to , and by (S4), is increasing in . Combining (S1)–(S3), we have

provided that . Moreover, it is easy to see that (2.8) holds when . Similarly, one can show that

Then, it follows that

Let

Then, using arguments similar to those in the proof of [7, Theorem ], one can show that has a unique fixed point in , and

We claim that is the unique fixed point of in . In fact, let be a fixed point of in , and such that . By the above proof, has a unique fixed point in , which means that . In addition, it follows from

that .

Step 3.

By Step 2, we can define an operator by

Let with and with . Denote by the corresponding sequences in the proof of Step 2. Then

Next, by induction and being increasing, one can show that for all . So

that is, . Thus, is increasing. By a similar method, one can prove that is decreasing. On the other hand, by (S3), for and ,

Let , and

By choosing in Step 1, we get . Then

As is increasing and is decreasing, it follows immediately that

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 .

Step 4.

By Steps 2 and 3, we get

Let such that . Then it follows from Step 2 that , that is, is a fixed point of in . Thus, by Step 3, , which means that is the unique fixed point of in .

Remark 2.2.

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.

First, let us recall some definitions and basic results about Hilbert's projection metric (for more details, see [6]).

Definition 3.1.

Elements and belonging to (not both zero) are said to be linked if there exist such that

This defines an equivalence relation on and divides into disjoint subsets which we call constituents of .

Let and be linked. Define

Then, the following holds.

Theorem 3.2.

defines a complete metric on each constituent of .

Proof.

See [6].

We will also need the following result.

Theorem 3.3.

[9] Let be a complete metric space and suppose that satisfies

where is upper semicontinuous from the right and satisfies for all . Then has a unique fixed point in .

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, [1] 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.

Theorem 3.4.

Let be an operator from to . Assume that there exist a constant and a function such that for all , and

for all and satisfying . Then has a unique fixed point in .

Proof.

We divided the proof into 2 steps.

Step 1.

Let , , and . Then, there exists such that

In view of

by the assumptions, we have

Similar to the above proof, since , one can deduce

Continuing by this way, one can get

Let

Then is continuous, for all , and

for all and satisfying .

Step 2.

Next, let with and

Then , , and Moreover, by Step 1, we have

On the other hand, since , we also have

Thus, we get

Now, by the definition of , we have

Let

Then, is a continuous function from to , and

Moreover, since for all , we get

On the other hand, is obviously a constituent of , and thus is complete by Theorem 3.2. Now, Theorem 3.3 yields that has a unique fixed point in .

Corollary 3.5.

Assume that is a mixed monotone operator, that is, is increasing and is decreasing. Moreover, there exist a constant and a function such that for all , and

for all and . Then has a unique fixed point in .

Proof.

Let . Then, since is a mixed monotone operator, we have

for all and satisfying . Then, Theorem 3.4 yields the conclusion.

Remark 3.6.

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 .

In this section, we give an example to illustrate Theorem 3.4. Let us consider the following nonlinear delay integral equation:

which is a classical model for the spread of some infectious disease (cf. [12]). In fact, (4.1) has been of great interest for many authors (see, e.g., [3, 8] and references therein).

In the rest of this paper, let and

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 [13]).

Definition 4.1.

A continuous function is called almost periodic if for each there exists such that every interval of length contains a number with the property that

Denote by the set of all such functions.

Lemma 4.2.

Assume that , . Then the following hold.

(a)The range is precompact in , and so is bounded.

(b) provided that is continuous on .

(c), . Moreover, provided that .

(d) Equipped with the sup norm

turns out to be a Banach space.

Now, let , and is defined by . It is not difficult to verify that is a normal cone in , and

Define a nonlinear operator on by

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

that is, for all and with . Then, by Theorem 3.4, has a unique fixed point in , that is, (4.1) has a unique almost periodic solution with positive infimum.

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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