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Fixed Point Theorems for Nonlinear Operators with and without Monotonicity in Partially Ordered Banach Spaces
Fixed Point Theory and Applications volume 2010, Article number: 108343 (2010)
Abstract
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.
1. Introduction
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:

2. Monotonic Operators
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.
3. Nonmonotonic Case
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
.
4. An Example
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.
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Acknowledgments
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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Ding, HS., Liang, J. & Xiao, TJ. Fixed Point Theorems for Nonlinear Operators with and without Monotonicity in Partially Ordered Banach Spaces. Fixed Point Theory Appl 2010, 108343 (2010). https://doi.org/10.1155/2010/108343
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DOI: https://doi.org/10.1155/2010/108343