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:
(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
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, .
The proof is divided into 4 steps.
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.
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
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
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 .
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 .
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 ).
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.
defines a complete metric on each constituent of .
We will also need the following result.
 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,  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.
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 .
We divided the proof into 2 steps.
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
Then is continuous, for all , and
for all and satisfying .
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
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 .
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 .
Let . Then, since is a mixed monotone operator, we have
for all and satisfying . Then, Theorem 3.4 yields the conclusion.
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:
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 ).
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.
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).
Chen, YZ: Thompson's metric and mixed monotone operators. Journal of Mathematical Analysis and Applications. 177(1), 31–37 (1993). Publisher Full Text
Ding, H-S, Xiao, T-J, Liang, J: Existence of positive almost automorphic solutions to nonlinear delay integral equations. Nonlinear Analysis: Theory, Methods & Applications. 70(6), 2216–2231 (2009). PubMed Abstract | Publisher Full Text
Li, K, Liang, J, Xiao, T-J: New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation. Journal of Mathematical Analysis and Applications. 328(2), 753–766 (2007). Publisher Full Text
Zhang, Z, Wang, K: On fixed point theorems of mixed monotone operators and applications. Nonlinear Analysis: Theory, Methods & Applications. 70(9), 3279–3284 (2009). PubMed Abstract | Publisher Full Text
Ait Dads, E, Cieutat, P, Lhachimi, L: Positive pseudo almost periodic solutions for some nonlinear infinite delay integral equations. Mathematical and Computer Modelling. 49(3-4), 721–739 (2009). Publisher Full Text
Boyd, DW, Wong, JSW: On nonlinear contractions. Proceedings of the American Mathematical Society. 20, 458–464 (1969). Publisher Full Text
Kirk, WA: Fixed points of asymptotic contractions. Journal of Mathematical Analysis and Applications. 277(2), 645–650 (2003). Publisher Full Text
Cooke, KL, Kaplan, JL: A periodicity threshold theorem for epidemics and population growth. Mathematical Biosciences. 31(1-2), 87–104 (1976). Publisher Full Text