Abstract
In this paper, some fixed point theorems for monotone operators in partially ordered complete metric spaces are proved. Especially, a sufficient and necessary condition for the existence of a fixed point for a class of monotone operators is presented. The main results of this paper are generalizations of the recent results in the literature. Also, the main results can be applied to solve the nonlinear elliptic problems and the delayed hematopoiesis models.
MSC:47H10, 54H25.
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1 Introduction
In the last decades, the fixed point theorems for the contraction mappings have been improved and generalized in different directions. During the extensive applications to the nonlinear integral equations, there were many researchers to investigate the existence of a fixed point for contraction-type mappings in partially ordered metric spaces. In 2006, Bhaskar and Lakshmikantham [1] introduced the notion of coupled fixed point and proved some coupled fixed point theorems for mixed monotone mappings. Later, Lakshmikantham and Ciric presented a coincidence point theorem for a mapping with g-monotone property in [2]. Also, the concepts of tripled fixed point and quadruple fixed point were introduced by the authors in [3] and [4], respectively. Meanwhile, they proved the corresponding fixed point theorems. More details on the direction of the coupled fixed point theory and its applications can be found in the literature (see, e.g., [5–27]).
In this manuscript, we give a common method to deal with the existence of a coupled fixed point and the coincidence point for a class of mixed monotone mappings in a partially ordered complete metric space. Indeed, we establish some fixed point theorems for the monotone operators in the partially ordered complete metric space. Especially, we present the sufficient and necessary condition for the existence of a fixed point for a class of monotone operators. Our results improve and generalize the main results in the literature [1–4, 10].
In the rest of this section, we recall some basic definitions.
Let be a partially ordered set, a subset is said to be a totally ordered subset if either or holds for all . We say the elements x and y are comparable if either or holds. It is said that a triple is a partially ordered complete metric space if is a partially ordered set and is a complete metric space. Let Φ denote all the functions which satisfy that and for all . We should mention that Agarwal et al. [28] considered the non-decreasing functions satisfying for all and established some fixed point theorems.
Definition 1.1 (Bhaskar and Lakshmikantham [1])
Let be a partially ordered set and . The mapping F is said to have the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any ,
Definition 1.2 (Bhaskar and Lakshmikantham [1])
An element is said to be a coupled fixed point of the mapping if and .
2 Fixed points theorems for monotone operators
Theorem 2.1 Let be a partially ordered complete metric space and let be a monotone non-decreasing operator with respect to the order ⪯ on X. Assume that
-
(i)
there is a such that
(1) -
(ii)
there exists an such that ;
-
(iii)
either (a) G is a continuous operator, or (b) if a non-decreasing monotone sequence in X tends to , then for all n.
Then the operator G has a fixed point in X.
Proof Definite a sequence in by
Considering the operator is non-decreasing monotone for the order ⪯ and , we have
If there exists such that , then and is a fixed point of G. Then the result of Theorem 2.1 trivially holds.
Suppose now that for all n. Let , noting that the sequence is a non-decreasing sequence in , we conclude that
Thus we obtain that
This implies that both sequences and are convergent. Set . If , noting for all , we have
This is a contradiction. Thus .
Now, we shall prove that is a Cauchy sequence in . In fact, by , we can choose a positive sequence with and . For a fixed m, there exists a large enough positive number N satisfying . Let , then by the triangle inequality
Also, . This means that the set Ω is invariant for the operator G. Clearly, . Thus for all . So, the sequence is a Cauchy sequence in . Since is a complete metric space, there exists a point such that .
Suppose that G is a continuous operator. Then, by definition of , we have
Let us assume that the assumption (b) holds, then for all . Thus from the assumption (i), we have
So, . The proof of Theorem 2.1 is complete. □
Let be a mapping having the mixed monotone property on X and define the operator by
It is easy to see that the coupled fixed points of F is the fixed points of G in . Also, for , we introduce a partial order ⪯ in given by
Thus, if F has the mixed monotone property on X, then the operator G is non-decreasing monotone for the order ⪯. For , let , then is a complete metric space provided is a complete metric space. Then, as a consequence of Theorem 2.1, we achieve the following corollary.
Corollary 2.1 Let be a partially ordered complete metric space and let be a mapping having the mixed monotone property on X. Assume that
-
(i)
there is a such that satisfying
-
(ii)
there exists an such that ;
-
(iii)
one of (a) and (b) holds:
-
(a)
G is a continuous operator;
-
(b)
if a non-decreasing monotone sequence in tends to , then for all n.
Then the operator G has a fixed point in , that is, there exist such that
Let , then we have the following theorem.
Theorem 2.2 Let be a partially ordered complete metric space and let be a mapping having the mixed monotone property on X. Assume that (i) in Theorem 2.1 and one of following conditions holds:
-
(a)
G is a continuous operator;
-
(b)
if a monotone sequence in tends to , then and are comparable for all n.
Then the operator G has a fixed point in if and only if . Furthermore, if D is a totally ordered nonempty subset, then the operator G has a unique fixed point in .
Proof It is easy to see that all the fixed points of G fall in the set D. Thus if the operator G has a fixed point in , then .
We suppose . If the condition (a) holds and , then there are two cases: or . For the first case, following Theorem 2.1, we claim that the operator G has a fixed point in . For the other case: , noting the symmetry of the metric, we see that the formula (1) holds for . Thus
Constructing the same sequence in by , for , we have
For a mini-revise to the proof of Theorem 2.1 and resetting , we conclude that the sequence tends to a fixed point of G.
Now we assume the condition (b) holds. Similar to the case (a), we see that the monotone sequence is a Cauchy sequence and denote as the limit point. Thus is comparable with for all . Then we have
Thus the operator G has a fixed point in .
Next, we suppose that D is a totally ordered nonempty subset. It is sufficient to prove the uniqueness of a fixed point of G. Let and be two fixed points of G, then is comparable with , and . Following the assumption (i), we have
Thus , that is, . The proof of Theorem 2.2 is complete. □
Following Theorem 2.1, we have the next two corollaries.
Corollary 2.2 ([1], Theorem 2.1)
Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists a with
If there exist such that and , then there exist such that
Proof Taking for , , , if then
Thus Corollary 2.1 is an immediate consequence of Theorem 2.1. □
Corollary 2.3 ([1], Theorem 2.2)
Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Assume that X has the following property:
-
(i)
if a non-decreasing sequence , then for all n;
-
(ii)
if a non-increasing sequence , then for all n.
Let be a mapping having the mixed monotone property on X. Assume that there exists a with
If there exist such that and , then there exist such that
Proof It follows from Theorem 2.1 immediately. □
Let be a real Banach space and let K be a cone. The relation holds if and only if . Denote and for a given . Let and for . Then d defines a metric on which is known as the Thompson metric [15]. More details about the Thompson metric can be found in the references [29–32]
At this stage, we state our main results in the real Banach space.
Theorem 2.3 Let be a real Banach space, let a cone and . Suppose that are two mixed monotone maps satisfying and for , with . Let and assume that there exists a point such that
Then A has a unique fixed point in , that is, there exists a unique point such that .
In order to prove this result, we need some technique lemmas.
Lemma 2.1 ([10], Lemma 3.1)
Under the assumptions of Theorem 2.3, there exists such that
where
Lemma 2.2 Under the assumptions of Theorem 2.3, then
Proof Noting that for all , we have
On the other hand, since for , then we have
Thus . □
Lemma 2.3 Under the assumptions of Theorem 2.3, the successive sequences and are Cauchy sequences, where
Proof Since , it follows by an induction argument that
Noting that
we have, for all n,
Thus
Next, we claim that
In fact, it holds for . For arbitrary n, by induction argument, we have
Thus (3) holds for all n.
On the other hand, since
we obtain that, for all n,
Thus
Then following Lemma 2.2, we have
Similarly, we have
Let , then and
Thus, for all n,
Furthermore, for any , we have
This shows that both successive sequences and are Cauchy sequences. □
Proof of Theorem 2.3 By Lemma 2.3, there are such that and . Obviously, and for all . Thus . Next, noting that
and
and letting n go to infinity, we get that and . It follows from
that and . The uniqueness is obvious. Thus A has a unique fixed point in , that is, there exists a unique point such that . The proof is complete. □
Remark 2.1 Our result in Theorem 2.3 improved the corresponding result in [10] (Theorem 3.4) and removed some restriction conditions: the successive sequences have convergent subsequences.
3 Application to the nonlinear elliptic problems
Let Ω be the open unit ball in , , with center at the origin. We consider positive solutions of the Dirichlet problem
When , and , it is well known that (4) has no positive solution if , and that the positive solution of (4) is unique if , see [22] and [23]. Also, in this case when , (4) has a unique positive radial solution [24].
In this section, we assume that , and are constants, and are positive and continuous for . Our result is as follows.
Theorem 3.1 Problem (4) has a unique positive radial solution if , where and .
To this end, we should establish a technique lemma.
Lemma 3.1 The function u is a positive radial solution of problem (4) if and only if u is a positive solution of the integral equation
where
Proof Assuming solutions to be functions of r, the radial distance from the origin, (4) reduces to
where . Then the Green function for problem (5) is
which is positive on . Thus the function u is a positive radial solution of problem (4) if and only if u is a positive solution of the integral equation
□
Proof of Theorem 3.1 Let K denote the cone of nonnegative functions in , the relation holds if and only if for all , and for , then and . Denote .
Now we introduce the maps defined by
For , then there exist , , , such that and . By direct computation, we have
Thus the map is well defined and for . Also, . Obviously, is a mixed monotone map in and and .
Since , we choose a positive number k (large enough) satisfying
Let , , , then and
and
Thus
Applying Theorem 2.3 to the operator A, we conclude that there is a unique point u in such that . On the other hand, for all , we have
This means that . Thus problem (4) has a unique positive radial solution. □
4 Application to the delayed hematopoiesis models
In this section, we consider the positive periodic solution of the following hematopoiesis model with delays:
where are positive T-periodic functions and for all , q is a nonnegative constant (). In the case when , Wu [26] proved that (6) had a unique positive T-periodic solution.
Here we assume that and our result is as follows.
Theorem 4.1 Problem (6) has a unique positive T-periodic solution.
Proof Let K denote the cone of nonnegative T-periodic functions in , the relation holds if and only if for all , and for . Denote . It is easy to show that the function x is a positive T-periodic solution of problem (6) if and only if x is a positive solution of the integral equation
where is a constant. Define the maps by
For arbitrary , there are positive numbers , , , such that and for all . Furthermore, we deduce that
and
Similarly, we have
This means and . Thus and are well defined and . Also, . Obviously, and are mixed monotone maps in and and .
Since , we can choose a constant satisfying
Let , and for all , then we have
and
Thus
Applying Theorem 2.3 to the operator A, we conclude that there is a unique point x in such that . On the other hand, for all , we have
This means that . Thus problem (6) has a unique positive T-periodic solution. □
Remark 4.1 Using similar ideas, it is possible to extend our results to investigate the existence and uniqueness of nonlinear singular boundary value problems and fractional differential equation boundary value problems, which are mentioned extensively in the literature [10, 11, 25].
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Acknowledgements
The authors are grateful to the reviewers for their valuable comments and suggestions. This work was partly supported by the National Natural Science Foundation of China (11201481) and Hunan Provincial Education Department Foundation (12B007).
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Wu, J., Liu, Y. Fixed point theorems for monotone operators and applications to nonlinear elliptic problems. Fixed Point Theory Appl 2013, 134 (2013). https://doi.org/10.1186/1687-1812-2013-134
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DOI: https://doi.org/10.1186/1687-1812-2013-134