Fixed point theorems for monotone operators and applications to nonlinear elliptic problems
© Wu and Liu; licensee Springer 2013
Received: 12 February 2013
Accepted: 8 May 2013
Published: 27 May 2013
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.
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  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 . Also, the concepts of tripled fixed point and quadruple fixed point were introduced by the authors in  and , 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.  considered the non-decreasing functions satisfying for all and established some fixed point theorems.
Definition 1.1 (Bhaskar and Lakshmikantham )
Definition 1.2 (Bhaskar and Lakshmikantham )
An element is said to be a coupled fixed point of the mapping if and .
2 Fixed points theorems for monotone operators
- (i)there is a such that(1)
there exists an such that ;
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.
If there exists such that , then and is a fixed point of G. Then the result of Theorem 2.1 trivially holds.
This is a contradiction. Thus .
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 .
So, . The proof of Theorem 2.1 is complete. □
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.
- (i)there is a such that satisfying
there exists an such that ;
one of (a) and (b) holds:
G is a continuous operator;
if a non-decreasing monotone sequence in tends to , then for all n.
Let , then we have the following theorem.
G is a continuous operator;
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 .
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.
Thus the operator G has a fixed point in .
Thus , that is, . The proof of Theorem 2.2 is complete. □
Following Theorem 2.1, we have the next two corollaries.
Corollary 2.2 (, Theorem 2.1)
Thus Corollary 2.1 is an immediate consequence of Theorem 2.1. □
Corollary 2.3 (, Theorem 2.2)
if a non-decreasing sequence , then for all n;
if a non-increasing sequence , then for all n.
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 . 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.
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 (, Lemma 3.1)
Thus . □
Thus (3) holds for all n.
This shows that both successive sequences and are Cauchy sequences. □
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  (Theorem 3.4) and removed some restriction conditions: the successive sequences have convergent subsequences.
3 Application to the nonlinear elliptic problems
When , and , it is well known that (4) has no positive solution if , and that the positive solution of (4) is unique if , see  and . Also, in this case when , (4) has a unique positive radial solution .
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.
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 .
Thus the map is well defined and for . Also, . Obviously, is a mixed monotone map in and and .
This means that . Thus problem (4) has a unique positive radial solution. □
4 Application to the delayed hematopoiesis models
where are positive T-periodic functions and for all , q is a nonnegative constant (). In the case when , Wu  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.
This means and . Thus and are well defined and . Also, . Obviously, and are mixed monotone maps in and and .
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].
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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