New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales
- Huiye Xu^{1}Email author
Received: 3 July 2010
Accepted: 13 December 2010
Published: 21 December 2010
Abstract
Some new existence and uniqueness theorems of fixed points of mixed monotone operators are obtained, and then they are applied to a nonlinear singular second-order three-point boundary value problem on time scales. We prove the existence and uniqueness of a positive solution for the above problem which cannot be solved by using previously available methods.
1. Introduction
The study of mixed monotone operators has been a matter of discussion since they were introduced by Guo and Lakshmikantham [1] in 1987, because it has not only important theoretical meaning but also wide applications in microeconomics, the nuclear industry, and so on (see [1–4]). Recently, some new and interesting results about these kinds of operators have emerged, and they are used extensively in nonlinear differential and integral equations (see [5–9]).
where with , , and . The functions and ( ) are continuous. Our nonlinearity may have singularity at and/or and ( ) may have singularity at .
To understand the notations used in (1.1), we recall that is a time scales, that is, is an arbitrary nonempty closed subset of . For each interval of , we define . For more details on time scales, one can refer to [10–12].
In recent years, there is much attention paid to the existence of positive solutions for nonlocal boundary value problems on time scales, see [13–18] and references therein. Dynamic equations have been applied in the study of insect population models, stock market and heat transfer and so on [19–22]. Time scales can be used in microeconomics models to study behavior which is sometimes continuous and sometimes discrete. A simple example of this continuous-discrete behavior is seen in suppliers short-run decisions and long-run decisions. Unifying both continuous and discrete model can avoid repeat research and has the capacity to get some different types of models which neither continuous models nor discrete models can effectively describe.
On the other hand, singular boundary value problems on time scales have also been investigated extensively (see [23–27]). We would like to mention some results of DaCunha et al. [23], Hao et al. [25], Luo [26], and Hu [27], which motivated us to consider problem (1.1).
where is fixed and is singular at and possible at , . The authors claimed that "we note that this is the first work (to our knowledge) that deals with singular boundary value problems in a general time scales setting." The results on existence of positive solutions were obtained by means of a fixed point theorem due to Gatica, Oliker and Waltman for mappings that are decreasing with respect to a cone.
where is rl-continuous and may be singular at and/or . With suitable growth and limit conditions, an existence theorem of positive solutions was established by using the Krasnoselskii fixed point theorem.
where is singular at and . The author obtained eigenvalue intervals in which there exists at least one positive solution of problem (1.4) by making use of the fixed point index theory.
where and are continuous. The nonlinearity may have singularity at and/or and may have singularity at . With the aid of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to (1.5) were obtained in the bounded set.
From the above research, we note that there is no result on the uniqueness of solutions and convergence of the iterative sequences for singular boundary value problems on time scales. As we know, completely continuity condition is crucial for the above discussion. However, it is difficult to verify for singular problems on time scales, in particular, in order to remove the singularity in at , more restricted conditions are required. For instance, condition of Theorem 2.3 in [23] and condition of Theorem 3.1 in [27]. In our abstract results on mixed monotone operators, since the compactness and continuity conditions are not required, they can be directly applied to singular boundary value problem (1.1).
The purpose of this paper is to present some conditions for problem (1.1) that have a unique solution, the iterative sequences yielding approximate solutions are also given. Our main result generalizes and improves Theorem 2.3 in [18].
2. Preliminaries and Abstract Theorems
Let the real Banach space be partially ordered by a cone of , that is, if and only if . is said to be a mixed monotone operator if is increasing in and decreasing in , that is, , , implies . Element is called a fixed point of if .
Recall that cone is said to be solid if the interior is nonempty and we denote if . is said to be normal if there exists a positive constant , such that , the smallest is called the normal constant of . For all , the notation means that there exist and such that . Clearly, ~ is an equivalence relation. Given (i.e., and ), we denote by the set . It is easy to see that is convex and for all . If and , it is clear that .
All the concepts discussed above can be found in [1, 2, 4]. For more results about mixed monotone operators and other related concepts, the reader is referred to [3, 5–9] and some of the references therein.
In [9], Zhai and Cao introduced the following definition of - -concave operators.
Definition 2.1 (see [9]).
Let be a real Banach space and be a cone in . We say an operator is - -concave if there exist two positive-valued functions on interval such that
They obtained the following result.
Theorem 2.2 (see [9]).
Let be a real Banach space and be a normal cone in . Suppose that an operator is increasing and - -concave. In addition, suppose that there exists such that . Then
(i)there are and such that , ;
(ii)operator has a unique fixed point in ;
(iii)for any initial , constructing successively the sequence , , we have .
We can extend Theorem 2.2 to mixed monotone operators, our main results can be stated as follows.
Theorem 2.3.
Let be a normal cone in a real Banach space , and a mixed monotone operator. Assume that for all , there exist two positive-valued functions on interval such that
In addition, suppose that there exists such that . Then
(i)there are and such that , ;
(ii)operator has a unique fixed point in ;
(iii)for any initial , constructing successively the sequences , , , we have and as .
Corollary 2.4.
Let be a real Banach space, a normal, solid cone in . Suppose is a mixed monotone operator and satisfies the conditions of Theorem 2.3. Then
(i)there are and such that , ;
(ii)operator has a unique fixed point in ;
(iii)for any initial , constructing successively the sequences , , , we have and as .
Remark 2.5.
In Theorem 2.3, if with is a solid cone, we can know that is automatically satisfied. Therefore, we can deduce that Corollary 2.4 holds from Theorem 2.3. For simplicity, we only present the proof of Theorem 2.3.
Proof of Theorem 2.3.
It is clear that and , . In general, we obtain , .
- (i)
- (ii)
which is also a contradiction. Thus, .
Furthermore, similarly to the proof of Theorem 2.1 in [9], there exits such that , and is the fixed point of operator .
By the uniqueness of fixed points of operator in , we have . Taking into account that is normal, we deduce that . This completes the proof.
3. Applications to Singular BVP (1.1) on Time Scales
A Banach space is the set of real-valued continuous (in the topology of ) function defined on with the norm .
It is clear that is a normal cone of which the normality constant is 1.
In order to obtain our main result, we need the following lemmas.
Lemma 3.1 (see [18]).
Lemma 3.2 (see [18]).
Our main result is the following theorem.
Theorem 3.3.
Assume that
Proof of Theorem 3.3.
Thus . Therefore, all the conditions of Theorem 2.3 are satisfied. By Theorem 2.3, we can obtain the conclusions of Theorem 3.3.
Now, let us end this paper by the following example.
Example 3.4.
Set , , , , . Then is a surjection and for .
Remark 3.5.
Example 3.4 indicates that Theorem 3.3 generalizes and complements Theorem 2.3 in [18] at the following aspects. Firstly, in our proof, we only need to check the conditions "there exists such that ", in fact, the author has shown that " " in the proof of Theorem 2.3 in [18]. It is clear that our hypotheses are weaker than those imposed in Theorem 2.3 in [18]. According to Lemma 3.2, we can know that the condition is automatically satisfied. Secondly, we have considered the case that the condition " and ( )" is not satisfied, therefore, the condition incorporates the more comprehensive functions than the condition in Theorem 2.3 in [18]. Thirdly, the more general conditions are imposed on our nonlinear term, they can be the sum of nondecreasing functions and nonincreasing functions.
Declarations
Acknowledgment
H. Xu was supported financially by the Science Foundation of North University of China.
Authors’ Affiliations
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