- Open Access
A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations
© Yan et al.; licensee Springer 2012
Received: 28 April 2012
Accepted: 30 August 2012
Published: 18 September 2012
The aim of this paper is to extend the results of Harjani and Sadarangani and some other authors and to prove a new fixed point theorem of a contraction mapping in a complete metric space endowed with a partial order by using altering distance functions. Our theorem can be used to investigate a large class of nonlinear problems. As an application, we discuss the existence of a solution for a periodic boundary value problem.
The Banach contraction principle is a classical and powerful tool in nonlinear analysis. Weak contractions are generalizations of Banach’s contraction mapping studied by several authors. In [1–8], the authors prove some types of weak contractions in complete metric spaces respectively. In particular, the existence of a fixed point for weak contraction and generalized contractions was extended to partially ordered metric spaces in [2, 9–18]. Among them, the altering distance function is basic concept. Such functions were introduced by Khan et al. in , where they present some fixed point theorems with the help of such functions. Firstly, we recall the definition of an altering distance function.
ψ is continuous and nondecreasing.
if and only if .
Recently, Harjani and Sadarangani proved some fixed point theorems for weak contraction and generalized contractions in partially ordered metric spaces by using the altering distance function in [11, 19] respectively. Their results improve the theorems of [2, 3].
Theorem 1.1 
where is continuous and nondecreasing function such that ψ is positive in , and . If there exists with , then f has a fixed point.
Theorem 1.2 
where ψ and ϕ are altering distance functions. If there exists with , then f has a fixed point.
Let ℜ denote the class of functions which satisfies the condition .
Theorem 1.3 
Assume that either f is continuous or M is such that if an increasing sequence , then , ∀n. Besides, if for each , there exists which is comparable to x and y, then f has a unique fixed point.
The purpose of this paper is to extend the results of [10, 11, 19] and to obtain a new contraction mapping principle in partially ordered metric spaces. The result is more generalized than the results of [10, 11, 19] and other works. The main theorems can be used to investigate a large class of nonlinear problems. In this paper, we also present some applications to first- and second-order ordinary differential equations.
2 Main results
We first recall the following notion of a monotone nondecreasing function in a partially ordered set.
Definition 2.1 If is a partially ordered set and , we say that T is monotone nondecreasing if , .
This definition coincides with the notion of a nondecreasing function in the case where and ≤ represents the usual total order in R.
We shall need the following lemma in our proving.
Lemma 2.1 If ψ is an altering distance function and is a continuous function with the condition for all , then .
This finishes the proof. □
In what follows, we prove the following theorem which is the generalized type of Theorem 1.1-1.3.
where ψ is an altering distance function and is a continuous function with the condition for all . If there exists such that , then T has a fixed point.
and this proves that z is a fixed point. This completes the proof. □
where ψ is an altering distance function and is a continuous function with the conditions for all . If there exists such that , then T has a fixed point.
Using Lemma 2.1, we have , which implies . Thus or equivalently, . □
Now, we present an example where it can be appreciated that the hypotheses in Theorems 2.1 and Theorems 2.2 do not guarantee the uniqueness of the fixed point. The example appears in .
Let and consider the usual order , . Thus, is a partially ordered set whose different elements are not comparable. Besides, is a complete metric space and is the Euclidean distance. The identity map is trivially continuous and nondecreasing, and the condition (9) of Theorem 2.2 is satisfied since the elements in X are only comparable to themselves. Moreover, and T has two fixed points in X.
Theorem 2.3 Adding the condition (11) to the hypotheses of Theorem 2.1 (resp. Theorem 2.2), we obtain the uniqueness of the fixed point of T.
Proof Suppose that there exist which are fixed points. We distinguish the following two cases:
By the condition for , we obtain and this implies .
This and the condition of Theorem 2.1 implies , and consequently, .
the uniqueness of the limit gives us . This finishes the proof. □
Remark 2.1 Under the assumption of Theorem 2.3, it can be proved that for every , , where z is the fixed point (i.e., the operator f is Picard).
Remark 2.2 Theorem 1.1 is a particular case of Theorem 2.1 for ψ, the identity function, and .
Theorem 1.2 is a particular case of Theorem 2.1 for , is an altering function in Theorem 1.2. Theorem 1.3 is a particular case of Theorem 2.1 for ψ, the identity function, and .
3 Application to ordinary differential equations
Clearly, satisfies the condition (10) since for , the functions and are the least upper and the greatest lower bounds of x and y, respectively. Moreover, in  it is proved that with the above mentioned metric satisfies the condition (9).
Now, we give the following definition.
Then the existence of a lower solution for (13) provides the existence of a unique solution of (13).
Finally, Theorems 2.2 and 2.3 give that F has an unique fixed point. □
Then our problem (20) has a unique nonnegative solution.
where is the Green function appearing in (21).
and this proves that T is a nondecreasing operator.
Theorems 2.2 and 2.3 tell us that F has a unique nonnegative solution. □
This project is supported by the National Natural Science Foundation of China under the grant (11071279).
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