Abstract
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.
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1 Introduction
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 [1], where they present some fixed point theorems with the help of such functions. Firstly, we recall the definition of an altering distance function.
Definition 1.1 An altering distance function is a function which satisfies
-
(a)
ψ is continuous and nondecreasing.
-
(b)
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 [11]
Let be a partially ordered set, and suppose that there exists a metric such that is a complete metric space. Let be a continuous and nondecreasing mapping such that
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 [19]
Let be a partially ordered set, and suppose that there exists a metric such that is a complete metric space. Let be a continuous and nondecreasing mapping such that
where ψ and ϕ are altering distance functions. If there exists with , then f has a fixed point.
Subsequently, Amini-Harandi and Emami proved another fixed point theorem for contraction type maps in partially ordered metric spaces in [10]. The following class of functions is used in [10].
Let ℜ denote the class of functions which satisfies the condition .
Theorem 1.3 [10]
Let be a partially ordered set, and suppose that there exists a metric d such that is a complete metric space. Let be an increasing mapping such that there exists an element with . Suppose that there exists such that
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 .
Proof Since and ϕ, ψ are continuous, we have
This finishes the proof. □
In what follows, we prove the following theorem which is the generalized type of Theorem 1.1-1.3.
Theorem 2.1 Let X be a partially ordered set and suppose that there exists a metric d in x such that is a complete metric space. Let be a continuous and nondecreasing mapping such that
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.
Proof Since T is a nondecreasing function, we obtain, by induction, that
Put . Then for each integer , as the elements and are comparable, from (1) we get
Using the condition of Theorem 2.1, we have
Hence the sequence is decreasing, and consequently, there exists such that
as . Letting in (2), we get
By using the condition of Theorem 2.1, we have , and hence
as . In what follows, we will show that is a Cauchy sequence. Suppose that is not a Cauchy sequence. Then, there exists for which we can find subsequences with such that
for all . Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (5). Then
From (5) and (6), we have
Letting and using (4), we get
By using the triangular inequality, we have
Letting in the above two inequalities and (4) and (7), we have
As and and are comparable, using (1), we have
Letting and taking into account (7) and (8), we have
From the condition of Theorem 2.1, we get , which is a contradiction. This shows that is a Cauchy sequence and, since X is a complete metric space, there exists such that as . Moreover, the continuity of T implies that
and this proves that z is a fixed point. This completes the proof. □
In what follows, we prove that Theorem 2.1 is still valid for T not necessarily being continuous, assuming the following hypothesis in X:
Theorem 2.2 Let be a partially ordered set and suppose that there exists a metric d in X such that is a complete metric space. Assume that X satisfies (9). Let be a nondecreasing mapping such that
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.
Proof Following the proof of Theorem 2.1, we only have to check that . As is a nondecreasing sequence in X and , the condition (9) gives us that for every , and consequently,
Letting and taking into account that ψ is an altering distance function, we have
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 [17].
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.
In what follows, we give a sufficient condition for the uniqueness of the fixed point in Theorems 2.1 and 2.2. This condition is as follows:
In [17], it is proved that the condition (10) is equivalent to
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:
Case 1. If y is comparable to z, then is comparable to for and
By the condition for , we obtain and this implies .
Case 2. If y is not comparable to z, then there exists comparable to y and z. Monotonicity of T implies that is comparable to and to , for . Moreover,
Hence, ψ is an altering distance function and the condition of for . This gives us that is a nonnegative decreasing sequence, and consequently, there exists γ such that
Letting in (12) and taking into account that ψ and Φ are continuous functions, we obtain
This and the condition of Theorem 2.1 implies , and consequently, .
Analogously, it can be proved that
Finally, as
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
In this section, we present two examples where our Theorems 2.2 and 2.3 can be applied. The first example is inspired by [17]. We study the existence of a solution for the following first-order periodic problem:
where and is a continuous function. Previously, we considered the space () of continuous functions defined on I. Obviously, this space with the metric given by
is a complete metric space. can also be equipped with a partial order given by
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 [17] it is proved that with the above mentioned metric satisfies the condition (9).
Now, we give the following definition.
Definition 3.1 A lower solution for (13) is a function such that
Theorem 3.1 Consider the problem (13) with continuous, and suppose that there exist with
such that for with ,
Then the existence of a lower solution for (13) provides the existence of a unique solution of (13).
Proof The problem (13) can be written as
This problem is equivalent to the integral equation
where is the Green function given by
Define by
Note that if is a fixed point of F, then is a solution of (13). In what follows, we check that the hypotheses in Theorems 2.2 and 2.3 are satisfied. The mapping F is nondecreasing for ; using our assumption, we can obtain
which implies, since , that for ,
Besides, for , we have
Using the Cauchy-Schwarz inequality in the last integral, we get
The first integral gives us
The second integral in (15) gives us the following estimate:
Taking into account (14)-(17), we have
and from the last inequality, we obtain
or equivalently,
By our assumption, as
the last inequality gives us
and hence,
Put and . Obviously, ψ is an altering distance function, and satisfy the condition of for . From (18), we obtain for ,
Finally, let be a lower solution for (13). We claim that . In fact
Multiplying by
we get
As , the last inequality gives us
and so
This and (19) give us
and consequently,
Finally, Theorems 2.2 and 2.3 give that F has an unique fixed point. □
The second example where our results can be applied is the following two-point boundary value problem of the second-order differential equation
It is well known that , a solution of (20), is equivalent to , a solution of the integral equation
where is the Green function given by
Theorem 3.2 Consider the problem (20) with continuous and nondecreasing with respect to the second variable, and suppose that there exists such that for with ,
Then our problem (20) has a unique nonnegative solution.
Proof Consider the cone
Obviously, with is a complete metric space. Consider the operator given by
where is the Green function appearing in (21).
As f is nondecreasing with respect to the second variable, then for with and , we have
and this proves that T is a nondecreasing operator.
Besides, for and taking into account (22), we can obtain
It is easy to verify that
and that
These facts, the inequality (23), and the hypothesis give us
Hence
Put , ; obviously, ψ is an altering distance function, ψ and ϕ satisfy the condition , for . From the last inequality, we have
Finally, as f and G are nonnegative functions,
Theorems 2.2 and 2.3 tell us that F has a unique nonnegative solution. □
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This project is supported by the National Natural Science Foundation of China under the grant (11071279).
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Yan, F., Su, Y. & Feng, Q. A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl 2012, 152 (2012). https://doi.org/10.1186/1687-1812-2012-152
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DOI: https://doi.org/10.1186/1687-1812-2012-152