Abstract
The aim of this paper is to present the definition of a generalized altering distance function and to extend the results of Yan et al. (Fixed Point Theory Appl. 2012:152, 2012) and some others, and to prove a new fixed point theorem of generalized contraction mappings in a complete metric space endowed with a partial order by using generalized altering distance functions. The results of this paper 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.
Similar content being viewed by others
1 Introduction
The Banach contraction mapping principle is a classical and powerful tool in nonlinear analysis. Weak contractions are generalizations of the Banach contraction mapping, which have been 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 contractions and generalized contractions was extended to partially ordered metric spaces in [2, 9–22]. Among them, some involve altering distance functions. Such functions were introduced by Khan et al. in [1], where they present some fixed point theorems with the help of such functions. First, 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 non-decreasing.
-
(b)
if and only if .
Recently, Harjani and Sadarangani proved some fixed point theorems for weak contractions and generalized contractions in partially ordered metric spaces by using the altering distance function in [11, 23], respectively. Their results improve the theorems of [2, 3].
Theorem 1.2 ([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 non-decreasing mapping such that
where is continuous and non-decreasing function such that ψ is positive in , and . If there exists with , then f has a fixed point.
Theorem 1.3 ([23])
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 non-decreasing 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 those functions which satisfy the condition: .
Theorem 1.4 ([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.
In 2012, Yan et al. proved the following result.
Theorem 1.5 ([24])
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 non-decreasing 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.
The aim of this paper is to present the definition of generalized altering distance function and to extend the results of Yan et al. [24] and some others, and to prove a new fixed point theorem of generalized contraction mappings in a complete metric space endowed with a partial order by using generalized altering distance functions. The results of this paper 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.
2 Main results
We first give the definition of generalized altering distance function as follows.
Definition 2.1 A generalized altering distance function is a function which satisfies:
-
(a)
ψ is non-decreasing;
-
(b)
if and only if .
We first recall the following notion of a monotone non-decreasing function in a partially ordered set.
Definition 2.2 If is a partially ordered set and , we say that T is monotone non-decreasing if , .
This definition coincides with the notion of a non-decreasing function in the case where and ≤ represents the usual total order in R.
In what follows, we prove the following theorem, which is the generalized type of Theorems 1.2-1.5.
Theorem 2.3 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 non-decreasing mapping such that
where ψ is a generalized altering distance function and is a right upper semi-continuous function with the condition: for all . If there exists such that , then T has a fixed point.
Proof Since T is a non-decreasing function, we obtain by induction that
Put . Then, for each integer , from (1) and, as the elements and are comparable, we get
Using the condition of Theorem 2.3 we have
Hence the sequence is decreasing and, consequently, there exists such that
as . Consider the properties of ψ and ϕ, letting in (2) we get
By using the condition: for all , 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 and 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 using (4) and (7), we have
As and and are comparable, using (1) we have
Consider the properties of ψ and ϕ, letting and taking into account (7) and (8), we have
From the condition for all , 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 2.3 is still valid for T being not necessarily continuous, assuming the following hypothesis in X:
Theorem 2.4 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 non-decreasing mapping such that
where ψ is a generalized altering distance functions and ϕ: is a right upper semi-continuous function with the condition for all . If there exists such that , then T has a fixed point.
Proof Following the proof of Theorem 2.3 we only have to check that . As is a non-decreasing 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 condition of theorem we have , this implies . Thus, or equivalently, . □
Now, we present an example where it can be appreciated that the hypotheses in Theorems 2.3 and Theorems 2.4 do not guarantee uniqueness of the fixed point. An example appears in [12].
Let and consider the usual order , . Thus, is a partially ordered set whose different elements are not comparable. Besides is a complete metric space considering the Euclidean distance. The identity map is trivially continuous and non-decreasing and condition (1) of Theorem 2.4 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 point in Theorems 2.3 and 2.4. This condition is:
In [12] it is proved that condition (10) is equivalent to:
Theorem 2.5 Adding condition (11) to the hypotheses of Theorem 2.3 (resp. Theorem 2.4) we obtain the uniqueness of the fixed point of T.
Proof Suppose that there exist which are fixed points. We distinguish two cases.
Case 1. If y is comparable to z then is comparable to for and
As we have 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 a generalized altering distance function and we have the condition for , this gives us that is a non-negative decreasing sequence and, consequently, there exists γ such that
Letting in (12) and, taking into account the properties of ψ and ϕ, we obtain
This and the condition for imply . Analogously, it can be proved that
Finally, as
the uniqueness of the limit gives us . This finishes the proof. □
Remark 2.6 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 2.7 Theorem 1.2 is a particular case of Theorem 2.3 for ψ being the identity function, and . Theorem 1.3 is a particular case of our Theorem 2.3 for being replaced by . Theorem 1.4 is a particular case of Theorem 2.3 for ψ being the identity function, and . Theorem 1.5 is also a particular case of Theorem 2.3 for ψ and ϕ being continuous.
Example 2.8 The following are some generalized altering distance functions:
where is a constant.
where is a constant.
We choose and
where is a constant. By using Theorem 2.3, we can get the following result.
Theorem 2.9 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 non-decreasing mapping such that
for any . If there exists such that , then T has a fixed point.
3 Application to ordinary differential equations
In this section we present two examples where our Theorems 2.3 and 2.4 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 condition (10), since for the functions and are least upper and greatest lower bounds of x and y, respectively. Moreover, in [17] it is proved that with the above mentioned metric satisfies condition (9).
Now we give the following definition.
Definition 3.1 A lower solution for (13) is a function such that
Theorem 3.2 Consider problem (13) with continuous and suppose that there exist with
such that for with
where is a light upper semi-continuous function with , , . Then the existence of a lower solution for (13) provides the existence of an unique solution of (13).
Proof 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 2.3 and 2.4 are satisfied. The mapping F is non-decreasing, since we have , and 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 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 a generalized 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
We multiply by ,
and this gives us
As , the last inequality gives us
and so
This and (19) give us
and, consequently,
Finally, Theorems 2.3 and 2.4 show that F has an unique fixed point. □
Example 3.3 In Theorem 3.2, we can choose the function as follows:
-
(1)
;
-
(2)
-
(3)
The functions , are continuous and non-decreasing. The function is right upper semi-continuous. If we choose in Theorem 3.2, we obtain the result of [5].
Example 3.4 Consider the following first-order periodic problem:
Let
then is continuous. Further, for , we have
We chose such that
Taking for all , we have
By using Theorem 3.2, we know that the first-order periodic problem (20) has a unique solution.
A 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 is a solution of (20) that is equivalent to being a solution of the integral equation
where is the Green function given by
Theorem 3.5 Consider problem (21) with continuous and non-decreasing with respect to the second variable and suppose that there exists such that for with
where is a light upper semi-continuous function with , , . Then our problem (21) has a unique non-negative solution.
Proof Consider the cone
Obviously, with is a complete metric space. Consider the operator given by
where is the Green function appearing in (22).
As f is non-decreasing with respect to the second variable, for with and , we have
and this proves that T is a non-decreasing operator.
Besides, for and taking into account (23), we obtain
It is easy to verify that
and that
These facts, the inequality (24), and the hypothesis give us
Hence
Put , , obviously ψ is an altering distance function, ψ and ϕ satisfy the condition of , for . From the last inequality, we have
Finally, as f and G are non-negative functions
and Theorems 2.3 and 2.4 tell us that F has a unique non-negative solution. □
Remark 3.6 In Theorem 3.5, we can choose as , , and as well as in Theorem 3.2.
Example 3.7 Consider the following two-point boundary value problem of the second order differential equation:
Let
then is continuous and non-decreasing with respect to the second variable. Further, for , we have
Taking for all . By using Theorem 3.2, we know that the two-point boundary value problem (25) has a unique non-negative solution.
References
Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull. Aust. Math. Soc. 1984, 30(1):1–9. 10.1017/S0004972700001659
Dhutta P, Choudhury B: A generalization of contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151
Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal. 2001, 47: 2683–2693. 10.1016/S0362-546X(01)00388-1
Aydi H, Karapınar E, Bessem S: Fixed point theorems in ordered abstract spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 76
Nieto JJ, Pouso RL, Rodríguez-López R: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Proc. Am. Math. Soc. 2007, 135: 2505–2517. 10.1090/S0002-9939-07-08729-1
Gordji ME, Baghani H, Kim GH: Coupled fixed point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 981517
Sintunavarat W, Cho YJ, Kumam P: Existence, regularity and stability properties of periodic solutions of a capillarity equation in the presence of lower and upper solutions. Fixed Point Theory Appl. 2012., 2012: Article ID 128
Obersnel F, Omari P, Rivetti S: A generalization of Geraghty’s theorem in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal., Real World Appl. 2012, 13: 2830–2852. 10.1016/j.nonrwa.2012.04.012
Sastry K, Babu G: Some fixed point theorems by altering distance between the points. Indian J. Pure Appl. Math. 1999, 30: 641–647.
Amini-Harandi A, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 2010, 72: 2238–2242. 10.1016/j.na.2009.10.023
Harjani J, Sadarangni K: Fixed point theorems for weakly contraction mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240
Burgić D, Kalabušić S, Kulenović M: Global attractivity results for mixed monotone mappings in partially ordered complete metric spaces. Fixed Point Theory Appl. 2009., 2009: Article ID 762478
Ćirić L, Cakić N, Rajović M, Uma J: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 131294
Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020
Nieto JJ, Rodriguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5
Nieto JJ, Rodriguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. 2007, 23: 2205–2212. 10.1007/s10114-005-0769-0
O’Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026
Shatanawi W, Postolache M: Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 271
Shatanawi W, Postolache M: Common fixed point results of mappings for nonlinear contraction of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60
Karapınar E, Shatanawi W, Mustafa Z: Quadruple fixed point theorems under nonlinear contractive conditions in partially ordered metric spaces. J. Appl. Math. 2012., 2012: Article ID 951912
Shatanawi W: Some fixed point results for a generalized-weak contraction mappings in orbitally metric spaces. Chaos Solitons Fractals 2012, 45: 520–526. 10.1016/j.chaos.2012.01.015
Harjani J, Sadarangni K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003
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., 2012: Article ID 152
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Su, Y. Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations. Fixed Point Theory Appl 2014, 227 (2014). https://doi.org/10.1186/1687-1812-2014-227
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-227