Abstract
A class of weak -contractions satisfying the -condition is defined on metric spaces. The existence and uniqueness of fixed points of such maps are discussed both on metric spaces and on partially ordered metric spaces. The results are applied to a first order periodic boundary value problem.
MSC: 47H10, 54H25.
Similar content being viewed by others
1 Introduction and preliminaries
Recent developments in fixed point theory have been encouraged by the applicability of the results in the area of boundary value problems for differential and integral equations. Especially in the last few years, a lot of publications in fixed point theory have presented results directly related to specific initial or boundary value problems. These problems include not only ordinary and partial differential equations, but also fractional differential equations.
In 2004 Ran and Reurings [1] investigated the existence of fixed points in partially ordered metric spaces. The importance of this study presented itself in the area of boundary value problems. Nieto and Lopez [2] discussed the applications of the fixed point theorems to the problem of existence and uniqueness of solutions of first order boundary value problems. The results of Ran and Reurings and Nieto and Lopez have been followed soon by numerous studies concerning fixed points on partially ordered metric spaces [3]–[6]. In the case of partially ordered spaces the continuity condition is no longer needed, however, the map should be nondecreasing.
In a recent paper, Popescu [7] proved two generalizations of a result given by Bogin [8] for a class of non-expansive mappings on complete metric spaces. The idea behind his work was to replace the non-expansiveness condition with the weaker -condition introduced by Suzuki [9]–[11]. The existence and uniqueness of fixed points of maps satisfying the -condition have also been extensively studied; see [12]–[14]. We state first the definition of a non-expansive map and a map satisfying the -condition on a metric space.
Definition 1
A mapping on a metric space is called a non-expansive mapping if
for all .
Definition 2
A mapping on a metric space satisfies the -condition if
for all .
Popescu [7] stated and proved the following fixed point theorem.
Theorem 3
Let be a nonempty complete metric space and be a mapping satisfying
which implies
where, , and. Thenhas a unique fixed point.
In this paper, we investigate the existence and uniqueness of fixed points of maps satisfying the -condition on metric spaces and on partially ordered metric spaces. As an application, we study the existence and uniqueness of solutions of a first order periodic boundary value problem under certain conditions.
2 Existence and uniqueness of fixed points on metric spaces
Our main results can be considered as a generalization of the result of Popescu [7].
We first prove fixed point theorems on complete metric spaces and then we formulate these results on complete metric spaces endowed with a partial order.
Theorem 4
Letbe a complete metric space, be a map, andbe a continuous nondecreasing function such thatandfor. Suppose that
where
for all. Then the mappinghas a unique fixed point.
Proof
Let and define the sequence as follows:
If for some , then is the fixed point of . Assume that , for all .
Substituting and in (2.1) we get
where
From the triangle inequality we have
Therefore, can be either or . If , then (2.3) implies
so that and hence, which contradicts the assumption , for all . Thus, , which results in
Therefore, the sequence is non-increasing and bounded below by 0. Hence,
However, letting in (2.7) we get
and we conclude that , since , and therefore
We shall prove next that the sequence is a Cauchy sequence. Assume the contrary, that is, is not Cauchy. Then there exists for which one can find subsequences and in ℕ such that
for where is the smallest index satisfying (2.11), that is,
From the triangle inequality we have
Taking the limit as in (2.13) and using (2.10) we get
On the other hand, the convergence of implies that for this , there exists such that , for all . Let . Then, for all , we have
where and, hence,
Then from (2.1) with and we obtain
where
Regarding (2.10) and (2.14), we see that
Letting in (2.17) we get
which implies and hence, . This contradicts the assumption that is not a Cauchy sequence. Therefore, is Cauchy and by the completeness of it converges to a limit, say .
Assume now that there exists such that
Then we have
which is a contradiction. Hence, we must have or , for all . Therefore, for a subsequence of ℕ we have
for all which implies
where
Obviously,
Letting in (2.22) we get
and, hence, , that is, .
Finally, we prove the uniqueness of the fixed point. Assume that and and . Then
which implies
where
Thus, (2.26) becomes
and, clearly, , that is, . □
We next define a contractive condition similar to that in Theorem 4. The reason for introducing this new contraction is that in the framework of partially ordered metric spaces uniqueness of a fixed point requires an additional condition on the space. However, this condition is not sufficient for the uniqueness of the fixed point for a map satisfying contractive condition defined in Theorem 4.
Theorem 5
Letbe a complete metric space, be a map, andbe a continuous nondecreasing function such thatandfor. Suppose that
where
for all. Then the mappinghas a unique fixed point.
The proof of Theorem 5 can be done by following the lines of the proof of Theorem 4 and, hence, is omitted.
3 Fixed points on metric spaces with a partial order
In this section the fixed point theorems, Theorems 4 and 5, are formulated in the framework of partially ordered metric spaces. In what follows, we define a partial order ⪯ on the metric space .
Our first result is a counterpart of Theorem 4 on a partially ordered metric space.
Theorem 6
Letbe a partially ordered complete metric space, be a nondecreasing map, andbe a continuous nondecreasing function such thatandfor. Suppose that
where
for allwith. If there existssatisfying, thenhas a fixed point in.
Proof
Let satisfy . Define the sequence as follows:
If for some , then is the fixed point of . Assume that , for all . Since and is nondecreasing, then obviously
Substituting and in (3.1) we get
where
For the rest of the existence proof one can follow the lines of the proof of Theorem 4, since they are similar. □
Assume now that the space satisfies the condition
Our last result shows that the map given in Theorem 5 has a unique fixed point whenever it is defined on a partially ordered space , satisfying the condition (U).
Theorem 7
Letbe a partially ordered complete metric space satisfying the condition (U), be a nondecreasing map, andbe a continuous nondecreasing function such thatandfor. Suppose that
where
for allwith. If there existssatisfying, thenhas a unique fixed point in.
Proof
The existence proof is done by mimicking the proofs of Theorem 6 and Theorem 4. To prove the uniqueness we assume that there are two different fixed points, and , that is, and and . We consider the following cases:
Case 1. Suppose that and are comparable and, without loss of generality, that . Then
which implies
where
Thus, (3.10) becomes
and, clearly, , that is, .
Case 2. Assume that and are not comparable. From the condition (U) there exists satisfying and . Define the sequence as
Notice that since is nondecreasing and , we have
If for some , then and, hence, , for all . Thus, the sequence converges to the fixed point , that is, . Assume that , for all . Then we have
which implies that the contractive condition
where
holds for all . Observe that
can be either or due to the fact that
by the triangle inequality. If , then we have for some . In this case, since for , the inequality (3.12) implies
which is not possible. Then we must have , for all and, thus, the inequality (3.12) implies
that is, the sequence is positive and decreasing and, therefore, convergent. Let . Taking the limit as in (3.14) we get
from which it follows that , and, thus, we deduce
In a similar way we obtain
From (3.16) and (3.17) it follows that , which completes the proof. □
Some consequences of Theorem 7 are given next. If we choose as a specific function we get the following result.
Corollary 8
Letbe a partially ordered complete metric space satisfying the condition (U) andbe a nondecreasing map. Suppose that the condition
where
holds, for allwithand some constant. If there existssatisfying, thenhas a unique fixed point in.
Proof
Choose . Then the maps and satisfy the conditions of Theorem 7 and, thus, has a unique fixed point in . □
The next result is the analog of Theorem 2.1 in [7] on partially ordered metric spaces.
Corollary 9
Letbe a partially ordered complete metric space satisfying the condition (U) andbe a nondecreasing map. Suppose that
where
for allwith. If there existssatisfying, thenhas a unique fixed point in.
Proof
Define
Then, clearly,
Then the map satisfies the conditions of the Corollary 8 and, thus, has a unique fixed point in . □
4 Applications
In this section we investigate the existence and uniqueness of solutions of periodic boundary value problems of first order. These problems have been studied under different conditions in [2], [15]–[18]. However, the existence and uniqueness conditions obtained here are weaker than those in the previous studies.
Define the partial ordering and the metric in as follows:
The space satisfies the condition (U). Indeed, it is obvious that for every pair , in , we have and . We will consider the following first order periodic boundary value problem:
Definition 10
A lower solution of the problem (4.2) is a function satisfying
An upper solution to the problem (4.2) is a function satisfying
Observe that the problem (4.2) can be written as
This problem is equivalent to the integral equation
where is the Green function defined by
In what follows, we give a theorem for the existence and uniqueness of a solution of the problem (4.3).
Theorem 11
Consider the periodic boundary value problem (4.2). Assume thatis continuous and that there existssuch that, for allsatisfying, the following condition holds:
for some, such that. If the problem (4.2) has a lower solution, then it has a unique solution.
Proof
Define the map as follows:
where is the Green’s function given in (4.7). Then the solution of the problem (4.2) is the fixed point of . Assume that are functions in satisfying (4.8). Rewrite the inequality as
Multiplying both sides by and integrating from 0 to we obtain
which, due to the condition , gives
Hence,
Employing this inequality and (4.10) we get
Hence, we obtain
which can be written as
This implies
or, in terms of the metric,
Moreover, since satisfies (4.8), we have
that is, is nondecreasing. Consider now
where . Choosing in a way that we see that the nondecreasing map satisfies the condition (2.21) of Corollary 8. We next show that for some . Since the problem (4.2) has a lower solution, there exists satisfying (4.3). Hence, we have
Multiplying both sides by and then integrating from 0 to we obtain
Employing the inequality we get
or equivalently,
Combining (4.18) and (4.19) we get
where is the Green’s function given in (4.8). Hence, we have
for the lower solution of (4.2). Then, by the Corollary 8, the map has a unique fixed point; thus, the boundary value problem (4.2) has a unique solution. □
Example 12
Consider the BVP
It can easily be verified by direct calculation that the unique solution is
For this specific example the function satisfies the condition
not only for , with
but for all , where . Indeed,
for . Observe that is a lower solution of the BVP. Clearly,
and
By Theorem 11, the BVP has a unique solution.
Next, we give the following example of a nonlinear equation.
Example 13
Consider the BVP
The function satisfies the condition
for where and are nonnegative functions continuous on , the positive constant is defined as , and is chosen such that . The existence of is verified by the fact that and are continuous on the closed interval . Observe that is a lower solution of the BVP. Clearly,
By Theorem 11, the BVP has a unique solution.
References
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435-1443. 10.1090/S0002-9939-03-07220-4
Nieto JJ, Lopez RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223-239. 10.1007/s11083-005-9018-5
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1-8. 10.1016/j.jmaa.2007.03.105
Bhaskar TG, Lakshmikantham V: Fixed point theory in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379-1393. 10.1016/j.na.2005.10.017
Lakshmikantham V, Ćirić LB: 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
Nashine HK, Samet B:Fixed point results for mappings satisfying weakly contractive condition in partially ordered metric spaces. Nonlinear Anal. 2011, 74: 2201-2209. 10.1016/j.na.2010.11.024
Popescu O: Two generalizations of some fixed point theorems. Comput. Math. Appl. 2011, 62: 3912-3919. 10.1016/j.camwa.2011.09.044
Bogin J: A generalization of a fixed point theorem of Goebel, Kirk and Shimi. Can. Math. Bull. 1976, 19: 7-12. 10.4153/CMB-1976-002-7
Suzuki T: Some results on recent generalization of Banach contraction principle. Proc. of the 8th Int. Conf. of Fixed Point Theory and Its Appl. 2007, 751-761.
Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 163: 1861-1869.
Suzuki T: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 2008, 340: 1088-1095. 10.1016/j.jmaa.2007.09.023
Suzuki T: A new type of fixed point theorem on metric spaces. Nonlinear Anal. 2009, 71: 5313-5317. 10.1016/j.na.2009.04.017
Karapınar E, Taş K: Generalized (C)-conditions and related fixed point theorems. Comput. Math. Appl. 2011, 61: 3370-3380. 10.1016/j.camwa.2011.04.035
Karapınar, E: Remarks on Suzuki (C)-condition.
Nieto JJ, Pouso RL, Rodríguez-López R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc. 2007, 135: 2505-2517. 10.1090/S0002-9939-07-08729-1
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
Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403-3410. 10.1016/j.na.2009.01.240
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
Acknowledgements
The authors are thankful to the referees for careful reading of the manuscript and the valuable comments and suggestions for the improvement of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to this work. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Karapınar, E., Erhan, I.M. & Aksoy, Ü. Weak -contractions on partially ordered metric spaces and applications to boundary value problems. Bound Value Probl 2014, 149 (2014). https://doi.org/10.1186/s13661-014-0149-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-014-0149-8